Bijections of silting complexes and derived Picard groups
aa r X i v : . [ m a t h . R T ] J a n BIJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS
FLORIAN EISELEA
BSTRACT . We introduce a method that produces a bijection between the posets silt - ðŽ and silt - ðµ formedby the isomorphism classes of basic silting complexes over ï¬nite-dimensional ð -algebras ðŽ and ðµ , bylifting ðŽ and ðµ to two ð [[ ð ]] -orders which are isomorphic as rings. We apply this to a class of algebrasgeneralising Brauer graph and weighted surface algebras, showing that their silting posets are multiplicity-independent in most cases. Under stronger hypotheses we also prove the existence of large multiplicity-independent subgroups in their derived Picard groups as well as multiplicity-invariance of ðð«ðð¢ððð§ð . Asan application to the modular representation theory of ï¬nite groups we show that if ðµ and ð¶ are blocks with | IBr( ðµ ) | = | IBr( ð¶ ) | whose defect groups are either both cyclic, both dihedral or both quaternion, then theposets tilt - ðµ and tilt - ð¶ are isomorphic (except, possibly, in the quaternion case with | IBr( ðµ ) | = 2 ) and ðð«ðð¢ððð§ð ( ðµ ) â ðð«ðð¢ððð§ð ( ð¶ ) (except, possibly, in the quaternion and dihedral cases with | IBr( ðµ ) | = 2 ). C ONTENTS
1. Introduction 12. Preliminaries 43. Generalised weighted surface algebras 84. Lifts for twisted Brauer graph algebras 125. Lifts for generalised weighted surface algebras 146. Bijections of silting complexes 187. Tilting bijections for algebras of dihedral, semi-dihedral and quaternion type 228. Multiplicity-independence of derived Picard groups 24References 411. I
NTRODUCTION
The notion of a silting complex over a ï¬nite-dimensional algebra was ï¬rst introduced by Keller andVossieck [KV88], and is closely related to Rickardâs stronger notion of a tilting complex [Ric89]. It waslater discovered by Aihara and Iyama [AI12] that silting complexes have a well-behaved mutation theory,which kindled a wider interest in this class objects. One of the most obvious problems to consider in thiscontext would be their classiï¬cation over a given algebra ðŽ . For most algebras a classiï¬cation of eithersilting or tilting complexes is entirely out of reach (although there are exceptions, e.g. [AM17]), but two-term silting (and tilting) complexes are much more accessible thanks to Adachi, Iyama and Reitenâs theoryof ð -tilting modules [AIR14], and classiï¬cation results in this area include Brauer tree and Nakayamaalgebras [Zvo14, AAC18, Ada16], algebras of dihedral, semi-dihedral and quaternion type [EJR18], andmore [IZ20, Miz14]. The present paper aims to generalise some of the results of [EJR18], which wasjoint work of Janssens, Raedschelders and the author, to silting complexes of arbitrary length.The main tool of [EJR18] was a one-to-one correspondence between two-term silting complexes over ðŽ and ðŽ â ð§ðŽ , where ðŽ is a ï¬nite-dimensional algebra over a ï¬eld ð and ð§ â rad( ð ( ðŽ )) is arbitrary.Unsurprisingly, this fails for complexes of length greater than two, the problem being that neithercomplexes nor the morphisms between them lift from ðŽ â ð§ðŽ to ðŽ in general. A key idea of the present Mathematics Subject Classiï¬cation. paper is to consider a ð [[ ð ]] -order Î , by which we mean a ð [[ ð ]] -algebra which is free and ï¬nitelygenerated as a ð [[ ð ]] -module, such that ðŽ â Îâ ð Î . It is known [Ric91b] that pre-silting complexes over ðŽ always lift to Î , even uniquely, and the converse holds as well (see Proposition 6.1). Taking into accountthat silting complexes also need to generate, we obtain that silt - ðŽ and a certain set t-silt - Î â silt - Î (seeDeï¬nition 6.2) are in bijection. Now one just needs to realise that Î can be turned into a ð [[ ð ]] -order inmany diï¬erent ways, and therefore has many diï¬erent âreductions modulo ð â, while t-silt - Î (like silt - Î )only depends on the structure of Î as a ring. Hence, if ð , ð ⶠð [[ ð ]] ⪠ð (Î) are two diï¬erent ways ofturning Î into a ð [[ ð ]] -order, with reductions modulo ð being ï¬nite-dimensional ð -algebras ðŽ and ðµ ,then we get a diagram t-silt - Î g g ⌠' ' âââââââââââ ⌠w w â£â£â£â£â£â£â£â£â£â£â£ silt - ðŽ silt - Îâ ð ( ð )Î o o ⌠/ / silt - Îâ ð ( ð )Î silt - ðµ. These bijections do not alter the terms of complexes in a non-trivial way, but the eï¬ect of lifting andsubsequent reduction on diï¬erentials is less straightforward. The relationship between ðŽ and ðµ is notobvious either, and in particular they may have diï¬erent ð -dimensions since the ð [[ ð ]] -rank of Î dependson the chosen ð [[ ð ]] -algebra structure. This principle, formally stated in Corollary 6.5, is quite versatilesince we are not imposing any structural restrictions on the ð [[ ð ]] -order Î , and it should have applicationsbeyond what we do in the present article.Now we need to identify some families of algebras that arise as the reduction of a single ring Î with respect to diï¬erent ð [[ ð ]] -algebra structures. A ï¬rst example are Brauer tree and certain Brauergraph algebras, which Gnedin [Gne19] showed to have lifts, called â Ribbon graph orders â, whose ringstructure is manifestly independent of the multiplicities involved. In the present article we will deï¬nea much larger class of algebras, comprising Brauer graph algebras and weighted surface algebras asdeï¬ned by Erdmann and SkowroÅski [ES18, ES20c]. This class is similar to what is outlined underthe heading âthe general contextâ in [ES20c]. We call an algebra in this class a generalised weightedsurface algebra , denoted Î( ð, ð , ð â , ð â , ð¡ â , î ) . We then construct lifts of these algebras to ð [[ ð ]] -orders provided the multiplicities are big enough, and use that to establish multiplicity-independence of silt - Î( ð, ð , ð â , ð â , ð¡ â , î ) .Our main results, which are Theorems 6.6, 8.7 and 8.9, are stated in terms of these generalisedweighted surface algebras and twisted Brauer graph algebras , which the reader may not be familiarwith. What we will do in this introduction is state explicit versions of these results for Brauer graphalgebras and certain blocks of group algebras of ï¬nite groups, two widely studied classes of algebraswhich were actually the main intended application. Since these algebras are symmetric, the notions ofsilting and tilting coincide and we even get correspondences of tilting complexes.Recall that a
Brauer graph is a ï¬nite undirected graph ðº equipped with a cyclic order on the set ofhalf-edges incident to ð£ for each vertex ð£ â ðº . Once we assign a multiplicity to each vertex by meansof a function ð â ⶠðº ⶠ†> , we can deï¬ne the Brauer graph algebra ðŽ ( ðº, ð â ) , which is symmetricand special biserial. See [Sch18] for a survey on these algebras. Theorem A (Tilting bijections in Brauer graph algebras) . Let ð be an algebraically closed ï¬eld and let ðº be a Brauer graph with two sets of multiplicities ð (1) â , ð (2) â ⶠðº ⶠ†> . Assume that either(1) char( ð ) = 2 , or(2) ðº is bipartite, or(3) ð (1) â and ð (2) â only take values â©Ÿ .Then there is a bijection between the isomorphism classes of (pre-)tilting complexes over ðŽ ( ðº, ð (1) â ) andthose over ðŽ ( ðº, ð (2) â ) , inducing a poset isomorphism tilt - ðŽ ( ðº, ð (1) â ) ⌠ⷠtilt - ðŽ ( ðº, ð (2) â ) . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 3
Theorem B (Tilting bijections in blocks) . Let ð be an algebraically closed ï¬eld of characteristic ð > ,and let ðŽ and ðµ be blocks of group algebras of ï¬nite groups deï¬ned over ð such that | IBr( ðŽ ) | = | IBr( ðµ ) | .Assume that the defect groups of ðŽ and ðµ are(1) cyclic groups of orders ð ð and ð ð , or(2) dihedral groups of orders ð and ð , or(3) quaternion groups of orders ð and ð and | IBr( ðŽ ) | = | IBr( ðµ ) | â ,for arbitrary ð, ð â©Ÿ (in the cyclic case) or ð, ð â©Ÿ (in the other two cases). Then there is a bijectionbetween the isomorphism classes of (pre-)tilting complexes over ðŽ and those over ðµ , inducing a posetisomorphism tilt - ðŽ ⌠ⷠtilt - ðµ. It should be noted that the ð [[ ð ]] -orders used to prove Theorem A do not usually have semisimple ð (( ð )) -span and are not canonical in any way. By contrast, the lifts used in Theorem B happen to beorders in semisimple ð (( ð )) -algebras, and in many ways look like equicharacteristic versions of blockalgebras over an extension î» of the ð -adic integers. This fact is exploited in the second half of the paper,which is devoted to derived Picard groups.The derived Picard group of a ï¬nite-dimensional ð -algebra ðŽ , denoted ðð«ðð¢ð ð ( ðŽ ) , is the groupof standard auto-equivalences of î° ð ( ðŽ ) . This object was ï¬rst considered by Yekutieli [Yek99] andRouquier and Zimmermann [RZ03]. The group ðð«ðð¢ð ð ( ðŽ ) is a locally algebraic group [Yek04], whoseidentity component is ðð¢ð ð ( ðŽ ) , the identity component of the ordinary Picard group. What really putderived Picard groups into the limelight was the discovery by Seidel and Thomas [ST01] that elementscalled âspherical twistsâ satisfy Braid relations and give rise to embeddings of Braid groups into thederived Picard groups of certain dg -versions of Brauer tree algebras (a similar result was obtainedindependently in [RZ03]). This led to a number of papers proving the existence and faithfulness ofBraid actions on derived categories of Brauer tree algebras, using that to fully determine their derivedPicard groups [Zim01, SZI02, MA08, Zvo15, VZ17]. Of course there are results for other classes ofalgebras as well [MY01, BPP17, NV20].The group ðð«ðð¢ð ð ( ðŽ ) acts on silt - ðŽ and tilt - ðŽ , with kernel containing ðð¢ð ð ( ðŽ ) . ThereforeTheorems A and B should have some implications for derived Picard groups. The group ðð¢ð ð ( ðŽ ) is certainly not multiplicity-independent, but it might have a multiplicity-independent complement orsupplement. The rough idea is that for a ð [[ ð ]] -order Î which reduces to an algebra ðŽ we have a grouphomomorphism ðð«ðð¢ð ð [[ ð ]] (Î) ⶠðð«ðð¢ð ð ( ðŽ ) . The group ðð«ðð¢ð ð [[ ð ]] (Î) still depends on the ð [[ ð ]] -algebra structure on Î , though, and in general the aforementioned homomorphism is neither surjective(even modulo ðð¢ð ð ( ðŽ ) ) nor injective. Therefore we must rely on a number of favourable properties of ourchosen lift Î to prove Theorems C and D below. Our results give a conceptual explanation of multiplicity-independence results such as [Zim01, MA08], but more importantly also cover cases where there are noâspherical objectsâ in î° ð ( ðŽ ) that would allow to apply the results of [ST01]. We also obtain very elegantmultiplicity-independence results for ðð«ðð¢ððð§ð ( ðŽ ) , the subgroup of ðð«ðð¢ð ð ( ðŽ ) acting trivially on ð ( ðŽ ) . Theorem C.
Let ð be an algebraically closed ï¬eld and let ðº be a Brauer graph with two sets ofmultiplicities ð (1) â , ð (2) â ⶠðº ⶠ†> . Assume ðº is a simple graph, and if char( ð ) â then alsoassume that ðº is bipartite. Then ðð«ðð¢ððð§ð ( ðŽ ( ðº, ð (1) â )) â ðð«ðð¢ððð§ð ( ðŽ ( ðº, ð (2) â )) . If the multiplicities have the property that ð (1) ð£ = ð (1) ð€ if and only if ð (2) ð£ = ð (2) ð€ for all ð£, ð€ â ðº , thenthere are subgroups îŽ ð â©œ ðð«ðð¢ð ð ( ðŽ ( ðº, ð ( ð ) â )) for ð â {1 , , such that îŽ â îŽ and ðð«ðð¢ð ð ( ðŽ ( ðº, ð ( ð ) â )) = ðð¢ð ð ( ðŽ ( ðº, ð ( ð ) â )) â îŽ ð for ð â {1 , . The subgroups îŽ ð can be identiï¬ed with a certain subgroup îŽ of ï¬nite index in the derived Picardgroup of some ð [[ ð ]] -order Î , and in that setting it is reasonable to identify îŽ = îŽ = îŽ , as we FLORIAN EISELE do in Theorems 8.7 and 8.9. The latter, together with Proposition 4.3, implies Theorem C. It is worthmentioning that in certain situations one can modify the decomposition
ðð¢ð ð ( ðŽ ( ðº, ð ( ð ) â )) â îŽ ð slightly toobtain a semi-direct product decomposition (e.g. in Proposition 8.11). Theorem D.
Let ð be an algebraically closed ï¬eld of characteristic ð > , and let ðµ and ðµ be blocksof group algebras of ï¬nite groups deï¬ned over ð such that | IBr( ðµ ) | = | IBr( ðµ ) | . Assume that the defectgroups of ðµ and ðµ are(1) cyclic groups of orders ð ð and ð ð , or(2) dihedral groups of orders ð and ð and | IBr( ðµ ) | = | IBr( ðµ ) | â , or(3) quaternion groups of orders ð and ð and | IBr( ðµ ) | = | IBr( ðµ ) | â ,for arbitrary ð, ð â©Ÿ ð , where we set ð = 1 in the cyclic case and ð = 3 in the other two cases. Then ðð«ðð¢ððð§ð ( ðµ ) â ðð«ðð¢ððð§ð ( ðµ ) . If ð = ð = ð or ð, ð > ð then there are îŽ ð â©œ ðð«ðð¢ð ð ( ðµ ð ) for ð â {1 , such that îŽ â îŽ and ðð«ðð¢ð ð ( ðµ ð ) = ðð¢ð ð ( ðµ ð ) â îŽ ð for ð â {1 , . Again one can do slightly better in some cases. For blocks of quaternion defect we get
ðð«ðð¢ð ð ( ðµ ð ) = ðð¢ð ð ð ( ðµ ð ) â îŽ ð , where ðð¢ð ð ð ( ðµ ð ) is the subgroup of the Picard group that ï¬xes the isomorphism classes ofall simple modules. In the cyclic defect case we even have ðð«ðð¢ð ð ( ðµ ð ) = ðð¢ð ð ð ( ðµ ð ) Ã îŽ ð (which, however,already follows from [VZ17]).It would be interesting to see if there are other families of blocks which arise as the various reductionsmodulo ð of a ring Î endowed with diï¬erent ð [[ ð ]] -algebra structures. The proof of Theorem D usesonly one such ring Î for each ð , type of defect group and number of simple modules. In this framework itwould therefore be possible to formulate much more radical ï¬niteness conjectures. Given the importanceof derived equivalences in modular representation theory it would also be interesting to see if one canuse ð [[ ð ]] -orders to construct derived equivalences between whole families of blocks. Relation to other work.
After a ï¬rst preprint of this paper had appeared online, the author was informedthat W. Gnedin has also, independently, studied bijections of silting posets. In particular, Gnedinhas a version of Propositions 6.1 and 6.3 as well as Corollary 6.5, mentioned in an Oberwolfachreport [ACBIK20], which will appear in an upcoming paper. Gnedin can also show a version ofTheorem A without restrictions on the multiplicities [Gne21].
Conventions.
Modules are right modules by default. For a quiver ð we let ð denote its set of verticesand ð its set of arrows. Two-sided ideals in a ring ðŽ are denoted by (âŠ) ðŽ , or (âŠ) when the choice of ðŽ is unambiguous. If ð is a discrete valuation ring with ï¬eld of fractions ðŸ , then we call an ð -algebrawhich is free and ï¬nitely-generated as an ð -module an ð -order . Given a ðŸ -algebra ðŽ we say that Î â ðŽ is an ð -order in ðŽ if Î is an ð -order which also spans ðŽ as a vector space.2. P RELIMINARIES
Silting and tilting complexes.
Let ðŽ be a ring for which the Krull-Schmidt theorem holds in î° ð ( ðŽ ) ,which is true for example if ðŽ is an algebra over a ï¬eld or an order over a complete discrete valuationring. Deï¬nition 2.1.
A complex ð â â î· ð ( proj - ðŽ ) is called(1) pre-silting if ððšðŠ î° ð ( ðŽ ) ( ð â , ð â [ ð ]) = 0 for all ð > ,(2) pre-tilting if ððšðŠ î° ð ( ðŽ ) ( ð â , ð â [ ð ]) = 0 for all ð â .A pre-silting (or pre-tilting) complex ð â with the property thick( ð â ) = î· ð ( proj - ðŽ ) is called silting (or tilting ). Deï¬ne silt - ðŽ = { basic silting complexes over ðŽ } / isomorphism , tilt - ðŽ = { basic tilting complexes over ðŽ } / isomorphism . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 5
We will sometimes refer to tilting complexes as deï¬ned above as one-sided tilting complexes . It wasshown by Rickard [Ric91a] that, under mild hypotheses on ðŽ , every one-sided tilting complex is therestriction of a two-sided tilting complex as deï¬ned in the subsection below. The set silt - ðŽ is partiallyordered by deï¬ning (see [AI12, Deï¬nition 2.10]) ð â â©Ÿ ð â if and only if ððšðŠ î° ð ( ðŽ ) ( ð â , ð â [ ð ]) = 0 for all ð > . (1)Of course, this restricts to a partial order on tilt - ðŽ . There is also a mutation theory for silt - ðŽ (see [AI12]),while mutation in tilt - ðŽ is not always possible. In general silting complexes tend to be better behavedthan tilting complexes. Fortunately, most algebras we are interested in are symmetric, and for symmetricalgebras the notions of silting and tilting coincide. Proposition 2.2. If ðŽ is a symmetric algebra over a ï¬eld or a symmetric order over a complete discretevaluation ring then partial silting and partial tilting complexes coincide. (cid:3) For algebras this is well-known, see for instance [AI12, Example 2.8] (or [HK02, Lemma 3.1] for aproof of the relevant version of Auslander-Reiten duality). For ð -orders we can use the fact that pre-silting and pre-tilting complexes coincide over the reduction to the residue ï¬eld of ð , and then useProposition 6.1 in conjunction with [Ric91b, Proposition 3.1 and Theorem 3.3] to get the same overthe order. See also [Zim99].2.2. Derived Picard groups.
Let ð be a commutative ring, and let ðŽ and ðµ be ð -algebras that areprojective as ð -modules. We call an object ð â â î° ð ( ðŽ op â ð ðµ ) invertible if there is a ð â â î° ð ( ðµ op â ð ðŽ ) such that ð â â ð ðµ ð â â ðŽ in î° ð ( ðŽ op â ð ðŽ ) and ð â â ð ðŽ ð â â ðµ in î° ð ( ðµ op â ð ðµ ) . We call ð â a two-sidedtilting complex if ð â is invertible and restricts to a bounded complex of projective left ðŽ -modules andto a bounded complex of projective right ðµ -modules. As mentioned in [Ric91a, Deï¬nition 4.2] everyinvertible complex is isomorphic to a two-sided tilting complex. If ð â is a two-sided tilting complex,then the derived tensor product â ð â â ð ðµ â â and the ordinary tensor product of complexes â ð â â ðµ â âcoincide, which is why we will not need to use left derived tensor products in the remainder of this article. Deï¬nition 2.3.
The derived Picard group of ðŽ is deï¬ned as ðð«ðð¢ð ð ( ðŽ ) = { two-sided tilting complexes in î° ð ( ðŽ op â ð ðŽ ) } / isomorphism . The product in this group is induced by â â â ðŽ = â.For basic properties of derived Picard groups refer to [RZ03]. We will make extensive use of the factthat a two-sided tilting complex ð â induces an isomorphism ðŸ ð ⶠð ( ðŽ ) ⌠ⶠð ( ðµ ) . This can be seen in a number of ways. For instance, the functor ð â1 â ðŽ â â ðŽ ð sends the ðŽ - ðŽ -bimodule ðŽ to the ðµ - ðµ -bimodule ðµ , and therefore induces a homomorphism between the endomorphism rings ofthese bimodules, which are ð ( ðŽ ) and ð ( ðµ ) , respectively. In particular ðð«ðð¢ð ð ( ðŽ ) acts on the centreof ðŽ . It also acts on the Grothendieck group ð ( ðŽ ) , as we will see in the next subsection. Deï¬nition 2.4.
Deï¬ne the following subgroups of
ðð«ðð¢ð ð ( ðŽ ) : ðð«ðð¢ððð§ð ( ðŽ ) = { ð â â ðð«ðð¢ð ð ( ðŽ ) | ðŸ ð = id ð ( ðŽ ) } , ðð¢ð ð ( ðŽ ) = { ð â â ðð«ðð¢ð ð ( ðŽ ) | ð â is isomorphic to an ðŽ - ðŽ -bimodule } , ðð¢ð ð ð ( ðŽ ) = { ð â ðð¢ð ð ( ðŽ ) | ð â ðŽ ð â ð for all projective ðŽ -modules ð } , ðð¢ððð§ð ( ðŽ ) = ðð¢ð ð ( ðŽ ) â© ðð«ðð¢ððð§ð ( ðŽ ) . If ð is a ï¬eld and ðŽ is ï¬nite-dimensional and basic then ðð¢ð ð ( ðŽ ) â ðð®ð ð ( ðŽ ) , which is the group of outer automorphisms of ðŽ , and ðð¢ððð§ð ( ðŽ ) â ðð®ðððð§ð ( ðŽ ) , the group of outer central automorphisms . FLORIAN EISELE
In the present paper we will often consider diï¬erent ð -algebra structures on the same ring ðŽ . Thisraises some questions regarding well-deï¬nedness, which the following remark addresses. Remark . (1) Let ðŽ be a ring and let ð , ð â ð ( ðŽ ) be two commutative subrings such that ðŽ is projective both as an ð -module and as an ð -module. Then, technically, the elements of ðð«ðð¢ð ð ( ðŽ ) are represented by complexes whose terms are ð -linear bimodules, and those of ðð«ðð¢ð ð ( ðŽ ) by complexes whose terms are ð -linear bimodules. However, we can always embed ðð«ðð¢ð ð ( ðŽ ) , ðð«ðð¢ð ð ( ðŽ ) âª î° ð ( ðŽ op â †ðŽ ) , showing that expressions like â ðð«ðð¢ð ð ( ðŽ ) â© ðð«ðð¢ð ð ( ðŽ ) â are well-deï¬ned. If one wants to stickcloser to the setting of [Ric91a] one could embed into ðð«ðð¢ð ð ( ðŽ ) , provided there is a commonsubï¬eld ð â ð , ð (which is the case in all examples we are interested in). In either case, theseare embeddings because an element ð â â ðð«ðð¢ð ð ( ðŽ ) is trivial if and only if ð» ð ( ð â ) = 0 for all ð â and ð» ( ð â ) â ðŽ as an ðŽ - ðŽ -bimodule (this follows, for example, from [RZ03, Proposition2.3]). This can be detected in î° ð ( ðŽ op â †ðŽ ) .(2) If the automorphism ðŸ ð â ðð®ð ð ( ð ( ðŽ )) induced by a ð â â ðð«ðð¢ð ð ( ðŽ ) happens to be ð -linearas well, then there is an ð â â ðð«ðð¢ð ð ( ðŽ ) such that ð â â ð â in î° ð ( ðŽ op â †ðŽ ) . To ï¬nd ð â ï¬rst pick a two-sided tilting complex ð â â î° ð ( ðŽ op â ð ðŽ â² ) , for a suitable ð -algebra ðŽ â² , whoserestriction to the left is the restriction to the left of ð â . Then, by [RZ03, Proposition 2.3], wehave ð â â ðŽ â² ðŒ ðŽ â ð â in î° ð ( ðŽ op â †ðŽ ) for some ring isomorphism ðŒ ⶠðŽ ⲠⶠðŽ . Note that,technically, [RZ03, Proposition 2.3] asks for ðŽ and ðŽ â² to be projective over a commutative basering. In all cases we are interested in we could in principle use a ï¬eld ð for this. But the proofof [RZ03] goes through regardless since in our situation the existence of inverses of ð â and ð â is guaranteed by other means. It follows that ðŒ | ð ( ðŽ â² ) ⊠ðŸ ð = ðŸ ð (where ðŸ ð ⶠð ( ðŽ ) ⶠð ( ðŽ â² ) is induced by ð â ), which shows that ðŒ | ð ( ðŽ â² ) must be ð -linear, implying that ðŒ ðŽ is an ð -linearbimodule. We can therefore choose ð â = ð â â ðŽ â² ðŒ ðŽ .2.3. Grothendieck groups and derived equivalences.
The contents of this subsection are standard(except perhaps Lemma 2.8), but both deï¬nitions and notation vary a lot across the literature. Assume Î is an ð -algebra, free and ï¬nitely-generated as an ð -module, where ð is either a ï¬eld or a completediscrete valuation ring. By ð (Î) we denote the Grothendieck group of Î , which is spanned by symbols [ ð ] , where ð is a ï¬nitely-generated projective Î -module, and [ ð â²â² ] = [ ð ] + [ ð â² ] whenever there is ashort exact sequence â ð â ð â²â² â ð â² â . This free abelian group is equipped with a bilinear form (â , =) Πⶠð (Î) à ð (Î) ⶠ†ⶠ( [ ð ] , [ ð ] ) ⊠rank ð ððšðŠ Î ( ð , ð ) . ð (Î) comes with a distinguished basis consisting of the symbols [ ð ] for indecomposable projectivemodules ð .Similar to the above we also get a Grothendieck group ð ( î· ð ( proj - Î)) , which is isomorphic to ð (Î) by means of the Euler characteristic ð ([ ð¶ â ]) = â ð (â1) ð [ ð¶ ð ] . In general, we have ([ ð â ] , [ ð â ]) Î = â ð â †(â1) ð rank ð ððšðŠ î· ð ( proj - Î) ( ð â , ð â [ ð ]) . See [Hap88, Chapter III.1] for a reference (but note that Happel deï¬nes ð ( ðŽ ) as ð ( mod - ðŽ ) ). It followsthat if ð â â î° ð (Î op â ð Î) is a two-sided tilting complex (where Î is another ð -algebra), then ([ ð â Î ð â ] , [ ð â Î ð â ]) Î = ([ ð ] , [ ð ]) Î for any two ï¬nitely-generated projective Î -modules ð and ð . When the form (â , =) Î is symmetric (e.g.for orders in semisimple algebras) this means that a derived equivalence induces an isometry betweenGrothendieck groups. IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 7
Deï¬nition 2.6 (âDecomposition mapâ) . Let ð be a complete discrete valuation ring with residue ï¬eld ð = ð â ðð and ï¬eld of fractions ðŸ . Let Î be an ð -order in a semisimple ðŸ -algebra ðŽ , and set Ì Î = Îâ ð Î .Then we can deï¬ne a †-linear map ð· Πⶠð ( Ì Î) ⶠð ( ðŽ ) given by the composition of the canonical isomorphism between ð ( Ì Î) and ð (Î) followed by the mapinduced by â ðŸ â ð â â.These maps satisfy the identity (â , =) Ì Î = ( ð· Î (â) , ð· Î (=)) ðŽ . We could call ð· Î a decomposition map and its matrix a decomposition matrix , but these terms usually refer to the adjoint of ð· Î and its matrix.We will therefore refrain from using this terminology. Proposition 2.7.
Let ð be a complete discrete valuation ring with residue ï¬eld ð = ð â ðð and ï¬eld offractions ðŸ . Moreover, let Î and Î be ð -orders in semisimple ðŸ -algebras ðŽ and ðµ , respectively. Write Ì Î and Ì Î for the reductions of Î and Î modulo ð . If ð â â î° ð (Î op â ð Î) is a two-sided tilting complex,then there is a commutative diagram ð ( Ì Î) ð Ìð / / ð· Î (cid:15) (cid:15) ð ( Ì Î) ð· Î (cid:15) (cid:15) ð ( ðŽ ) ð ðŸð / / ð ( ðµ ) . where ð ðŸð and ð Ìð are the isometries induced by the functor â â â Î ð â â. Let ð , ⊠, ð ð and ð , ⊠, ð ð denote representatives for the simple ðŽ - and ðµ -modules, and let ð , ⊠, ð ð and ð â²1 , ⊠, ð â² ð denote thecorresponding primitive idempotents in ð ( ðŽ ) and ð ( ðµ ) . The following hold:(1) There are a ð â ð ð and signs ð ⶠ{1 , ⊠, ð } ⶠ{±1} such that ð ðŸð ([ ð ð ]) = ð ( ð ) â [ ð ð ( ð ) ] . (2) There is an isomorphism ðŸ ðŸð ⶠð ( ðŽ ) ⶠð ( ðµ ) such that ðŸ ðŸð ( ð ð ) = ð â² ð ( ð ) for all ð â {1 , ⊠, ð } and ðŸ ðŸð restricts to the isomorphism ðŸ ð ⶠð (Î) ⶠð (Î) induced by ð â . (cid:3) In the situation of Proposition 2.7 the isomorphism between the centres of Î and Î induced by thetwo-sided tilting complex ð â is determined by the isomorphism between the centres of ðŽ and ðµ inducedby ðŸ â ð ð â . If ðŽ and ðµ are split, then this is even determined by the induced map on Grothendieckgroups. However, we will also deal with orders where ð ( Ì Î) is bigger than the reduction of ð (Î) modulo ð . The following lemma helps determining the induced isomorphism between the centres of Ì Î and Ì Î inthose cases. Lemma 2.8.
Let ð be an algebraically closed ï¬eld. Let ðŽ be a basic ï¬nite-dimensional symmetric ð -algebra, and let ð , ⊠, ð ð â ðŽ ( ð â â ) denote a full system of orthogonal primitive idempotents. Let ð¡ ⶠðŽ ⶠð be a symmetrising form. Assume moreover that soc( ð ( ðŽ )) = soc( ðŽ ) , and for all â©œ ð â©œ ð let ð ð â ð ð soc( ðŽ ) ð ð be the unique element such that ð¡ ( ð ð ) = 1 . Let ð â â ðð«ðð¢ð ð ( ðŽ ) be a two-sided tiltingcomplex, and deï¬ne ð¶ â ðð ð ( †) such that [ ð ð ðŽ â ðŽ ð â ] = ð â ð =1 ð¶ ð,ð â [ ð ð ðŽ ] for all â©œ ð â©œ ð holds in ð ( ðŽ ) . If ðŸ ð ⶠð ( ðŽ ) ⶠð ( ðŽ ) is the automorphism induced by ð â , then âš ðŸ ð ( ð ð ) â© ð = âš ð â ð =1 ( ð¶ â1 ) ð,ð â ð ð â© ð for all â©œ ð â©œ ð . FLORIAN EISELE
Proof.
For each projective ðŽ -module ð set ð¡ ð = ð¡ ⊠Tr ð , where Tr ð is the composition of the naturalisomorphism ðð§ð ðŽ ( ð ) ⶠð â ðŽ ððšðŠ ðŽ ( ð , ðŽ ) and the evaluation map ð â ðŽ ððšðŠ ðŽ ( ð , ðŽ ) ⶠðŽ . Notethat ð¡ ð deï¬nes a symmetrising form on ðð§ð ðŽ ( ð ) . Then deï¬ne forms ð¡ ð ð ðŽâ ðŽ ð â ( ðŒ ) = â ð â †(â1) ð â ð¡ ð ð ðŽâ ðŽ ð ð ( ðŒ ð ) = ð¡ âš ð ð ð ðŽâ ðŽ ð ð ( â ð â †(â1) ð â ðŒ ð ) for â©œ ð â©œ ð , where ðŒ â ðð§ð î° ð ( ðŽ ) ( ð ð ðŽ â ðŽ ð â ) . Note that this sum is ï¬nite and it is well-deï¬ned (therightmost expression shows it vanishes on null-homotopic maps).Let ð ð¿ , ð ð ⶠð ( ðŽ ) ⶠðð§ð î° ð ( ðŽ ) ( ð ð ðŽ â ðŽ ð â ) be the maps sending ð§ â ð ( ðŽ ) to the endomorphisminduced by left and right multiplication by ð§ , respectively (we use the same name for all ð ). By deï¬nition, ð ð¿ ( ð§ ) and ð ð ( ðŸ ð ( ð§ )) are equal (as elements of ðð§ð î° ð ( ðŽ ) ( ð ð ðŽ â ðŽ ð â ) ). Now note that ð¡ ð ð ðŽâ ðŽ ð ð ( ð ð ( ð ð )) for â©œ ð, ð â©œ ð and ð â †counts the number of times ð ð ðŽ occurs as a summand of ð ð ðŽ â ðŽ ð ð and therefore ð¡ ð ð ðŽâ ðŽ ð â ( ð ð ( ð ð )) = ð¶ ð,ð . We get ð¡ ð ð ðŽâ ðŽ ð â ( ð â ð =1 ( ð¶ â1 ) ð,ð â ð ð ( ð ð ) ) = (id ð à ð ) ð,ð (2)for all â©œ ð, ð â©œ ð . At the same time ð¡ ð ð ðŽâ ðŽ ð â ( ð ð¿ ( ð ð )) = 0 when ð â ð since then ð ð ð ð = 0 . If we had ð¡ ð ð ðŽâ ðŽ ð â ( ð ð¿ ( ð ð )) = 0 as well (for some ð ), then ð¡ ð ð ðŽâ ðŽ ð â ( ð ð¿ (soc( ðŽ ))) = ð¡ ð ð ðŽâ ðŽ ð â ( ð ð (soc( ðŽ ))) = 0 , whichis impossible by equation (2). Therefore there is a multiple ð â² ð of ð ð such that ð¡ ð ð ðŽâ ðŽ ð â ( ð ð¿ ( ð â² ð )) = ð¡ ð ð ðŽâ ðŽ ð â ( ð ð ( ðŸ ð ( ð â² ð ))) = (id ð à ð ) ð,ð . In particular, given a linear combination of the elements ðŸ ð ( ð â² ð ) for â©œ ð â©œ ð , applying ð¡ ð ð ðŽâ ðŽ ð â ⊠ð ð recovers the coeï¬cient of ðŸ ð ( ð â² ð ) . By considering equation (2) again we see that ðŸ ð ( ð â² ð ) = ð â ð =1 ( ð¶ â1 ) ð,ð â ð ð which proves the claim. (cid:3)
3. G
ENERALISED WEIGHTED SURFACE ALGEBRAS
In this section we will introduce a class of algebras which was ï¬rst studied by Erdmann and SkowroÅski[ES18, ES20c] to get a uniï¬ed description of both Brauer graph algebras and the algebras of dihedral,semi-dihedral and quaternion type classiï¬ed in [Erd90]. The paper [ES20c] already lays out a frameworkthat allows uniï¬ed treatment of most of these algebras. We will make a slightly more general deï¬nitionthat also encompasses the socle deformations studied in [ES20a, ES19]. In particular, the class of generalised weighted surface algebras deï¬ned below contains all Brauer graph algebras and their socledeformations, as well as almost all algebras from [Erd90] up to derived equivalence. In this section ð denotes an arbitrary ï¬eld. Set-up.
The combinatorial data to specify a generalised weighted surface algebra consists of thefollowing:(1) A ï¬nite -regular quiver ð . Such a quiver comes with an involution Ì â¶ ð ⶠð on its set of arrows such that ÌðŒ (for ðŒ â ð ) is the unique arrow in ð ⧵ { ðŒ } sharing its sourcewith ðŒ .(2) A permutation ð ⶠð ⶠð such that the source of ð ( ðŒ ) is the target of ðŒ for all ðŒ â ð . This deï¬nes another permutation ð ⶠð ⶠð ⶠðŒ ⊠ð ( ðŒ ) . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 9
For ðŒ â ð deï¬ne ð ðŒ = | ðŒ âš ð â©| (the cardinality of the ð -orbit of ðŒ ) . (3) Functions on ð -orbits ð â ⶠð â âš ð ⩠ⶠ†> and ð â ⶠð â âš ð ⩠ⶠð à representing multiplicities and certain scalars occurring in socle relations, respectively.(4) A function on ð -orbits ð¡ â ⶠð â âš ð ⩠ⶠðð ⶠðŒ ⊠ð¡ ðŒ, + ð¡ ðŒ, â ðŒ, with ð¡ ðŒ, â {0 , and ð¡ ðŒ, â ð . We allow ð¡ ðŒ, = 1 only if ð ( ðŒ ) = ðŒ and ð ÌðŒ ð ÌðŒ â©Ÿ , and weallow ð¡ ðŒ, â only if ð ( ðŒ ) = ðŒ and ð ÌðŒ ð ÌðŒ â©Ÿ (in which case ðŒ is a loop and ð ( ðŒ ) = ÌðŒ ). We willwrite â ð¡ ðŒ â¡ â if ð¡ ðŒ, = 0 and â ð¡ ðŒ â¡ â if ð¡ ðŒ, = 1 .(5) A set î â { ðŒð ( ðŒ ) ð ( ð ( ðŒ )) , ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð } of additional relations.The function ð¡ â is not present in [ES20c], but here we want to allow algebras with mixed special biserialand quaternion relations (to include algebras of semi-dihedral type). The purpose of ð¡ â is to control theshape of the relation the monomial ðŒð ( ðŒ ) is involved in. Informally, ð¡ ðŒ â¡ corresponds to a âspecialbiserial relationâ, ð¡ ðŒ â¡ to a âquaternion type relationâ, and a non-zero ð¡ ðŒ, corresponds to a âsocledeformationâ. Note that ð¡ ðŒ â is only possible if ð ÌðŒ ð ÌðŒ â©Ÿ . We should also point out that the conditionsin the speciï¬cation of ð¡ â need to be satisï¬ed for all ðŒ â ð , it is not suï¬cient to verify them for atransversal of the ð -orbits.The set î is also not present in [ES20c]. The idea is that working with completed path algebras as in[Lad14] seems more elegant, and it will be necessary anyway once we consider lifts to ð [[ ð ]] later. But[Lad14] excludes some cases of small quivers with small multiplicities which occur as block algebras(an application we have in mind). The above set-up allows us to throw in the relations of the form ðŒð ( ðŒ ) ð ( ð ( ðŒ )) from [ES18, ES20c] if needed. But there is also the case of Proposition 3.5 where weexplicitly do not want these relations, hence why we do not include them by default. Deï¬nition 3.1 (Additional notation) . Given the data above we deï¬ne ðµ ðŒ = ðŒð ( ðŒ ) ð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðŒ ) (a circular path of length ð ðŒ ð ðŒ ) ,ðŽ ðŒ = ðŒð ( ðŒ ) ð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â2 ( ðŒ ) (a path of length ð ðŒ ð ðŒ â 1 ) , for all ðŒ â ð . If ð ðŒ ð ðŒ = 1 then ðŽ ðŒ = ð ( ðŒ ) is the source of ðŒ (this will not matter in the sequel, though).When given ð , ð , ð â , ð â , ð¡ â and î as above we will always use the notations â ð â, â â, â ð ðŒ â, â ðµ ðŒ â andâ ðŽ ðŒ â without explicit reintroduction. We will avoid the use of the notation â ðµ ðŒ â and â ðŽ ðŒ â where it mightbe ambiguous (e.g. when we are dealing with more than one multiplicity function). In the followingdeï¬nition, Ìðð denotes the completion of ðð with respect to the ideal generated by ð , and we use ahorizontal bar to indicate completions of ideals. Deï¬nition 3.2 (âGeneralised weighted surface algebrasâ) . Let ð , ð , ð â , ð â , ð¡ â and î be as above. Deï¬ne Î = Î(
ð, ð , ð â , ð â , ð¡ â , î ) as Î = Ìðð â( ðŒð ( ðŒ ) â ð ÌðŒ ðŽ ÌðŒ ð¡ ðŒ , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ , î | ðŒ â ð ) . We call Î a generalised weighted surface algebra if(1) dim ð Î = â ðŒ âš ð â© â ð â âš ð â© ð ðŒ ð ðŒ , and(2) for all ðŒ â ð with ð ðŒ ð ðŒ â©Ÿ the relation ( ðŒð ( ðŒ ) ⯠ð ðð ðŒ â1 ( ðŒ ) + ÌðŒð ( ÌðŒ ) ⯠ð ðð ÌðŒ â1 ( ÌðŒ ) ) ðŽ ðŒ = 0 (3)holds in Î for all ð â©Ÿ (if ð ðŒ = 1 ) or ð â©Ÿ (if ð ðŒ > ). Our goal is not per se to extend the combinatorial description of [ES20c, ES20b] (while that wouldbe interesting, it would also be a sizeable undertaking unrelated to the ideas presented in this paper).Therefore the preceding deï¬nition includes as axioms only the key properties we want from our algebras.Let us now give suï¬cient criteria for when Î( ð, ð , ð â , ð â , ð¡ â , î ) is a generalised weighted surface algebra.For ð¡ â = 1 this is contained in [ES20c]. Proposition 3.3. (1) The algebra
Î = Î(
ð, ð , ð â , ð â , ð¡ â , { ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð }) is a generalisedweighted surface algebra if ð ðŒ ð ðŒ â©Ÿ for all ðŒ â ð with ð¡ ÌðŒ â¡ .(2) The algebra Î = Î(
ð, ð , ð â , ð â , ð¡ â , î ) (for any admissible î ) is a generalised weighted surfacealgebra if ð ðŒ ð ðŒ â©Ÿ for all ðŒ â ð with ð¡ ÌðŒ â¡ .In both cases all elements ðŒð ( ðŒ ) ð ( ð ( ðŒ )) and ðŒð ( ðŒ ) ð ( ð ( ðŒ )) for ðŒ â ð become zero in Î .Proof. (1) Write ð¡ ðŒ = ð¡ ðŒ, + ð¡ ðŒ, â ðŒ for ðŒ â ð . We have ðŒð ( ðŒ ) ð ( ð ( ðŒ )) â¡ ð ð ( ðŒ ) ð¡ ð ( ðŒ ) , â ðŒðŽ ð ( ðŒ ) + ð ð ( ðŒ ) ð¡ ð ( ðŒ ) , â ðŒðµ ð ( ðŒ ) â¡ ð ð ( ðŒ ) ð¡ ð ( ðŒ ) , â ðŒðŽ ð ( ðŒ ) (4)modulo the relations of Î . This uses the fact that if ð¡ ðŒ, â then ðŒ = ð ( ðŒ ) and therefore ð ( ðŒ ) = ÌðŒ , which entails ðŽ ÌðŒ ðŒ = ðµ ÌðŒ . If ð¡ ð ( ðŒ ) , = 0 , then the right hand side of (4) is zero. If ð¡ ð ( ðŒ ) , = 1 then our assumption implies ð ð ( ðŒ ) ð ð ( ðŒ ) â©Ÿ , which means that ðŽ ð ( ðŒ ) = ðŽ ð ( ðŒ ) containsthe initial subword ð ( ðŒ ) ð ( ð ( ðŒ )) . Hence the right hand side of (4) is zero modulo the relations of Î in this case as well.It follows that any path of length at least three containing a subword of the form ðŒð ( ðŒ ) either becomes zero in Î , or is of the form ðŒð ( ðŒ ) ð ( ðŒ ) , which can be rewritten as ð ÌðŒ ð¡ ðŒ, ðµ ÌðŒ .Using the relation ð ðŒ ðµ ðŒ = ð ÌðŒ ðµ ÌðŒ it also follows that each ðµ ðŒ lies in the socle of Î , whichimplies the condition from equation (3) we need to show. Hence Î is spanned by the initialsubwords of the elements ðµ ðŒ for ðŒ â ð . We should also note that we have in fact shown that Ìðð â( ðŒð ( ðŒ )â ð ÌðŒ ðŽ ÌðŒ ð¡ ðŒ , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ , ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð ) is ï¬nite-dimensional without takingthe completion, which shows that the ideal we are modding out here is in fact already complete.Note that the assumption ð ðŒ ð ðŒ â©Ÿ whenever ð¡ ÌðŒ â¡ implies that all paths involved in thedeï¬ning relations of Î have length at least two. A linear combination of paths involving paths oflength less than two can therefore not become zero modulo the relations of Î . Any non-triviallinear dependence between initial subwords of one or more of the ðµ ðŒ involving only paths oflength â©Ÿ implies that one of the ðµ ðŒ is zero in Î (by multiplying such a linear dependence byanother monomial, turning the shortest occurring initial subword of some ðµ ðŒ into ðµ ðŒ itself whilstannihilating all other terms). Hence we just need to show that all of the ðµ ðŒ are non-zero in Î ,since it will then follow that the proper initial subwords of the ðµ ðŒ together with one element ofthe pair { ðµ ðŒ , ðµ ÌðŒ } for each ðŒ form a basis of Î , which counting reveals to have size â ðŒ âš ð â© ð ðŒ ð ðŒ .Let us now show that, in fact, all ðµ ðŒ are non-zero. To this end, let us consider the ideal in ðŽ = ðð â( ðŒð ( ðŒ ) ð ( ð ( ðŒ )) , ðŒð ( ðŒ ) ð ( ð ( ðŒ )) , ðµ ðŒ ðŒ, ðµ ðŒ ÌðŒ, ÌðŒðµ ðŒ | ðŒ â ð ) (rather than Ìðð ) given by ðŒ = ( ðŒð ( ðŒ ) â ð ÌðŒ ðŽ ÌðŒ ð¡ ðŒ , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ | ðŒ â ð ) ðŽ . Since the deï¬ning relations of ðŽ are monomial,the algebra ðŽ has a ð -basis consisting of all paths not containing any of the relations as a subpath.By multiplying by all possible monomials from the left and from the right we can then write down ð -vector space generators of ðŒ : ðŒð ( ðŒ ) â ð ÌðŒ ð¡ ðŒ, ðŽ ÌðŒ â ð ÌðŒ ð¡ ðŒ, ðµ ÌðŒ , ðŒð ( ðŒ ) ð ( ðŒ ) â ð ðŒ ð¡ ðŒ, ðµ ðŒ ,ðŒð ( ðŒ ) ð ( ðŒ ) â ð ÌðŒ ð¡ ðŒ, ðµ ÌðŒ , ðŒð ( ðŒ ) ⯠ð ð ( ðŒ ) , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ , running over all ð â©Ÿ and ðŒ â ð . Moreover, all paths involved in these elements are linearlyindependent in ðŽ . It follows that ðŒ â© âš ðµ ðŒ | ðŒ â ð â© ð is equal to âš ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ | ðŒ â ð â© ð , whichcontains none of the ðµ ðŒ . Therefore each ðµ ðŒ is non-zero in ðŽ â ðŒ , which implies that it is non-zeroin Î . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 11 (2) We can show that in this case the paths ðŒð ( ðŒ ) ð ( ð ( ðŒ )) become zero in Î even though we did notexplicitly add them as relations. Once that is done, we can apply the ï¬rst part of this proposition.Let us assume that we have a path of length ð â©Ÿ containing the subword ðŒð ( ðŒ ) ð ( ð ( ðŒ )) ,and let us assume ð¡ ðŒ â and therefore ð ( ðŒ ) = ðŒ (otherwise the path is zero in Î anyway). Then this path is equivalent to the sum of (if ð¡ ðŒ â¡ ) a multiple of a path oflength ð + ð ÌðŒ ð ÌðŒ â 3 > ð with subword ðŽ ÌðŒ ð ( ð ( ðŒ )) , and (if ð¡ ðŒ, â ) a multiple of a path oflength ð + ð ÌðŒ ð ÌðŒ â 2 > ð with subword ðµ ÌðŒ ð ( ð ( ðŒ )) . The terminal subword of length threeof ðŽ ÌðŒ ð ( ð ( ðŒ )) is ð ð ÌðŒ ð ÌðŒ â3 ( ÌðŒ ) ð ð ÌðŒ ð ÌðŒ â2 ( ÌðŒ ) ð ( ð ð ÌðŒ ð ÌðŒ â2 ( ÌðŒ )) , since ð ( ð ( ðŒ )) = ð ( ðŒ ) , and ð ( ð ( ðŒ )) â ð ( ð ( ðŒ )) = ðŒ , which implies ð ( ðŒ ) = ð ð ÌðŒ ð ÌðŒ â1 ( ÌðŒ ) and therefore ð ( ðŒ ) = ð ð ÌðŒ ð ÌðŒ â1 ( ÌðŒ ) = ð ( ð ð ÌðŒ ð ÌðŒ â2 ( ÌðŒ )) . The element ðµ ÌðŒ ð ( ð ( ðŒ )) is equal to ðµ ÌðŒ ÌðŒ , assuming ð ( ðŒ ) = ðŒ . Modulothe relations of Î this is a multiple of ðµ ðŒ ÌðŒ , which has the same length and terminates in ð ð ðŒ ð ðŒ â2 ( ðŒ ) ð ð ðŒ ð ðŒ â1 ( ðŒ ) ð ( ð ð ðŒ ð ðŒ â1 ( ðŒ )) .Similarly, if we have a path of length ð â©Ÿ containing the subword ðŒð ( ðŒ ) ð ( ð ( ðŒ )) (with ð¡ ð ( ðŒ ) â ), then this path can be rewritten as a sum of multiples of paths of greater lengthcontaining a subword of the form ðœð ( ðœ ) ð ( ð ( ðœ )) . It thus follows that any path of length â©Ÿ containing a subword of the form ðŒð ( ðŒ ) ð ( ð ( ðŒ )) can be rewritten as a path of arbitrarily largelength, and such paths converge to zero in the completed path algebra Ìðð . Hence the originalpath was zero in Î . (cid:3) Proposition 3.4 (Alternative presentations) . Assume
Î = Î(
ð, ð , ð â , ð â , ð¡ â , î ) is a generalised weightedsurface algebra in the sense of Deï¬nition 3.2, and assume ð ðŒ ð ðŒ â©Ÿ for all ðŒ with ð¡ ÌðŒ â¡ . For each ðŒ â ð let ð¶ ðŒ and ð¶ â² ðŒ be linear combinations of paths each containing a subword of the form ðœð ( ðœ ) ð ( ð ( ðœ )) or ðœð ( ðœ ) ð ( ð ( ðœ )) , and ð¶ â²â² ðŒ a linear combination of paths each containing a subword ðœð ( ðœ ) with ð¡ ðœ = 0 .If ðŒ and ð ( ðŒ ) lie in the same ð -orbit then also assume that all paths involved in ð¶ â² ðŒ and ð¶ â²â² ðŒ have lengthequal to or bigger than that of ðµ ðŒ . Then Î = Ìðð â( ðŒð ( ðŒ ) â ð ÌðŒ ðŽ ÌðŒ ð¡ ðŒ + ð¶ ðŒ , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ + ð¶ â² ðŒ + ð¶ â²â² ðŒ , î | ðŒ â ð ) . (5) Proof.
By the proof of the second part of Proposition 3.3 we know that the ð¶ ðŒ , ð¶ â² ðŒ and ð¶ â²â² ðŒ become zeroin Î . Hence, the ideal being factored out on the right hand side of (5) is contained in the deï¬ning ideal of Î . We can also eliminate the ð¶ â²â² ðŒ from the presentation, by replacing the subwords ðœð ( ðœ ) by â ð¶ ðœ . Nowwe can use the same argument as in the proof of the second part of Proposition 3.3 (which is where weneed the length condition) to show that the ideal on the right hand side of (5) contains all elements ofthe form ðœð ( ðœ ) ð ( ð ( ðœ )) and ðœð ( ðœ ) ð ( ð ( ðœ )) , which shows that it contains all ð¶ ðŒ âs and ð¶ â² ðŒ âs. Therefore it isalso included in the deï¬ning ideal of Î , proving equality. (cid:3) Proposition 3.5.
Assume char( ð ) = 2 . Let ð be the quiver â â â ðœ k k ðŒ + + ðŒ | | ðœ < < ðŒ U U ðœ (cid:21) (cid:21) and let ð be the permutation ( ðŒ ðŒ ðŒ )( ðœ ðœ ðœ ) (i.e. ð ðŸ = 2 for all ðŸ â ð ). Set ð¡ ðŸ = 1 , ð ðŸ = 1 and ð ðŸ = ð for some ï¬xed ð â ð à , for all ðŸ â ð . Then Î = Î(
ð, ð , ð â , ð â , ð¡ â , â ) is a generalised weighted surface algebra.Proof. This is easy to verify. First note that for all ð â©Ÿ and all ðŸ â ð ðŸð ( ðŸ ) ⯠ð ðð ðŸ â1 ( ðŸ ) + ÌðŸð ( ÌðŸ ) ⯠ð ðð ÌðŸ â1 ( ÌðŸ ) = ðµ ððŸ + ðµ ðÌðŸ . Now ð ððŸ = ð ðÌðŸ = â ð ðÌðŸ gives that the above is zero in Î , implying the condition in equation (3).We claim that Î is a split-semisimple ð -algebra, and more speciï¬cally Î â ð â ð â ð â ð ( ð ) , (6)which, once shown, will imply the dimension condition immediately.Let ðž , ðž , ðž denote the respective unit elements in the ï¬rst three summands on the right hand sideof (6), and let ðž ( ð, ð ) denote the ( ð, ð ) -matrix unit in the rightmost summand in (6). One can easily verifythat Πⶠð â ð â ð â ð ( ð ) ⶠð ð ⊠ðž ð + ðž ( ð, ð ) , ðŒ ð ⊠ð â ðž ( ð, ð ( ð )) , ðœ ð ⊠ð â ðž ( ð, ð â1 ( ð )) (7)deï¬nes a homomorphism (where ð = (1 , ,
3) â ð ), by checking that the images satisfy the deï¬ningrelations of Î . It follows that dim ð Î â©Ÿ . On the other hand, Î is spanned by paths along ð -orbits oflength at most two, since any path involving a ðŸð ( ðŸ ) for ðŸ â ð , and likewise any path of length greaterthan two, can be rewritten as a multiple of a shorter path. Hence dim ð Î is at most , which implies thatthe map in (7) is an isomorphism. (cid:3)
4. L
IFTS FOR TWISTED B RAUER GRAPH ALGEBRAS
In this section we will have a look at a class of algebras closely related to Brauer graph algebras, butwith slightly more well-behaved lifts to ð [[ ð ]] -orders. These lifts were ï¬rst studied in [Gne19]. Note thatBrauer graph algebras are just generalised weighted surface algebras in the sense of Deï¬nition 3.2 where ð â is constant equal to one and ð¡ â is constant equal to zero. By ð we again denote an arbitrary ï¬eld. Deï¬nition 4.1 (see [Gne19, Deï¬niton 3.7]) . Let ð and ð be as in §3, and assume we are also given amap ð â ⶠð â âš ð ⩠ⶠ†> . We deï¬ne the twisted Brauer graph algebra Î tw ( ð, ð , ð â ) = ðð â( ðŒð ( ðŒ ) , ðµ ðŒ + ðµ ðŒ â² | ðŒ â ð ) , where ðµ ðŒ = ðŒð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðŒ ) for all ðŒ â ð , as before.Let us quickly explain how to associate a Brauer graph to the pair ( ð, ð ) . The Brauer graph encodesthe exact same information as ( ð, ð ) , and we will often switch back and forth between the two (with apreference for Brauer graphs in the statement of results, since that is the standard way of parametrisingBrauer graph algebras). Deï¬nition 4.2.
The
Brauer graph associated with ( ð, ð ) is the (undirected) graph whose vertices areindexed by the ð -orbits on ð and whose edges are indexed by the vertices in ð . The edge correspondingto ð â ð connects the vertices corresponding to ðŒ âš ð â© and ÌðŒ âš ð â© , where ðŒ and ÌðŒ are the two arrowswhose source is ð . The map ð induces a cyclic order on the half-edges incident to a vertex. Morespeciï¬cally, half-edges can be encoded as pairs ( ð, ðŒ ) â ð à ð where ð is the source of ðŒ . If ( ð, ðŒ ) corresponds to a half-edge, then the half-edge following ( ð, ðŒ ) in the cyclic order around ðŒ âš ð â© is deï¬nedto be ( ð ( ð ( ðŒ )) , ð ( ðŒ )) , where ð ( ð ( ðŒ )) denotes the source of ð ( ðŒ ) .There are some circumstances under which twisted Brauer graph algebras are isomorphic to theiruntwisted counterparts. This was studied in [Gne19], as were the lifts of these algebras given inProposition 4.4 below. In the present paper we want to deal with the class of algebras deï¬ned in §3,which properly contains Brauer graph algebras but not all twisted Brauer graph algebras. However, ifan algebra we are interested in happens to be a twisted Brauer graph algebra, then the lift provided inProposition 4.4 below will have nicer properties than the lifts constructed in Proposition 5.2, and we canprove slightly stronger results using them. Proposition 4.3 (see [Gne19, Proposition 3.16]) . If char( ð ) = 2 or if ð = Ìð and the Brauer graph of ( ð, ð ) is bipartite, then Î tw ( ð, ð , ð â ) â Î( ð, ð , ð â , ð â , ð¡ â , â ) where ð ðŒ = 1 and ð¡ ðŒ = 0 for all ðŒ â ð . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 13
Proposition 4.4 (see âRibbon Graph Ordersâ in [Gne19]) . Let ð and ð be as in §3, and assume we aregiven a map ð â ⶠð â âš ð ⩠ⶠ†> . Deï¬ne the ð -algebra Î tw ( ð, ð ) = Ìðð â( ðŒð ( ðŒ ) | ðŒ â ð ) and the element ð§ = â ðŒ â ð ðŒð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðŒ ) . The ð -algebra Î tw ( ð, ð ) becomes a ð [[ ð ]] -order by letting ð acts as ð§ (and extending this to thecompletion). We will denote this ð [[ ð ]] -order by Î = Î tw ( ð, ð , ð â ) . The following hold:(1) Î tw ( ð, ð , ð â )â ð Î tw ( ð, ð , ð â ) â Î tw ( ð, ð , ð â ) .(2) The ð (( ð )) -algebra ð (( ð )) â ð [[ ð ]] Î is semisimple.(3) The simple ð (( ð )) â ð [[ ð ]] Î -modules are labelled by the ð -orbits of arrows ðŒ âš ð â© â ð â âš ð â© .(4) The endomorphism algebra of the simple ð (( ð )) â ð [[ ð ]] Î -module labelled by ðŒ âš ð â© is ð (( ð ð ðŒ )) .(5) If ð· ð,ðŒ âš ð â© â {0 , , for ð â ð and ðŒ â ð is chosen such that ð is the source of exactly ð· ð,ðŒ âš ð â© arrows in the orbit ðŒ âš ð â© , then ð· Î ([ ð â Îâ ð Î]) = â ðŒ âš ð â© â ð â âš ð â© ð· ð,ðŒ âš ð â© â [ ð ðŒ âš ð â© ] for all ð â ð , where ð ðŒ âš ð â© denotes the simple ð (( ð )) â ð [[ ð ]] Î -module labelled by ðŒ âš ð â© .Proof. ðð â( ðŒð ( ðŒ ) | ðŒ â ð ) has a basis consisting of the vertices of ð and paths along ð -orbits, i.e. ðŒð ( ðŒ ) ⯠ð ð ( ðŒ ) for ð â©Ÿ . This follows from the fact that the ideal ( ðŒð ( ðŒ ) | ðŒ â ð ) is spanned by thepaths which have a subword of the form ðŒð ( ðŒ ) for ðŒ â ð .As a ð [ ð§ ] -module, ðð â( ðŒð ( ðŒ ) | ðŒ â ð ) is clearly spanned by initial subwords (including length zero)of ðŒð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðŒ ) for the various ðŒ â ð . In particular, it is ï¬nitely generated. Moreover, if we ï¬xa subset ð â ð such that for each ðŒ â ð exactly one of the two arrows ðŒ and ÌðŒ is contained in ð ,then the proper initial subwords (including length zero) of ðŒð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðŒ ) for the various ðŒ â ð together with the elements ðŒð ( ðŒ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðŒ ) for ðŒ â ð form a free generating set for the ð [ ð§ ] -module ðð â( ðŒð ( ðŒ ) | ðŒ â ð ) , showing that it is a ð [ ð§ ] -order. This carries over to the completion Î , turning itinto a free ð [[ ð§ ]] -module (or ð [[ ð ]] -module). The fact that Î tw ( ð, ð , ð â ) reduces to Î tw ( ð, ð , ð â ) is clearby deï¬nition.To show that ðµ = ð (( ð )) â ð [[ ð ]] Î is semisimple, let us ï¬rst write down (non-unital) embeddings of ð (( ð ð ðŒ )) into this algebra, which will turn out to correspond to the simple components of the centre.The image of under such an embedding gives a central idempotent ð ðŒ in ðµ , and we will write down a ð (( ð ð ðŒ )) -basis of ð ðŒ ðµ whose elements multiply like matrix units, proving semisimplicity and shape ofthe decomposition matrix in one go.Now, to an orbit ðŒ âš ð â© associate the (non-unital) embedding ð ðŒ ⶠð [[ ð ð ðŒ ]] ⶠð ( ðµ ) ⶠâ â ð =0 ð ð ( ð ð ðŒ ) ð ⊠ð â ðœ â ðŒ âš ð â© â â ð =0 ð ð ðœð ( ðœ ) ⯠ð ð ðŒ ( ð ðŒ + ð )â1 ( ðœ ) . Under this map, gets mapped to ð ðŒ = ð â ðœ â ðŒ âš ð â© ðœð ( ðœ ) ⯠ð ð ðŒ ð ðŒ â1 ( ðœ ) , and this turns ð ðŒ Î into a ð [[ ð ð ðŒ ]] -algebra. One easily checks that ð ðŒ and ð ðœ are orthogonal idempotents when ðŒ and ðœ lie indistinct ð -orbits. One also checks that ð (( ð ð ðŒ )) â ð [[ ð ððŒ ]] ð ðŒ Î has ð (( ð ð ðŒ )) -basis ð ( ðŒ ) ð,ð = 1 ð ð ðŒ ð ð â1 ( ðŒ ) ð ð ( ðŒ ) ⯠ð ð ðŒ + ð â2 ( ðŒ ) for â©œ ð, ð â©œ ð ðŒ . One veriï¬es that ð ( ðŒ ) ð,ð ð ( ðŒ ) ð,ð = ð ( ðŒ ) ð,ð for any â©œ ð, ð, ð â©œ ð ðŒ . That is, the ð ( ðŒ ) ð,ð multiply like matrixunits, thus proving that ð ðŒ ðµ â ð ð ðŒ ( ð (( ð ð ðŒ ))) . Moreover, the image in ð ðŒ Î of the idempotent in Î attached to a vertex ð£ â ð is given by â ðŒ ð ( ðŒ ) , , where ðŒ runs over all arrows whose source is ð£ (note that ð ( ðŒ ) ð,ð = ð ( ð ð â1 ( ðŒ )) , , so if two ðŒ âs in that sum lie in the same ð -orbit, then the correspondingidempotents lie in the same set of matrix units). This shows that the map ð· Î is as claimed, whichcompletes the proof. (cid:3) The above can be used to describe the order Î in slightly greater detail. For example, if for each ðŒ â ð the arrows ðŒ and ÌðŒ lie in diï¬erent ð -orbits, then the ð [[ ð ð ðŒ ]] -orders ð ðŒ Î are hereditary withbasis ð ( ðŒ ) ð,ð for ð â©œ ð and ð ð ðŒ ð ( ðŒ ) ð,ð for ð > ð . This is perfectly analogous to the lifts of Brauer treealgebras to ð -adic discrete valuation rings that occur in blocks (also called âGreen ordersâ [Rog92]).5. L IFTS FOR GENERALISED WEIGHTED SURFACE ALGEBRAS
In this section we will lift arbitrary generalised weighted surface algebras with suï¬ciently largemultiplicities to ð [[ ð ]] -orders. Such a lift will be a pullback of a lift as in Proposition 4.4 and an order ina generalised weighted surface algebra over ð (( ð )) (which is typically non-semisimple). We will see thatthese lifts will usually be independent of the multiplicities as rings, a fact which we then use to constructa bijection of silting complexes in Theorem 6.6. The latter is one of the main results of the present article.In this section, ð again denotes an arbitrary ï¬eld. Proposition 5.1.
Let ð be a ï¬nite quiver, and deï¬ne ðŽ = Ìðð as well as ðŽ = Ìð [ ð ] ð , the completionof the path algebra ð [ ð ] ð with respect to the ideal ( ð, ð ) ð [ ð ] ð . Consider maps ð ð ⶠ( ð, ð ) ðŽ ⶠðŽ given by ð ð ( ð ) = ð ð, + â â ð =1 ð ð â ð ð,ð for â©œ ð â©œ ð, where the ð ð,ð are elements of ðŽ and ð â â . For a given ð â ( ð, ð ) ðŽ let îŸ ( ð ) = { ð ( ð ) , ⊠, ð ð ( ð ) } .Let îµ be an ideal in ðŽ and let ð§ â ( ð ) ðŽ be chosen such that ð§ is central in ðŽ â( îŸ (0)) ðŽ . Deï¬ne ðµ = ðŽ â( îŸ ( ð§ ) , îµ â ð§ ) ðŽ , and assume that this is a ï¬nite-dimensional algebra. Assume moreover that all of the following hold:(1) dim ð ðµ = dim ð ðŽ â( îŸ (0) , îµ ) ðŽ + dim ð ðŽ â( îŸ (0) , ð§ ) ðŽ .(2) For each â©œ ð â©œ ð there is a Ìð§ ð â ( ð ) ðŽ such that Ìð§ ðð â ð ð,ð â ( îŸ ( ð ) , îµ ) ðŽ and Ìð§ ðð â ð ð,ð + ( îŸ (0)) ðŽ = ð§ ð â ð ð,ð + ( îŸ (0)) ðŽ (8) for all â©œ ð â©œ ð and ð â©Ÿ , and also ( ð ( Ìð§ ) , ⊠, ð ð ( Ìð§ ð ) , îµ â ð§, ð ) ðŽ = ( îŸ ( ð§ ) , îµ â ð§, ð ) ðŽ . (9) (3) ðŽ â( îŸ (0) , ð + ð§ ) ðŽ and ðŽ â( îŸ ( ð ) , îµ ) ðŽ are free and ï¬nitely generated as ð [[ ð ]] -modules.Let Î be the pullback ðŽ â( îŸ ( ð ) , îµ ) ðŽ / / / / ðŽ â( îŸ (0) , îµ , ð + ð§ ) ðŽ Î O O / / ðŽ â( îŸ (0) , ð + ð§ ) ðŽ , O O O O (10) where the maps into the top right term are the natural surjections. Then Î is a ð [[ ð ]] -order suchthat Îâ ð Î â ðµ .Proof. Our second assumption implies in particular that ð§ ð â ð ð,ð â ( îŸ (0) , îµ ) ðŽ , since substituting ð = 0 in the ï¬rst part of the condition yields Ìð§ ðð â ð ð,ð â ( îŸ (0) , îµ ) ðŽ . This shows that îŸ ( ð§ ) , îŸ (â ð§ ) â ( îŸ (0) , îµ ) ðŽ .Hence, the topmost horizontal arrow in the pullback diagram is well-deï¬ned, and the top right term, ðŽ â( îŸ (0) , îµ , ð + ð§ ) ðŽ , is isomorphic to a quotient of ðµ as a ð -algebra, and therefore is ï¬nite-dimensional.The ð [[ ð ]] -algebras ðŽ â( îŸ ( ð ) , îµ ) ðŽ and ðŽ â( îŸ (0) , ð + ð§ ) ðŽ are ð [[ ð ]] -orders by assumption, and theirreductions modulo ð are ðŽ â( îŸ (0) , îµ ) ðŽ and ðŽ â( îŸ (0) , ð§ ) ðŽ , respectively. From our assumptions it IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 15 follows that the dimensions of these quotients sum up to dim ð ( ðµ ) , from which it follows that Î has ð [[ ð ]] -rank dim ð ( ðµ ) . It therefore suï¬ces to show that Îâ ð Î is a quotient of ðµ . Note that ( îŸ ( ð ) , îµ ) ðŽ +( îŸ (0) , ð + ð§ ) ðŽ = ( îŸ (0) , îµ , ð + ð§ ) ðŽ . This shows that Î = ðŽ â( îŸ ( ð ) , îµ ) ðŽ â© ( îŸ (0) , ð + ð§ ) ðŽ . Hence weneed to show that ( îŸ ( ð ) , îµ ) ðŽ â© ( îŸ (0) , ð + ð§ ) ðŽ + ( ð ) ðŽ â ( îŸ ( ð§ ) , îµ â ð§ ) ðŽ + ( ð ) ðŽ , (11)which will imply the analogous inclusion for the completion as well (note that ( ð ) ðŽ = ( ð ) ðŽ , so the lefthand side is actually complete). We have ( îµ â ð§ ) ðŽ + ( ð ) ðŽ = îµ â ( ð + ð§ ) ðŽ + ( ð ) ðŽ , and îµ â ( ð + ð§ ) ðŽ isclearly contained in the intersection on the left hand side. Moreover, for all â©œ ð â©œ ð we have ð ð ( ð + Ìð§ ð ) â¡ ð ð ( Ìð§ ð ) mod ( ð ) ðŽ ,ð ð ( ð + Ìð§ ð ) â¡ ð ( ð + ð§ ) â¡ îŸ (0) , ð + ð§ ) ðŽ ,ð ð ( ð + Ìð§ ð ) â¡ ð ( ð ) â¡ îŸ ( ð ) , îµ ) ðŽ . In the last two lines we are using the assumptions of equation (8).In conclusion, each ð ð ( ð + Ìð§ ð ) lies in the intersection on the left hand side of (11), and at the sametime reduces to ð ( Ìð§ ð ) modulo ð . Now use the assumption of equation (9) to see that the left hand side of(11) actually contains the right hand side of (11). We have thus shown that Îâ ð Î is a quotient of ðµ , andtherefore, by virtue of dimensions, is isomorphic to ðµ . (cid:3) While perhaps not obvious at ï¬rst glance, the main point of Proposition 5.1 is that the pullback diagram(10) is often completely independent of the element ð§ if considered as a pullback diagram of rings.The bottom right term is isomorphic to ðŽ â( îŸ (0)) ðŽ as a ring, and the top right term is isomorphicto ðŽ â( îŸ (0) , îµ ) ðŽ . Neither of these depend on ð§ , nor does the map between them. There is a hiddendependence on ð§ in the topmost horizontal arrow, but we will see below that this dependence disappearsin most cases of interest. The point is that most admissible choices for ð§ give rise to the same ring Î lifting ðµ = ðµ ( ð§ ) , it is only the ð [[ ð ]] -algebra structure on this ring which varies. Proposition 5.2.
Let ð , ð , ð â , ð â , ð¡ â and î be as in §3. Assume that Î( ð, ð , ð â , ð â ð â , ð¡ â , î ) deï¬ned over ð (( ð )) is a generalised weighted surface algebra in the sense of Deï¬nition 3.2 (over ð (( ð )) rather than ð ). Ontop of that let ð â²â ⶠð â âš ð ⩠ⶠ†> be a function such that ð â² ðŒ ð ðŒ â©Ÿ for all ðŒ â ð with ð¡ ÌðŒ â¡ (which implies ( ð ðŒ + ð â² ðŒ ) ð ðŒ â©Ÿ in thosecases).Then there exists a ð [[ ð ]] -order Î = Î(
ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) such that Îâ ð Î â Î(
ð, ð , ð â + ð â²â , ð â , ð¡ â , î ) deï¬ned over ð. Proof.
Let us retain the notation ðŽ = Ìð [ ð ] ð and ðŽ = Ìðð . Deï¬ne the following: îŸ ( ð ) = { ðŒð ( ðŒ ) â ð â ð ÌðŒ ðŽ ÌðŒ ð¡ ðŒ âââââââââââââââââââââââââ = ð ðŒ ( ð ) | ðŒ â ð } ⪠î and îµ = ( ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ ) , where the ðŽ ðŒ âs and ðµ ðŒ âs are deï¬ned with respect to the multiplicity function ð â . Moreover, deï¬ne ð§ = â ðŒ â ð ðŒð ( ðŒ ) ⯠ð ð â² ðŒ ð ðŒ â1 ( ðŒ ) , as well as an element Ìð§ ðŒ = ðŒð ( ðŒ ) ⯠ð ( ðŒ ) ð â² ÌðŒ ð ðŒ â1 + ÌðŒð ( ÌðŒ ) ⯠ð ð â² ÌðŒ ð ÌðŒ â1 ( ÌðŒ ) for each ðŒ â ð (note the use of â ð â² ÌðŒ â instead of â ð â² ðŒ â in the ï¬rst summand). We need to check thatthe assumptions of Proposition 5.1 are satisï¬ed. Note that the ðŒ â ð replace the indices â©œ ð â©œ ð from Proposition 5.1 (technically we should also include the elements of î , but they are irrelevant forthe veriï¬cation). We have the following:(1) ðŽ â( îŸ (0) , îµ ) is isomorphic to the generalised weighted surface algebra Î( ð, ð , ð â , ð â , , â ) (i.e. all relations of special biserial type, rendering the relations in î redundant). Thereforeit has dimension â ðŒ âš ð â© ð ðŒ ð ðŒ . The algebra ðŽ â( îŸ (0) , ð§ ) is exactly the twisted Brauer graphalgebra Î tw ( ð, ð , ð â²â ) as in Deï¬nition 4.1. It is a special biserial algebra spanned by theinitial subwords of the elements ðŒð ( ðŒ ) ⯠ð ð ðŒ ð â² ðŒ â1 ( ðŒ ) for ðŒ â ð , with the added relation that ðŒð ( ðŒ ) ⯠ð ð ðŒ ð â² ðŒ â1 ( ðŒ ) = â ÌðŒð ( ÌðŒ ) ⯠ð ð ÌðŒ ð â² ÌðŒ â1 ( ÌðŒ ) (and such an element lies in the socle). Countingelements in the given basis yields that this algebra has dimension â ðŒ âš ð â© ð â² ðŒ ð ðŒ . Hence the sum ofthe dimensions is equal to â ðŒ âš ð â© â ð â âš ð â© ( ð ðŒ + ð â² ðŒ ) ð ðŒ , which is precisely the dimension of Î( ð, ð , ð â + ð â²â , ð â , ð¡ â , î ) . Now ðŽ â( îŸ ( ð§ ) , îµ â ð§ ) is another presentation of Î( ð, ð , ð â + ð â²â , ð â , ð¡ â , î ) by Proposition 3.4, which shows thedimension condition is satisï¬ed. Let us quickly outline why the assumptions of Proposition 3.4are fulï¬lled. What could potentially go wrong is that ð§ðŽ ÌðŒ or ð§ðµ ÌðŒ could contain a summand ðŒ ÌðŒ for some ðŒ with ð¡ ðŒ â . Now, this can only happen if ð ðŒ = 1 , which implies ðŒ â ð ( ðŒ ) , andtherefore ð¡ ðŒ â {0 , . If ð¡ ðŒ = 0 then there is no issue. If ð¡ ðŒ = 1 then ð ( ðŒ ) = ðŒ , which showsthat ðŒ and ð ( ðŒ ) have the same target. Then ð ( ð ( ðŒ )) cannot also have the same target (which isat the same time the source of ð ( ðŒ ) ), which forces ð ( ð ( ðŒ )) â ð ( ðŒ ) , that is, ð ð ( ðŒ ) = ð ÌðŒ > . Butthen ðŽ ÌðŒ has length â©Ÿ , and there is no issue.(2) ð (( ð )) â ð [[ ð ]] ðŽ â( îŸ ( ð ) , îµ ) ðŽ is the algebra Î( ð, ð , ð â , ð â ð â , ð¡ â , î ) deï¬ned over ð (( ð )) , which isa generalised weighted surface algebra in the sense of Deï¬nition 3.2 by assumption. Also, ðŽ â( îŸ ( ð ) , îµ , ð ) ðŽ â ðŽ â( îŸ (0) , îµ ) ðŽ is the generalised weighted surface algebra Î( ð, ð , ð â , ð â , , â ) deï¬ned over ð (here all ð¡ ðŒ âs arezero, which means that this automatically satisï¬es Deï¬nition 3.2). Since the aforementioned twogeneralised weighted surface algebras have the same dimension (over ð (( ð )) and ð , respectively),it follows that ðŽ â( îŸ ( ð ) , îµ ) is in fact a ð [[ ð ]] -order (which will also be useful on its own furtherbelow), and therefore any element which becomes zero upon tensoring with ð (( ð )) (that is, zeroin Î( ð, ð , ð â , ð â ð â , ð¡ â , î ) over ð (( ð )) ) was already zero in ðŽ â( îŸ ( ð ) , îµ ) .Since ð â² ÌðŒ > whenever ð ÌðŒ = 1 and ð¡ ðŒ â¡ by assumption, we can now use the second partof Deï¬nition 3.2 to get that Ìð§ ðŒ ðŽ ÌðŒ = 0 in ðŽ â( îŸ ( ð ) , îµ ) ðŽ whenever ð¡ ðŒ â (note that ð¡ ðŒ â and ð¡ ðŒ â¡ implies ð ÌðŒ > , so Deï¬nition 3.2 applies in all cases where ð¡ ðŒ â ), and therefore inparticular Ìð§ ðŒ ð ðŒ, â ( îŸ ( ð ) , îµ ) ðŽ in the notation of Proposition 5.1 for all ðŒ â ð (if ð¡ ðŒ = 0 then ð ðŒ, = 0 , so we have indeed covered all cases). We also have that both Ìð§ ðŒ ðŽ ÌðŒ and ð§ðŽ ÌðŒ becomeequal to ÌðŒð ( ÌðŒ ) ⯠ð ð â² ÌðŒ ð ÌðŒ â1 ( ÌðŒ ) ðŽ ÌðŒ modulo ( îŸ (0)) ðŽ if ð¡ ðŒ â , which implies the other conditionfrom equation (8) the Ìð§ ðŒ need to satisfy. Moreover, by Proposition 3.4 (which applies for thesame reasons as earlier) we have that both ðŽ â( îŸ ( ð§ ) , îµ â ð§, ð ) ðŽ and ðŽ â( ð ðŒ ( Ìð§ ðŒ ) , îµ â ð§, ð | ðŒ â ð ) ðŽ are isomorphic to Î( ð, ð , ð â + ð â²â , ð â , ð¡ â , î ) , which gives the condition from equation (9).(3) We have already seen further up that ðŽ â( îŸ ( ð ) , îµ ) ðŽ is a ð [[ ð ]] -order. By Proposition 4.4 thealgebra ðŽ â( îŸ (0) , ð + ð§ ) ðŽ is also a ð [[ ð ]] -order, since it is isomorphic to ðŽ â( ðŒð ( ðŒ ) | ðŒ â ð ) ðŽ with ð acting as â ð§ . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 17
It follows that all conditions of Proposition 5.1 are satisï¬ed, which gives us a ð [[ ð ]] -order Î as a pullbackof ðŽ â( îŸ ( ð ) , îµ ) ðŽ and ðŽ â( îŸ (0) , ð + ð§ ) ðŽ with Îâ ð Î â Î(
ð, ð , ð â + ð â²â , ð â , ð¡ â , î ) , as claimed. (cid:3) Proposition 5.3.
Assume we are in the situation of Proposition 5.2, and assume we are given another map ð â²â²â ⶠð â âš ð ⩠ⶠ†> subject to the same conditions as ð â²â . Then the orders Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) and Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â²â ) deï¬ned in Proposition 5.2 are isomorphic as rings if either one of the followingholds:(1) ð â² ðŒ > ð ðŒ and ð â²â² ðŒ > ð ðŒ for all ðŒ â ð for which ð â² ðŒ â ð â²â² ðŒ ,(2) ð ðŒ ð ðŒ â©Ÿ for all ðŒ â ð with ð¡ ÌðŒ â¡ , and { ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð } â î ,(3) ð ðŒ ð ðŒ â©Ÿ for all ðŒ â ð with ð¡ ÌðŒ â¡ ,(4) char ( ð ) = 2 , ð , ð , ð â , ð â as well as ð¡ â are as in Proposition 3.5, and î = â .Proof. As a pullback diagram of rings, the diagram (10) is isomorphic to the diagram ðŽ â( îŸ ( ð ) , îµ ) ðŽ ð / / / / ðŽ â( îŸ (0) , îµ ) ðŽ Î O O / / ðŽ â( îŸ (0)) ðŽ ð O O O O (12)where the map ð is the natural surjection, and the map ð is induced by the map from ðŽ to ðŽ whichsends ð to â ð§ (using the notation of Proposition 5.1) and induces the identity on all arrows, extendedto the completion. For both ð â²â and ð â²â²â we get a diagram as in (12), with two diï¬erent maps ð , say ð and ð , but otherwise identical. The maps ð ð are actually determined by the image of ð . Clearly thetwo diagrams will be isomorphic if both ð and ð send ð to , but this is not always the case. We willshow below to what extent the map ð in (12) is independent of ð â²â (or ð â²â²â ) under our four alternativeassumptions.Concretely, in the construction of Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) in Proposition 5.2 we have that ðŽ â( îŸ (0) , îµ ) ðŽ â Î( ð, ð , ð â , ð â , , â ) and ð§ = â ðŒ â ð ðŒð ( ðŒ ) ⯠ð ð â² ðŒ ð ðŒ â1 ( ðŒ ) . Now we have the following cases, corresponding to the various possible assumptions from the statement:(1) If ð â² ðŒ > ð ðŒ for all ðŒ â ð then ð§ becomes zero in Î( ð, ð , ð â , ð â , , â ) since it is a sum of pathseach properly containing a subword of the form ðµ ðŒ (with respect to this generalised weightedsurface algebra). Hence ð maps ð to . If instead we only have ð â² ðŒ â©Ÿ ð ðŒ for all ðŒ â ð , then ð maps ð to â â ðŒ â ð s.t. ð ðŒ = ð ðŒ â² ðµ ðŒ , which clearly only depends on the set { ðŒ â ð | ð ðŒ = ð â² ðŒ } .(2) If ð ðŒ ð ðŒ â©Ÿ for all ðŒ â ð with ð¡ ÌðŒ â¡ and { ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð } â î , then the ð (( ð )) -spanof ðŽ â( îŸ ( ð ) , îµ ) ðŽ , which is isomorphic to Î( ð, ð , ð â , ð â ð â , ð¡ â , î ) deï¬ned over ð (( ð )) ,is a generalised weighted surface algebra in the sense of Deï¬nition 3.2. Moreover, Proposition 3.3applies to this algebra, which implies that elements of the form ðŒð ( ðŒ ) ð ( ð ( ðŒ )) and ðŒð ( ðŒ ) ð ( ð ( ðŒ )) for ðŒ â ð are zero in this algebra, and therefore also in ðŽ â( îŸ ( ð ) , îµ ) ðŽ . From this it is easy tosee that ð§ is actually central in ðŽ â( îŸ ( ð ) , îµ ) ðŽ , and ð ðŒ ( ð + ð§ ) becomes zero in ðŽ â( îŸ ( ð ) , îµ ) ðŽ (since ð§ðŽ ÌðŒ ð¡ ðŒ becomes zero, as our assumptions ensure that ðŽ ÌðŒ ð¡ ðŒ has length â©Ÿ if ð¡ ðŒ â ). Thisimplies that there is an automorphism of ðŽ â( îŸ ( ð ) , îµ ) ðŽ sending ð to ð + ð§ . Since ð maps ð to â ð§ , precomposing ð with this automorphism yields the homomorphism which sends ð to . That is, the pullback diagram (12) is isomorphic in this case to the pullback diagram where ð isreplaced by the map which sends ð to and all arrows to themselves.(3) Using Proposition 3.3 we see that this case is identical to the previous case, since elements ofthe form ðŒð ( ðŒ ) ð ( ð ( ðŒ )) and ðŒð ( ðŒ ) ð ( ð ( ðŒ )) for ðŒ â ð become zero in Î( ð, ð , ð â , ð â ð â , ð¡ â , â ) without being added as relations explicitly.(4) In this case we deï¬ne an element Ìð§ = â ðŒ â ð ( ðŒð ( ðŒ )) ð â² ðŒ â ð ð â² ðŒ â ð ð ( ðŒ ) = ð§ â â ðŒ â ð ð ð â² ðŒ â ð ð ( ðŒ ) , where ð ( ðŒ ) denotes the source of the arrow ðŒ , and ð ð ( ðŒ ) is the corresponding idempotent.Proposition 3.5 gives an explicit isomorphism between the ð (( ð )) -span of ðŽ â( îŸ ( ð ) , îµ ) ðŽ , whichis Î( ð, ð , ð â , ð â ð â , ð¡ â , â ) deï¬ned over ð (( ð )) , and the ð (( ð )) -algebra ð (( ð )) â ð (( ð )) â ð (( ð )) â ð ( ð (( ð ))) . Under this isomorphism, the element Ìð§ gets mapped to an element with non-zero entries onlyin the ï¬rst three components, while all arrows get mapped to elements with non-zero entriesonly in the last component. Hence Ìð§ is central in ðŽ â( îŸ ( ð ) , îµ ) ðŽ , and it annihilates all arrows,which implies that ð ðŒ ( ð + Ìð§ ) is zero in ðŽ â( îŸ ( ð ) , îµ ) ðŽ for all ðŒ â ð . Moreover, ð + Ìð§ togetherwith ð and the idempotents generate a ð -algebra whose completion is ðŽ â( îŸ ( ð ) , îµ ) ðŽ , since Ìð§ is contained in the square of the radical of this algebra. It follows that we get an automorphismon ðŽ â( îŸ ( ð ) , îµ ) ðŽ which sends ð to ð + Ìð§ , and the composition of this automorphism with ð sends ð to â ðŒ â ð ð§ ð â² ðŒ â ð ð ( ðŒ ) . Now this is clearly zero in ðŽ â( îŸ (0) , îµ ) ðŽ â Î( ð, ð , ð â , ð â , , â ) ,since ð§ already lies in the socle of this algebra. (cid:3) In all except the ï¬rst of the alternative assumptions of Proposition 5.3 the ð [[ ð ]] -orders Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) and Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â²â ) end up being isomorphic as rings to a pullback asin (12) where ð sends ð to . For future reference, we give this ring explicitly below. Deï¬nition 5.4.
In the situation of Proposition 5.2 deï¬ne Î = Î ( ð, ð , ð â , ð â , ð¡ â , î ) as the pullback ofrings Ìð [ ð ] ð ( ðŒð ( ðŒ )â ð â ð ÌðŒ ðŽ ÌðŒ ð¡ ðŒ , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ , î | ðŒ â ð ) ð / / / / Ìðð ( ðŒð ( ðŒ ) , ð ðŒ ðµ ðŒ â ð ÌðŒ ðµ ÌðŒ | ðŒ â ð ) Î O O / / Ìðð ( ðŒð ( ðŒ ) | ðŒ â ð ) ð O O O O where ð is the natural surjection and ð is the ð -algebra homomorphism sending ð to , the arrows in ð to themselves, extended to the completion.6. B IJECTIONS OF SILTING COMPLEXES
We now have lifts of generalised weighted surface algebras to ð [[ ð ]] -orders in many cases. We wouldlike to use these lifts to prove a statement about their silting complexes. To this end, we need to understandhow silting complexes over a ð [[ ð ]] -order relate to silting complexes over its reduction modulo ð . Let ð denote a complete discrete valuation ring with maximal ideal ðð , and let Î be an ð -order. It wasobserved in [KZ96] that the endomorphism ring of a tilting complex ð â over Î is not necessarily an ð -order, and by [Ric91a, Theorem 2.1] and [Ric91b, Theorem 3.3] it is an ð -order if and only if ð â reducesto a tilting complex over Îâ ð Î . That is, tilting complexes do not necessarily reduce to tilting complexes.It turns out that silting complexes are better behaved in this regard. IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 19
Proposition 6.1.
Let ð be a complete discrete valuation ring with maximal ideal ðð , and let Î be an ð -order. Deï¬ne Ì Î = Îâ ð Î , and use â â to denote application of the functor â ð â ðð â ð â â. Let ð â and ð â be bounded complexes of ï¬nitely generated projective Î -modules. We claim that ððšðŠ î· ð ( proj - Î) ( ð â , ð â [ ð ]) = 0 for all ð > if and only if ððšðŠ î· ð ( proj - Ì Î) ( Ìð â , Ìð â [ ð ]) = 0 for all ð > . In particular, ð â is a pre-silting complex over Î if and only if Ìð â is a pre-silting complex over Ì Î .Proof. For the âifâ-direction assume
ððšðŠ î· ð ( proj - Ì Î) ( Ìð â , Ìð â [ ð ]) = 0 for all ð > . Let ð ⶠð â ⶠð â [ ð ] ,for some ð > , be a map (of †-graded modules) that commutes with the diï¬erential. Then Ìð ⶠÌð â ⶠÌð â [ ð ] must be null-homotopic by assumption, i.e. there is a map Ìâ ⶠÌð â ⶠÌð â [ ð â 1] suchthat Ìâ ⊠ð Ìð â ð Ìð [ ð â1] ⊠Ìâ = Ìð (note that ð Ìð [ ð â1] = â ð Ìð [ ð ] by the usual sign conventions). Since homotopiesare not required to commute with the diï¬erential, we can lift it to a map â ⶠð â ⶠð â [ ð â 1] such that ð â² = ð â â ⊠ð ð + ð ð [ ð â1] ⊠â â ð ððšðŠ î· ð ( proj - Î) ( ð â , ð â [ ð ]) . Hence we may represent the homotopy classof ð as ð â² = ð â ð for some other map ð commuting with the diï¬erential. But the same argument appliesto ð , and can be repeated indeï¬nitely thereafter. That is, ð â ð ð ððšðŠ î· ð ( proj - Î) ( ð â , ð â [ ð ]) for all ð > ,which implies that ð is in fact null-homotopic.Now let us prove the âonly ifâ-direction. Assume that ððšðŠ î· ð ( proj - Î) ( ð â , ð â [ ð ]) = 0 for all ð > . We want to show ððšðŠ î· ð ( proj - Ì Î) ( Ìð â , Ìð â [ ð ]) = 0 for arbitrary ð > . To this end, considera map Ìð ⶠÌð â ⶠÌð â [ ð ] that commutes with the diï¬erential. Such a map can be lifted to a map ð ⶠð â ⶠð â [ ð ] , but ð ⊠ð ð â ð ð [ ð ] ⊠ð is a potentially non-zero map from ð â into ð â [ ð + 1] commutingwith the diï¬erential and reducing to zero modulo ð . By assumption, we get a map â ⶠð â ⶠð â [ ð ] such that ð ( ð ⊠ð ð â ð ð [ ð ] ⊠ð ) = â ⊠ð ð â ð ð [ ð ] ⊠â. This tells us that ð â ð â â does commute with the diï¬erential, and is therefore null-homotopic byassumption. But since ð â ð â â reduces to Ìð modulo ð , it follows that Ìð is null-homotopic, too.Hence ððšðŠ î· ð ( proj - Ì Î) ( Ìð â , Ìð â [ ð ]) = 0 . (cid:3) Deï¬nition 6.2 (âtorsion-siltingâ) . Let Î be a ring. We call a pre-silting complex ð â â î· ð ( proj - Î) torsion-silting if { ð â â î° (Î) | ððšðŠ î° (Î) ( ð â [ ð ] , ð â ) = 0 â ð } â { ð â â î° (Î) | ð» ð ( ð â ) = ð» ð ( ð â ) â rad(Î) â ð } . We denote the set of isomorphism classes of basic torsion-silting complexes by t-silt - Î . Proposition 6.3.
In the situation of Proposition 6.1, the complex ð â is torsion-silting if and only if Ìð â is silting.Proof. Let î Î denote the smallest localising subcategory of î· (Î) = î· ( Mod - Î) containing î· â ( Proj - Î) .Deï¬ne î Ì Î analogously. By [BN93, Proposition 2.12] the canonical functors î Î â¶ î° (Î) and î Ì Î â¶ î° ( Ì Î) are equivalences, and in particular any complex in î° (Î) and î° ( Ì Î) is isomorphic to a complex withprojective terms. We will also repeatedly use that ððšðŠ î· (Î) ( ð â , â) â ððšðŠ î° (Î) ( ð â , â) , since ð â is abounded complex of projectives (also, the analogous statement for Ì Î and Ìð â ).Assume that ð â is torsion-silting. Let ð â â î° ( Ì Î) be an element such that ððšðŠ î° ( Ì Î) ( Ìð â [ ð ] , ð â ) = 0 forall ð â †. Then ððšðŠ î° ( Ì Î) ( Ìð â [ ð ] , ð â ) â ððšðŠ î· ( Ì Î) ( Ìð â [ ð ] , ð â ) = ððšðŠ î· (Î) ( ð â [ ð ] , ð â ) â ððšðŠ î° (Î) ( ð â [ ð ] , ð â ) for all ð . Since ð â is torsion-silting this implies that ð» ð ( ð â ) â rad(Î) = ð» ð ( ð â ) for all ð . By assumption, Î is an ð -order and therefore there is a ð â©Ÿ such that rad(Î) ð â ð Î . It follows that ðð» ð ( ð â ) = ð» ð ( ð â ) for all ð . But ð â is a complex of Ì Î -modules, on which ð acts as zero. Hence ð» ð ( ð â ) = 0 for all ð â †, which implies that ð â â 0 in î° ( Ì Î) . We conclude that add( Ìð â ) is a âcompact generating subcategoryâ of î° ( Ì Î) , which by [AI12, Proposition 4.2] implies that thick( Ìð â ) = î· ð ( proj - Ì Î) .For the other direction, assume that thick( Ìð â ) = î· ð ( proj - Ì Î) . Then { ð â â î° ( Ì Î) | ððšðŠ ð· ( Ì Î) ( Ìð â [ ð ] , ð â ) = 0 for all ð â †} = 0 . Assume that ð â â î Î has the property that ððšðŠ î° (Î) ( ð â [ ð ] , ð â ) = 0 for all ð â †. Consider the complex Ìð â , which lies in î Ì Î . We can argue as in the second part of Proposition 6.1. An Ìð â ððšðŠ î· ( Ì Î) ( Ìð â [ ð ] , Ìð â ) ,for some ð â †, lifts to a homomorphism of †-graded modules ð â ððšðŠ Î ( ð â [ ð ] , ð â ) . Then ð ⊠ð ð [ ð ] â ð ð ⊠ð commutes with the diï¬erential, and has image contained in ðð â . We can therefore ï¬nd a homotopy â ⶠð â [ ð ] ⶠð â such that ð â1 ( ð ⊠ð ð [ ð ] â ð ð ⊠ð ) = â ⊠ð ð [ ð ] â ð ð ⊠â , from which it follows that ð â ðâ is a homomorphism of chain complexes, which must be null-homotopic since ððšðŠ î° (Î) ( ð â [ ð ] , ð â ) = 0 .But then Ìð is null-homotopic. In particular, ððšðŠ î° ( Ì Î) ( Ìð â [ ð ] , Ìð â ) = 0 for all ð â †, which implies (usingthe assumption on Ìð â ) that Ìð â â 0 in î° ( Ì Î) . Now ð â ðð â ð ð» ð ( ð â ) ⪠ð» ð ( Ìð â ) = 0 , which implies ðð» ð ( ð â ) = ð» ð ( ð â ) . Since ð â rad(Î) it follows that ð» ð ( ð â ) â rad(Î) = ð» ð ( ð â ) , showing that ð â istorsion-silting. (cid:3) Remark . By [Ric91b, Theorem 3.3] it is also true that if Ìð â is tilting the ð â is tilting.As a consequence, we get bijections of silting complexes in many cases. Note that we do not knowwhether t-silt - Î is partially ordered, but we can still deï¬ne the relation â â©œ â as in equation (1). InCorollary 6.5, t-silt - Î being a poset is part of the assertion. In general, we do not know to what extent t-silt - Î can diï¬er from silt - Î . A torsion-silting complex which is not silting would correspond to a pre-silting complex which reduces to a silting complex over Îâ ð Î , but tensoring with the ï¬eld of fractions ðŸ of ð does not give a generator of î· ð ( proj - ðŸ â ð Î) . If, for example, ðŸ â ð Î is semisimple then thiscannot happen. Corollary 6.5.
Let Î be a ring and let ð , ð ⶠð [[ ð ]] ⶠð (Î) be two embeddings which both turn Î into a ð [[ ð ]] -order. Then the functor â â Î Îâ ð ð ( ð )Î for ð â {1 , induces a bijection between the isomorphism classes of pre-silting complexes over Î and those over Îâ ð ð ( ð )Î , and in particular induces isomorphisms of partially ordered sets silt - Îâ ð ( ð )Π⌠ⷠt-silt - Π⌠ⷠsilt - Îâ ð ( ð )Î . Proof.
It suï¬ces to prove the claim for the relationship between Îâ ð ( ð )Î and Î . So let us assumewithout loss that Î is given as a ð [[ ð ]] -order. By [Ric91b, Proposition 3.1], for any pre-silting complex Ìð â in î· ð ( proj - Îâ ð Î) there is a complex ð â , unique up to isomorphism in î· ð ( proj - Î) , such that ð â â Î Îâ ð Î â Ìð â . Moreover, by Proposition 6.1, the complex ð â in î· ð ( proj - Î) is pre-silting if and only if Ìð â is pre-silting over Îâ ð Î , and by Proposition 6.3 it is torsion-silting if and only if Ìð â is silting. Thisshows that the functor â â Î Îâ ð Î does indeed induce a bijection between basic torsion-silting complexesover Î and basic silting complexes over Îâ ð Î . The relation â â©Ÿ â is deï¬ned in equation (1), and, withthis deï¬nition in mind, the main assertion of Proposition 6.1 can be restated as saying that ð â â©Ÿ ð â ifand only if Ìð â â©Ÿ Ìð â ( ð and ð being as in that proposition), which shows that the functor â â Î Îâ ð Î preserves order (which also shows that t-silt - Î is a poset). (cid:3) Theorem 6.6 (Lifting theorem & silting bijection) . Let ð , ð , ð â , ð¡ â and î be as in §3. For ðŒ â ð set ð ðŒ = { if ð ðŒ > , if ð ðŒ = 1 .Assume that we are given two multiplicity functions ð (1) â , ð (2) â ⶠð â âš ð ⩠ⶠ†⩟ . Moreover, assume thatfor each ð â {1 , and all ðŒ â ð with ð¡ ÌðŒ â¡ we have one of the following:(1) ð ( ð ) ðŒ â©Ÿ ð ðŒ + ð ðŒ if { ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð } â î , or IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 21 (2) ð ( ð ) ðŒ â©Ÿ ð ðŒ + ð ðŒ .Then there are a ring Î and embeddings ð ð ⶠð [[ ð ]] ⶠð (Î) for ð â {1 , , each endowing Î with the structure of a ð [[ ð ]] -order, such that Î( ð, ð , ð ( ð ) â , ð â , ð¡ â , î ) â Îâ ð ð ( ð ) â Î for ð â {1 , . In particular, there are isomorphisms of partially ordered sets t-silt - Î i i ⌠) ) ââââââââââââââ ⌠u u â§â§â§â§â§â§â§â§â§â§â§â§â§â§ silt - Î( ð, ð , ð (1) â , ð â , ð¡ â , î ) o o ⌠/ / silt - Î( ð, ð , ð (2) â , ð â , ð¡ â , î ) where the arrows going down are induced by the functor â â â Î Îâ ð ð ( ð ) â Î â.Proof. Deï¬ne a map ð â ⶠð â âš ð ⩠ⶠ†> ⶠðŒ ⊠{ min{ ð (1) ðŒ , ð (2) ðŒ } â ð ðŒ if ð¡ ÌðŒ â¡ , min{ ð (1) ðŒ , ð (2) ðŒ } â 1 if ð¡ ÌðŒ â¡ .Pick an ð â {1 , and deï¬ne ð â²â = ð ( ð ) â â ð â . The deï¬nition of ð â ensures that ð â² ðŒ â©Ÿ and ð ðŒ â©Ÿ for all ðŒ â ð . Furthermore, for any ðŒ â ð with ð¡ ÌðŒ â¡ our assumptions ensure that ð â² ðŒ â©Ÿ ð ðŒ , which in turnensures ð â² ðŒ ð ðŒ â©Ÿ . Also, when ð¡ ÌðŒ â¡ we have ð ðŒ ð ðŒ = min{( ð (1) ðŒ â ð ðŒ ) ð ðŒ , ( ð (2) ðŒ â ð ðŒ ) ð ðŒ } â©Ÿ { if { ðŒð ( ðŒ ) ð ( ð ( ðŒ )) | ðŒ â ð } â î , otherwise. (13)It follows that the assumptions of Proposition 5.2 are satisï¬ed. Speciï¬cally, the inequality (13) impliesthat we can use Proposition 3.3 to verify that the algebra Î( ð, ð , ð â , ð â ð â , ð¡ â , î ) in the statementof Proposition 5.2 is a generalised weighted surface algebra. We checked the other condition ofProposition 5.2 verbatim further above. Hence we get a ð [[ ð ]] -order Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) reducing to Î( ð, ð , ð â + ð â²â , ð â , ð¡ â , î ) = Î( ð, ð , ð ( ð ) â , ð â , ð¡ â , î ) .The inequality (13) tells us that one of the alternative assumptions of Proposition 5.3 is satisï¬ed,which implies that the isomorphism type of Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) as a ring does not depend on ð â²â . Tobe speciï¬c, by the remarks following Proposition 5.3, we have Î( ð, ð , ð â , ð â , ð¡ â , î ; ð â²â ) â Î ( ð, ð , ð â , ð â , ð¡ â , î ) (from Deï¬nition 5.4)as rings. Our claims on the correspondence of silting complexes now follow from Corollary 6.5. (cid:3) The preceding theorem shows that generalised weighted surface algebras deï¬ned for the samecombinatorial data but diï¬erent multiplicities have common lifts, and therefore common silting posets,provided the multiplicities are big enough. It is inevitable that in some cases we could, in theory,allow slightly smaller multiplicities. The problem with incorporating these cases in a general theoremis that checking when a Î( ð, ð , ð â , ð â , ð¡ â , î ) is a generalised weighted surface algebra in the sense ofDeï¬nition 3.2 is tricky for multiplicities smaller than those allowed by Proposition 3.3 (for instance, theanswer may no longer be independent of the characteristic of ð ). That is, one will have to consider thiscase-by-case, and verify the assumptions of Propositions 5.2 and 5.3 by hand. Below we do this for onecase which is of interest in the modular representation theory of ï¬nite groups. Proposition 6.7.
Assume char( ð ) = 2 . Let ð , ð , ð â and ð¡ â be as in Proposition 3.5. Then the conclusionof Theorem 6.6 holds for any two functions ð (1) â , ð (2) â ⶠð â âš ð ⩠ⶠ†⩟ . Proof.
We can proceed as in the proof of Theorem 6.6. Deï¬ne ð ðŒ = 1 for all ðŒ â ð and, after pickingan ð â {1 , , set ð â²â = ð ( ð ) â â ð â . All veriï¬cations made in the proof of Theorem 6.6 carry over tothis case, apart from inequality (13). But the inequality (13) is only used to check the assumptions ofProposition 3.3, which we can replace by Proposition 3.5, and to check the assumptions of Proposition 5.3,which are also satisï¬ed in this case. (cid:3) The advantage of Theorem 6.6 is that it can be applied to arbitrary Brauer graph algebras, but in caseswhere they coincide with twisted Brauer algebras we can allow arbitrary multiplicities by using the liftsfrom Proposition 4.4 instead.
Proposition 6.8.
In the situation of Theorem 6.6, if ð¡ â = 0 , ð â = 1 and either char( ð ) = 2 or ð = Ìð andthe Brauer graph of ( ð, ð ) in the sense of Deï¬nition 4.2 is bipartite, then the conclusion of Theorem 6.6holds for any two functions ð (1) â , ð (2) â ⶠð â âš ð ⩠ⶠ†⩟ .Proof. By Proposition 4.3 our assumptions imply that Î( ð, ð , ð ( ð ) â , ð â , ð¡ â , î ) â Î tw ( ð, ð , ð ( ð ) â ) for ð â{1 , . By Proposition 4.4 we can therefore choose Î = Î tw ( ð, ð ) . (cid:3) Theorem 6.6, together with Proposition 6.8, implies Theorem A. Since it is well-known that blocks ofgroup algebras of cyclic defect are Morita equivalent to Brauer tree algebras, Proposition 6.8 also impliesthe part of Theorem B concerned with cyclic defect. Note that, in all of these cases, the algebras involvedare symmetric, which is why we get statements on tilting complexes.7. T
ILTING BIJECTIONS FOR ALGEBRAS OF DIHEDRAL , SEMI - DIHEDRAL AND QUATERNION TYPE
The aim of this section is to give a quick summary of what Theorem 6.6 implies for algebras of dihedral,semi-dihedral and quaternion type as classiï¬ed by Erdmann in [Erd90]. Not all of these algebras aregeneralised weighted surface algebras, but all of them are derived equivalent to one (bar one exceptionalfamily of -dimensional algebras), and all of them are symmetric. Apart from being a class of concreteexamples to apply Theorem 6.6 to, these algebras also contain all blocks of dihedral, semi-dihedral andquaternion defect, which are of interest in the modular representation theory of ï¬nite groups. The derivedequivalence classiï¬cation of the algebras classiï¬ed in [Erd90] was carried out in [Hol97] and [Hol99].We exclude all local algebras from our considerations, since their tilting complexes are just shifts ofprogenerators by [RZ03, Theorem 2.11]. Deï¬nition 7.1.
Deï¬ne quivers ð ðµ = â â ðŒ k k ðŒ + + ðœ ðœ g g ð ðŸ = â â â ðœ k k ðŒ + + ðŒ | | ðœ < < ðŒ U U ðœ (cid:21) (cid:21) ð ð = â â â ðŒ + + ðŒ | | ðŒ U U ðœ ðœ g g ðœ X X and corresponding permutations ð ðµ = ( ðŒ ðœ ðŒ ) , ð ðŸ = ( ðŒ ðŒ ðŒ )( ðœ ðœ ðœ ) , ð ð = ( ðŒ ðœ ðŒ ðœ ðŒ ðœ ) ,ð ðµ = ( ðœ ðŒ ðŒ ) , ð ðŸ = ( ðŒ ðœ )( ðŒ ðœ )( ðŒ ðœ ) , ð ð = ( ðŒ ðŒ ðŒ ) . Theorem 7.2 (see [Hol99]) . Let ð be an algebraically closed ï¬eld. If ðŽ is a non-local ð -algebra ofdihedral, semi-dihedral or quaternion type in the sense of Erdmann [Erd90] , then ðŽ is derived equivalentto one of the following algebras: IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 23 (1) î° (2 ðµ ) ð ,ð ( ð£ ) = Î( ð ðµ , ð ðµ , ð â , ð â , ð¡ â , â ) , with ð ðœ ð = ð ð for all ð , ð â = 1 , ð¡ ðœ = ð£ðœ and ð¡ ðœ = 0 ,where ð â©Ÿ ð â©Ÿ and ð£ â {0 , .(2) î° (3 ðŸ ) ð ,ð ,ð = Î( ð ðŸ , ð ðŸ , ð â , ð â , ð¡ â , â ) with ð ðŒ ð = ð ð for all ð , ð â = 1 and ð¡ â = 0 , where ð â©Ÿ ð â©Ÿ ð â©Ÿ .(3) î° (3 ð ) ð ,ð ,ð ,ð = Î( ð ð , ð ð , ð â , ð â , ð¡ â , â ) with ð ðŒ = ð and ð ðœ ð = ð ð for all ð , ð â = 1 and ð¡ â = 0 ,where ð â©Ÿ ð â©Ÿ ð â©Ÿ ð â©Ÿ and ð â©Ÿ .(4) î¿î° (2 ðµ ) ð ,ð ( ð£ ) = Î( ð ðµ , ð ðµ , ð â , ð â , ð¡ â , â ) , with ð ðœ ð = ð ð for all ð , ð â = 1 , ð¡ ðœ = 1 + ð£ðœ and ð¡ ðœ = 0 , where ð , ð â©Ÿ , ( ð , ð ) â (1 , and ð£ â {0 , .(5) î¿î° (2 ðµ ) ð ,ð ( ð£ ) = Î( ð ðµ , ð ðµ , ð â , ð â , ð¡ â , { ðŒ ðœ , ðœ ðŒ }) , with ð ðœ ð = ð ð for all ð , ð â = 1 , ð¡ ðœ = ð£ðœ and ð¡ ðœ = 1 , where ð â©Ÿ , ð â©Ÿ , ð + ð â©Ÿ and ð£ â {0 , .(6) î¿î° (3 ðŸ ) ð ,ð ,ð = Î( ð ðŸ , ð ðŸ , ð â , ð â , ð¡ â , â ) with ð ðŒ ð = ð ð for all ð , ð â = 1 , ð¡ ðŒ = 1 and ð¡ ðœ = 0 ,where ð â©Ÿ ð â©Ÿ ð â©Ÿ and ð â©Ÿ .(7) îœ (2 ðµ ) ð ,ð ( ð¢, ð£ ) = Î( ð ðµ , ð ðµ , ð â , ð â , ð¡ â , { ðœ ðŒ , ðŒ ðœ }) with ð ðœ ð = ð ð for all ð , ð ðœ = ð¢ , ð ðœ = ð¢ ð and ð¡ ðœ = 1 + ð£ðœ , ð¡ ðœ = 1 , where ð â©Ÿ , ð â©Ÿ , ð¢ â ð à and ð£ â ð .(8) îœ (3 ðŸ ) ð ,ð ,ð = Î( ð ðŸ , ð ðŸ , ð â , ð â , ð¡ â , { ðœ ðŒ ðŒ , ðŒ ðœ ðœ , ðŒ ðœ ðœ }) with ð ðŒ ð = ð ð for all ð , ð â = 1 and ð¡ â = 1 , where ð â©Ÿ ð â©Ÿ ð â©Ÿ , ð â©Ÿ and ( ð , ð , ð ) â (2 , , .(9) îœ (3 ðŽ ) , ( ð ) , where ð â ð ⧵ {0 , . These are algebras of dimension which are excludedin [Hol99] .In the names of these algebras, the letters â î° â, â î¿î° â and â îœ â indicate the type (dihedral, semi-dihedralor quaternion), and this type is preserved under derived equivalences. (cid:3) We should mention that in the case of îœ (2 ðµ ) ð ,ð ( ð¢, ð£ ) the presentation given in [Hol99] is diï¬erentfrom ours, and our parameters ð¢ and ð£ are not the same as Holmâs ð and ð . To recover Holmâs presentationone needs to replace the generator ðœ by ðœ â²2 = ð¢ â1 ðœ , and then the parameters correspond as ð = ð¢ and ð = ð¢ð£ . We also corrected the range of allowed parameters for î¿î° (2 ðµ ) ð ,ð ( ð£ ) . Corollary 7.3.
Let ð be an algebraically closed ï¬eld. Consider the following families of algebras: { î° (2 ðµ ) ð ,ð ( ð£ ) | ð , ð â©Ÿ for ð£ â {0 , , { î° (3 ðŸ ) ð ,ð ,ð | ð , ð , ð â©Ÿ , { ð· (3 ð ) ð ,ð ,ð ,ð | ð , ð , ð , ð â©Ÿ , { î¿î° (2 ðµ ) ð ,ð ( ð£ ) | ð â©Ÿ , ð â©Ÿ , { î¿î° (2 ðµ ) ð ,ð ( ð£ ) | ð â©Ÿ , ð â©Ÿ for ð£ â {0 , , { î¿î° (3 ðŸ ) ð ,ð ,ð | ð , ð , ð â©Ÿ , { îœ (2 ðµ ) ð ,ð ( ð¢, ð£ ) | ð â©Ÿ , ð â©Ÿ for ð¢ â ð à and ð£ â ð, { îœ (3 ðŸ ) ð ,ð ,ð | ð , ð , ð â©Ÿ . If char( ð ) = 2 then also consider the families { î° (2 ðµ ) ð ,ð (0) | ð , ð â©Ÿ , { î° (3 ðŸ ) ð ,ð ,ð | ð , ð , ð â©Ÿ , { îœ (3 ðŸ ) ð ,ð ,ð | ð , ð , ð â©Ÿ . If î² is any of the families listed above, and ðŽ, ðµ â î² are two algebras in this family, then the pre-tilting complexes over ðŽ and ðµ are in bijection, and the bijection induces an isomorphism of partiallyordered sets tilt - ðŽ ⌠ⷠtilt - ðµ. Proof.
For the families in arbitrary characteristic this is a straightforward application of Theorem 6.6. Forthe family { îœ (3 ðŸ ) ð ,ð ,ð | ð , ð , ð â©Ÿ in characteristic two this is an application of Proposition 6.7.For the two families of algebras of dihedral type in characteristic two it follows from Proposition 6.8. (cid:3) It was shown in [Hol97] that a block having quaternion defect group ð ð (for ð â©Ÿ ) with three simplemodules is derived equivalent to îœ (3 ðŸ ) , , ð â2 , and that a non-local block with dihedral defect group ð· ð is derived equivalent to either î° (3 ðŸ ) , , ð â2 or î° (2 ðµ ) , ð â2 ( ð£ ) with ð£ â {0 , . It was shown by theauthor in [Eis12] that no algebra of the form î° (2 ðµ ) , ð â2 (1) is derived equivalent to a block. This provesthe part of Theorem B concerned with blocks of dihedral or quaternion defect. We should note that the families deï¬ned above do not contain any blocks of semi-dihedral defect, sincethe multiplicities in such blocks are too small for Theorem 6.6. This cannot easily be ï¬xed, and it wasshown in [EJR18] that the poset of basic two-term tilting complexes over î¿î° (3 ðŸ ) ð ,ð ,ð is not the samefor all parameter choices (note that we allow ð ð = 1 , and the algebras so-obtained technically have adiï¬erent name in Erdmannâs classiï¬cation).In the case of blocks of quaternion defect with two simple modules there are two problems. The ï¬rstone is that we would need to allow â ð â©Ÿ â in the family { îœ (2 ðµ ) ð ,ð ( ð¢, ð£ ) | ð â©Ÿ , ð â©Ÿ to coverblocks with defect groups of order < (such blocks do not exist for the defect group ð , but they do for ð ). This is only a minor issue and one should be able to extend this family in characteristic two similarto Proposition 6.7. The second problem are the parameters ð¢ and ð£ , which are not fully determined forblocks of quaternion defect. This is a long-standing problem, and it means that Donovanâs conjecture isnot fully settled for these blocks. As long as this is not resolved we need to exclude these blocks fromTheorem B. 8. M ULTIPLICITY - INDEPENDENCE OF DERIVED P ICARD GROUPS
We have seen that if a ring Î carries two diï¬erent ð [[ ð ]] -algebra structures, both turning it into a ð [[ ð ]] -order, then it has two diï¬erent reductions modulo ð , say Ì Î and Ì Î . Isomorphism classes ofsilting complexes over Ì Î and Ì Î are in bijection. The derived Picard groups of Ì Î and Ì Î both act on therespective posets of basic silting complexes, and one might be tempted to ask what their relationship is. Inthe classes of algebras we study we will identify large common subgroups of ðð«ðð¢ð ð ( Ì Î ) and ðð«ðð¢ð ð ( Ì Î ) and show that ðð¢ððð§ð ( Ì Î ) â ðð¢ððð§ð ( Ì Î ) . This was motivated by the results on self-injective Nakayamaalgebras in [VZ17]. The results of this section, Theorems 8.7 and 8.9, imply Theorems C and D.Let us now quickly sketch the method used in this section. Denote the two copies of ð [[ ð ]] in ð (Î) by ð and ð (although later we will go with ð and ð ). The group ðð«ðð¢ð ð ( Ì Î ð ) (where ð â {1 , )acts on tilt - Ì Î ð , and ðð¢ð ð ( Ì Î ð ) is the stabiliser of the trivial basic tilting complex (the stalk complex of aprogenerator) under this action. In principle, each one-sided tilting complex Ìð â over Ì Î ð can be lifted to atilting complex ð â over Î . However, there is no guarantee that tilting complexes with endomorphism ring Ì Î ð lift to tilting complexes with endomorphism ring Î . This would only follow if Î was the unique ð ð -order reducing to Ì Î ð . We will generalise the ideas of [Eis16] to show that the algebras we are interested indo lift uniquely up to technicalities. This is proved in Lemma 8.1, which is bespoke for the algebras we areinterested in (but since it is separate from the rest of the proof one could easily replace it if one can provethe same statement for a diï¬erent class of algebras). Assuming one has such a unique lifting property, onecan always ï¬nd an element ð â â ðð«ðð¢ð ð ð (Î) which restricts to the one-sided complex ð â . This showsthat ðð«ðð¢ð ð ð (Î) and ðð¢ð ð ( Ì Î ð ) taken together generate ðð«ðð¢ð ð ( Ì Î ð ) . However, the group ðð«ðð¢ð ð ð (Î) stilldepends on the ð ð -algebra structure of Î . We will formulate conditions on the Grothendieck groups andthe centre of Î which ensure that we can even pick a lift in ð â â ðð«ðð¢ð ð (Î) â© ðð«ðð¢ð ð (Î) . The group ðð«ðð¢ð ð (Î) â© ðð«ðð¢ð ð (Î) is the same for Ì Î and Ì Î , so it independent of ð ð in the sense in which weneed it to be, but of course some further work is required to see how it interacts with ðð¢ð ð ( Ì Î ð ) . Lemma 8.1 (Unique lifting) . Let ð be a complete discrete valuation ring with algebraically closedresidue ï¬eld ð = ð â ðð and ï¬eld of fractions ðŸ . Moreover, let Î be an ð -order in a semisimple ðŸ -algebra ðŽ and set Ì Î = Îâ ð Î . Denote by ð , ⊠, ð ð ( ð â â ) a full set of orthogonal primitive idempotentsin Î . Now suppose that all of the following hold: ( ð² As a ð ( Ì Î) -module, ð ð Ì Î ð ð is generated by a single element, for all ð, ð â {1 , ⊠, ð } . ( ð² Either ð (Î) ⶠð ( Ì Î) is surjective or ð (Î) ð ð â ð ( ðŽ ) ð ð is not properly contained in any local ð -order with the same ðŸ -span for all ð â {1 , ⊠, ð } . ( ð² For all ð, ð, ð â {1 , ⊠, ð } with ð â ð and ð â ð dim ðŸ ( ð ð ðŽð ð ðŽð ð ) = max{dim ð ( ð ð Ì Î ð ð Ì Î ð ð ) , dim ð ( ð ð Ì Î ð ð Ì Î ð ð )} . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 25 ( ð² To each pair ð â ð â {1 , ⊠, ð } we can assign a simple ð ( ðŽ ) -module ð ð,ð in such a way that ð ð,ð = ð ð,ð and for all pair-wise distinct ð, ð, ð â {1 , ⊠, ð } the ð ( ðŽ ) -module ð ð ðŽð ð ðŽð ð is eitherzero or ð ð ðŽð ð ðŽð ð â ð ð,ð = ð ð,ð = ð ð,ð . ( ð² For every ð â ð ð such that dim ðŸ ( ð ð ( ð ) ðŽð ð ( ð ) ) = dim ðŸ ( ð ð ðŽð ð ) for all ð, ð â {1 , ⊠, ð } and dim ð ( ð ð Ì Î ð ð Ì Î â¯ Ì Î ð ð ð ) â â¹ dim ð ( ð ð ( ð ) ðŽð ð ( ð ) ðŽ ⯠ðŽð ð ( ð ð ) ) â for all ð â©Ÿ and ð , ⊠, ð ð â {1 , ⊠, ð } , there is a ðŸ â ðð®ð ðŸ ( ðŽ ) such that ðŸ ( ð ð ) = ð ð ( ð ) and ðŸ ( ð (Î)) = ð (Î) .If Î is an ð -order in ðŽ such that ( ðª there is a full set of orthogonal primitive idempotents ð , ⊠, ð ð â Î such that ð ð ðŽ â ð ð ðŽ as ðŽ -modules for all ð â {1 , ⊠, ð } , and ( ðª Ì Î = Îâ ð Î â Ì Î , and ( ðª ð (Î) = ð (Î) ,then Î â Î . If there is an isomorphism between Î and Î sending ð ð to ð ð for all ð , then Î is even conjugateto Î within the units of ðŽ .Proof. First of all, the idempotents ð , ⊠, ð ð are conjugate to ð , ⊠, ð ð (in that order) within the units of ðŽ , so we can replace Î by a conjugate and assume ð ð = ð ð for all ð . If we ï¬x an isomorphism between Ì Î and Ì Î sending ð ð + ð Î to ð ð ( ð ) + ð Î for some ð â ð ð then dim ðŸ ð ð ðŽð ð = dim ð ð ð Ì Î ð ð = dim ð ð ð ( ð ) Ì Î ð ð ( ð ) =dim ðŸ ð ð ( ð ) ðŽð ð ( ð ) , and given ð , ⊠, ð ð such that ð ð Ì Î ð ð Ì Î â¯ Ì Î ð ð ð â we have ð ð ( ð ) Ì Î ð ð ( ð ) Ì Î â¯ Ì Î ð ð ( ð ð ) â ,which implies ð ð ( ð ) ðŽð ð ( ð ) ðŽ ⯠ðŽð ð ( ð ð ) â . Hence there is a ðŸ â ðð®ð ðŸ ( ðŽ ) as in assumption ( ð² . If wereplace Î by ðŸ â1 (Î) then we can assume without loss of generality that there is an isomorphism between Ì Î and Ì Î sending ð ð + ð Î to ð ð + ð Î for all ð â {1 , ⊠, ð } . If, as in the very last sentence of the assertion,there is an isomorphism between Î and Î sending ð ð to ð ð for all ð then we can assume this withoutreplacing Î by ðŸ â1 (Î) . We will now show that Î is conjugate to Î , which will imply both parts of theclaim.Since ð (Î) = ð (Î) we have ð (Î) ð ð = ð (Î) ð ð for all ð . If some ð (Î) ð ð is not contained in any largerlocal order in ð ( ðŽ ) ð ð , then we must have ð (Î) ð ð = ð ð Î ð ð = ð ð Î ð ð . If ð (Î) ⶠð ( Ì Î) is surjective, then ð (Î) ⶠð ( Ì Î) must be surjective as well by virtue of dimensions. To see this note that ð (Î) is a puresublattice of Î (this holds for any ð -order), which implies that the rank of ð (Î) equals the dimension ofthe image of ð (Î) in Ì Î . Now it follows that each ð ð ð (Î) maps surjectively onto ð ð ð ( Ì Î) = ð ð Ì Î ð ð . Anyproper sublattice of ð ð Î ð ð maps to a proper subspace of ð ð Ì Î ð ð (again, by purity), whence ð ð Î ð ð = ð ð ð (Î) .Regardless of which of the two options given in ( ð² holds, we now know that ð ð Î ð ð = ð ð Î ð ð for all ð â {1 , ⊠, ð } , and all of these rings are commutative. Since the ð ð Ì Î ð ð are generated by a single elementas an ð ð Ì Î ð ð -module, the ð ð Î ð ð -lattice ð ð Î ð ð must also be generated by a single element. Because theanalogous statement is true for ð ð Î ð ð , and we have ð ð Î ð ð = ð ð Î ð ð , we must have ð ð Î ð ð = ð ð,ð â ð ð Î ð ð forcertain ð ð,ð in ð ( ðŽ ) . Of course only the image of ð ð,ð in ð ð ð ( ðŽ ) = ð ð ðŽð ð matters, and we can ask withoutloss of generality that the ð ð,ð should lie in ð ( ðŽ ) à . Also, set ð ð,ð = 1 for all ð â {1 , ⊠, ð } . We are freeto modify the ð ð,ð by units of ð . Now consider the simple ð ( ðŽ ) -modules ð ð,ð provided by ( ð² . Theendomorphism ring of ð ð,ð is a ï¬nite extension ðŸ ð,ð of ðŸ , in which the integral closure of ð is a totallyramiï¬ed (since ð = Ìð ) extension ð ð,ð of ð with uniformiser ð ð,ð . Make a choice such that ð ð,ð = ð ð â² ,ð â² whenever ð ð,ð = ð ð â² ,ð â² . Now we can write any element of ðŸ ð,ð as ð ð§ð,ð â ð for some ð§ â †and ð â ð .Hence we can stipulate that the ð ð,ð should act on ð ð,ð by multiplication by a power of ð ð,ð for all ð â ð .Our assumption ( ð² implies that for any ð, ð, ð â {1 , ⊠, ð } with ð â ð and ð â ð either ð ð Î ð ð Î ð ð isa pure sublattice of ð ð Î ð ð or ð ð Î ð ð Î ð ð is a pure sublattice of ð ð Î ð ð (or both; note that dim ðŸ ð ð ðŽð ð ðŽð ð =dim ðŸ ð ð ðŽð ð ðŽð ð follows from ( ð² ). Until further notice let us only consider triples ð, ð, ð such that ð ð Î ð ð Î ð ð is a pure sublattice of ð ð Î ð ð . Since we have seen that there is an isomorphism between Ì Î and Ì Î sending ð ð + ð Î to ð ð + ð Î for all ð , it follows that Î also satisï¬es (the analogue of) assumption ( ð² , whichmeans that ð ð Î ð ð Î ð ð is also a pure sublattice of ð ð Î ð ð . This is the same as saying that ð ð,ð ð ð,ð ð â1 ð,ð â ð ð Î ð ð Î ð ð is a pure sublattice of ð ð Î ð ð , and therefore ð ð,ð ð ð,ð ð â1 ð,ð â ð ð Î ð ð Î ð ð = ð ð Î ð ð Î ð ð (14)since pure sublattices are determined by their ðŸ -span.For ð = ð the ð (Î) -modules ð ð Î ð ð Î ð ð , ð ð Î ð ð and ð ð Î ð ð are isomorphic (they are generated by a singleelement, which means they are determined by their annihilator, which is the simultaneous annihilator of ð ð and ð ð in all cases), and therefore ð ð,ð â ð ð,ð = ð§ for some element ð§ â ð (Î) which acts invertibly on ð ð Î ð ð Î ð ð (w.l.o.g. ð§ â ð (Î) à ). We can replace ð ð,ð by ð§ â1 â ð ð,ð whenever ð > ð , thus making ð ð,ð â ð ð,ð act as the identity. This is compatible with the action on ð ð,ð that was ï¬xed earlier.Now assume that ð, ð and ð are pair-wise distinct. By our assumption, whenever ð ð Î ð ð Î ð ð is non-zero,the element ð ð,ð ð ð,ð ð â1 ð,ð acts by multiplication by a power of ð ð,ð on it, which in light of equality (14) aboveis only possible if ð ð,ð ð ð,ð ð â1 ð,ð acts as the identity. The element ð ð,ð ð ð,ð ð â1 ð,ð acts like the inverse of ð ð,ð ð ð,ð ð â1 ð,ð ,and therefore also acts as the identity.The upshot of the above is that ð ð Î ð ð = ð ð,ð â ð ð Î ð ð and ð ð,ð ð ð,ð ð â1 ð,ð acts as the identity on ð ð Î ð ð Î ð ð for all ð, ð, ð â {1 , ⊠, ð } (not subject to any restrictions), and therefore the map ΠⶠΠⶠð ð ð¥ð ð ⊠ð ð,ð â ð ð ð¥ð ð for all ð, ð â {1 , ⊠, ð } deï¬nes an isomorphism which restricts to the identity on ð (Î) = ð (Î) , thus showing that Î and Î areconjugate. (cid:3) Lemma 8.2.
Let ð be a complete discrete valuation ring with algebraically closed residue ï¬eld ð = ð â ðð and ï¬eld of fractions ðŸ . Let Î be an ð -order in a semisimple ðŸ -algebra ðŽ , set Ì Î = Îâ ð Î andassume that Ì Î is basic. Suppose furthermore that ð â ð (Î) is a complete discrete valuation ring withï¬eld of fractions
ð¿ â ð ( ðŽ ) such that Î is also an ð -order. Let ð , ⊠, ð ð ( ð â â ) denote representativesfor the simple ðŽ -modules and let ð , ⊠, ð ð â ð ( ðŽ ) be the corresponding central primitive idempotents.Assume that all of the following hold: (A1) Î fulï¬ls the assumptions ( ð² â ( ð² of Lemma 8.1. (A2) Every ðŸ -algebra automorphism of ð ( ðŽ ) lifts to a Morita auto-equivalence of ðŽ . (A3) If Î is an ð -order in a ðŸ -algebra ðµ such that Î is derived equivalent to Î , Ì Î = Îâ ð Î â Ì Î ,and there is an isometry between ð ( ðµ ) and ð ( ðŽ ) sending Im ð· Î to Im ð· Î , then there existisometries ð ⶠð ( Ì Î) ⌠ââââââââ ð ( Ì Î) and ð ⶠð ( ðµ ) ⌠ââââââââ ð ( ðŽ ) sending distinguished bases todistinguished bases such that the following diagram commutes ð ( Ì Î) ð / / ð· Î (cid:15) (cid:15) ð ( Ì Î) ð· Î (cid:15) (cid:15) ð ( ðµ ) ð / / ð ( ðŽ ) , where ð· Î and ð· Î are as in Deï¬nition 2.6. (A4) If ð ðŽ ⶠð ( ðŽ ) ⶠð ( ðŽ ) ⶠ[ ð ð ] ⊠(â1) ð ( ð ) â [ ð ð ( ð ) ] is a self-isometry, where ð ⶠ{1 , ⊠, ð } ⶠ{±1} and ð â ð ð , such that ð ðŽ (Im ð· Î ) â Im ð· Î then there is a ðŸ â ðð®ð ðŸ ( ð ( ðŽ )) such that ðŸ ( ð ð ) = ð ð ( ð ) for all ð â {1 , ⊠, ð } and ðŸ ( ð (Î)) = ð (Î) . IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 27 (A5)
If, for a ð ðŽ as in (A4) , there is a ðŸ â ðð®ð ð¿ ( ð ( ðŽ )) such that ðŸ ( ð ð ) = ð ð ( ð ) for all ð , then ðŸ ( ð (Î)) = ð (Î) . (A6) The images of
ðð®ð ðŸ ( ð ( ðŽ )) â© ðð®ð ð¿ ( ð ( ðŽ )) and ðð®ð ðŸ ( ð ( ðŽ )) in ðð®ð ( ð ( ðŽ )) are equal. (A7) The kernel of
ðð®ð ð¿ ( ð ( ðŽ )) ⶠðð®ð ( ð ( ðŽ )) is contained in ðð®ð ðŸ ( ð ( ðŽ )) .Deï¬ne îŽ as the full preimage of ðð¬ðšðŠ ( ð ( ðŽ )) (the group of self-isometries, which depends on ðŸ ) underthe group homomorphism ðð«ðð¢ð ð (Î) ⶠðð®ð ( ð ( ðŽ )) . Then îŽ â ðð«ðð¢ð ð (Î) , and if we let Ì îŽ denotethe image of îŽ in ðð«ðð¢ð ð ( Ì Î) , we have ðð«ðð¢ð ð ( Ì Î) =
ðð¢ð ð ( Ì Î) â Ì îŽ . (15) Proof.
Let Ìð â be an arbitrary element of ðð«ðð¢ð ð ( Ì Î) , and let Ìð â â î· ð ( Ì Î - proj ) be its restriction to theleft. By [Ric91b, Proposition 3.1] there is a tilting complex ð â â î· ð (Î - proj ) such that ð â ð ð â â Ìð â .By [Ric91b, Theorem 3.3] the endomorphism algebra Î =
ðð§ð î° ð (Î) ( ð â ) op is again an ð -order, and ð â ð Î â Ì Î . By [Ric91a, Proposition 3.1] there is a two-sided tilting complex ð â in î° ð (Î op â ð Î) whose restriction to the left is ð â . We get a commutative diagram ð ( Ì Î) ð Ìð / / ð· Î (cid:15) (cid:15) ð ( Ì Î) ð· Î (cid:15) (cid:15) ð / / ð ( Ì Î) ð· Î (cid:15) (cid:15) ð ( ðŽ ) ð ðŸð / / ð ( ðµ ) ð / / ð ( ðŽ ) , (16)where the left hand square comes from Proposition 2.7 and the right hand square comes fromassumption (A3) .Now assumption (A4) implies that there is an automorphism of ð ( ðŽ ) inducing the permutationon central primitive idempotents which corresponds to ð ⊠ð ðŸð , and mapping ð (Î) into itself. ByProposition 2.7 there exists an algebra homomorphism ðŸ ðŸð corresponding to ð ðŸð mapping ð (Î) into ð (Î) , which tells us that there is an algebra homomorphism ðŸ ⶠð ( ðµ ) ⶠð ( ðŽ ) which maps thecentral primitive idempotent corresponding to [ ð ] (for some simple ðµ -module ð ), to the central primitiveidempotent corresponding to ð ([ ð ]) , and ðŸ ( ð (Î)) = ð (Î) . By considering a Morita equivalencebetween ðŽ and ðµ , and applying assumption (A2) to the isomorphism of centres induced by the Moritaequivalence followed by ðŸ , we see that there is in fact a Morita equivalence between ðµ and ðŽ , given bya bimodule ð , such that [ ð â ðµ ð ] = ð ([ ð ]) for all ðµ -modules ð , and the induced isomorphism ofcentres ðŸ ðŸð is equal to ðŸ .Now note that ð ([ Ì Î]) = [ Ì Î] , since Ì Î is assumed to be basic which means that [ Ì Î] and [ Ì Î] are just thesums over the distinguished bases. Moreover, ð· Î ([ Ì Î]) = [ ðµ ] and ð· Î ([ Ì Î]) = [ ðŽ ] . Hence commutativityof the rightmost square in (16) implies that [ ðµ â ðµ ð ] = [ ðŽ ] , which shows that ð is induced by analgebra isomorphism ðŒ ⶠðµ ⶠðŽ . That is, ð â ðŒ ðŽ . Set Î â² = ðŒ (Î) â ðŽ . We have ðŒ | ð ( ðµ ) = ðŸ ðŸð = ðŸ ,which shows that ðŒ ( ð (Î)) = ð (Î) . It follows that ð (Î â² ) = ð (Î) . Applying Proposition 2.7 to thefunctor â â Î ðŒ Î â² (and taking into account that it sends simple modules to simple modules) gives usan isometry ð ⶠð ( Ì Î) ⶠð (Î â² â ð Î â² ) mapping the distinguished basis to the distinguished basissuch that ð ⊠ð· Î = ð· ΠⲠ⊠ð . Combining this with rightmost square in (16) gives ð· Î â² = ð· Π⊠ð ⊠ð â13 . Thismeans that the image of the distinguished basis of ð (Î â² â ð Î â² ) under ð· Î â² is the same (up to reordering)as the image of the distinguished basis of ð ( Ì Î) under ð· Î . It follows that the ðŸ -spans of the projectiveindecomposable Î â² -modules fulï¬l condition ( ðª in Lemma 8.1. Since we have checked ( ðª and ( ðª already, we can conclude by Lemma 8.1 that Î â² â Î . We can hence compose our original two-sidedtilting complex ð â with a bimodule corresponding to an isomorphism between Î and Î , to obtain anelement of ðð«ðð¢ð ð (Î) whose restriction to the left is ð â . It is well-known that any two two-sided tiltingcomplexes over Ì Î restricting to Ìð â diï¬er from one another only by an element of ðð¢ð ð ( Ì Î) (see [RZ03,Proposition 2.3]). We have thus shown that ðð«ðð¢ð ð ( Ì Î) =
ðð«ðð¢ð ð (Î) â ðð¢ð ð ( Ì Î) . Now suppose that ð â â îŽ â ðð«ðð¢ð ð (Î) . By Proposition 2.7 ð â induces maps ð ð¿ð ⶠð ( ðŽ ) ⶠð ( ðŽ ) and ðŸ ð¿ð â ðð®ð ð¿ ( ð ( ðŽ )) . By deï¬nition of îŽ the map ð ð¿ð is an isometryof ð ( ðŽ ) (equipped with the bilinear form coming from the ðŸ -algebra structure of ðŽ , rather thanthe ð¿ -algebra structure for which this would be trivial), and it maps Im ð· Î into itself. Therefore, byassumption (A4) , there is a map ðŸ â² â ðð®ð ðŸ ( ð ( ðŽ )) which induces the same permutation of the ð ð âs as ðŸ ð¿ð . By assumption (A6) there must be a ðŸ â²â² â ðð®ð ðŸ ( ð ( ðŽ )) â© ðð®ð ð¿ ( ð ( ðŽ )) which also induces thesame permutation, meaning that ðŸ â²â² ⊠ðŸ â1 ð¿ð is ð¿ -linear and acts trivially on ð ( ðŽ ) . By assumption (A7) itfollows that ðŸ â²â² ⊠ðŸ â1 ð¿ð is ðŸ -linear, but then ðŸ ð¿ð must be ðŸ -linear as well. By Remark 2.5 it follows that ð â lies in (or, is isomorphic to an element of) ðð«ðð¢ð ð (Î) .Now let us consider an arbitrary ð â â ðð«ðð¢ð ð (Î) . By assumption (A6) there must be a ðŸ - and ð¿ -linear automorphism ðŸ Ⲡⶠð ( ðŽ ) ⶠð ( ðŽ ) inducing the same permutation of idempotents as ðŸ ðŸð . Byassumption (A5) it follows that ðŸ â² ( ð (Î)) = ð (Î) . Set ðŸ â²â² = ðŸ â1 ðŸð ⊠ðŸ â² . Then ðŸ â²â² is ðŸ -linear, acts triviallyon ð ( ðŽ ) , and ðŸ â²â² ( ð (Î)) = ð (Î) . By assumption (A2) it follows that there is a Morita auto-equivalenceof ðŽ extending ðŸ â²â² , and since ðŸ â²â² acts trivially on ð ( ðŽ ) this must actually be induced by an ðŒ â ðð®ð ðŸ ( ðŽ ) .That is, ðŒ induces the identity on ð ( ðŽ ) and restricts to ðŸ â²â² . So, if we set Î = ðŒ (Î) , the assumptions ( ðª â ( ðª of Lemma 8.1 are satisï¬ed, as well as the one needed for conjugacy instead of isomorphism,and therefore there is a central automorphism ðœ of ðŽ such that ðœ ( ðŒ (Î)) = Î . Hence ð â = ð â â Î ðœ ⊠ðŒ Î induces the automorphism ðŸ ðŸð = ðŸ â² on ð ( ðŽ ) , which is ð¿ -linear. Hence ð â â ðð«ðð¢ð ð (Î) by Remark 2.5,and ð â clearly satisï¬es the condition deï¬ning îŽ . Therefore ð â â îŽ â ðð¢ð ð (Î) . We have now establishedthat ðð«ðð¢ð ð ( Ì Î) = Ì îŽ â ðð¢ð ð ( Ì Î) , and the equality (15) follows by inverting. (cid:3) Lemma 8.3 (Additional structure) . Assume that all assumptions of Lemma 8.2 hold.(1) If (B1) Ì Î is silting-connectedthen ðð¢ð ð ð ( Ì Î) is normal in ðð«ðð¢ð ð ( Ì Î) .(2) If (B1) holds and in addition (B2) the kernel of the action of ðð¢ð ð (Î) on ð (Î) â ð ( Ì Î) is trivial, and (B3) the image of ðð¢ð ð ( Ì Î) in ðð®ð ( ð ( Ì Î)) is contained in the image of
ðð¢ð ð (Î) â© ðð¢ð ð (Î) in ðð®ð ( ð ( Ì Î)) ,then Ì îŽ â îŽ and we can write ðð«ðð¢ð ð ( Ì Î) =
ðð¢ð ð ð ( Ì Î) â îŽ . Note that if ð ( Ì Î) = ð (Î)â ðð (Î) and ðð¢ððð§ð ( Ì Î) â©
ðð¢ð ð ð ( Ì Î) = 1 , then
ðð«ðð¢ððð§ð (Î) âŽ îŽ actstrivially on ðð¢ð ð ð ( Ì Î) .(3) If ðð«ðð¢ð ð ( Ì Î) =
ðð¢ð ð ð ( Ì Î) â Ì îŽ (e.g. due to (B1) â (B3) holding), and in addition (B4) ðð¢ð ð ð ( Ì Î) â©
ðð¢ððð§ð ( Ì Î) â Ì îŽ , and (B5) the images of Ì îŽ and ðð¢ð ð ð ( Ì Î) in ðð®ð ð ( ð ( Ì Î)) intersect trivially,then
ðð«ðð¢ððð§ð ( Ì Î) = Ker( Ì îŽ â¶ ðð®ð ð ( ð ( Ì Î))) . (17) Proof. If (B1) holds then any two silting complexes over Ì Î are linked by a ï¬nite sequence of irreduciblemutations. It is clear from the deï¬nition of irreducible silting mutations by minimal left or rightapproximations (see [Aih13, Deï¬nition-Theorem 2.3]) that if a Morita auto-equivalence ï¬xes theisomorphism classes of all indecomposable summands of a silting complex, then it also ï¬xes allindecomposable summands of an irreducible mutation of that complex. Hence, in the connected case,the group ðð¢ð ð ð ( Ì Î) ï¬xes all isomorphism classes of silting complexes, since it ï¬xes the projectiveindecomposable modules and every silting complex is connected via mutation to their direct sum. Inparticular, the restriction of ð â â Î ð to the right is isomorphic to the restriction of ð â to the right, forany ð â â ðð«ðð¢ð ð ( Ì Î) and ð â ðð¢ð ð ð ( Ì Î) . It follows that ð â â Î ð â Î ( ð â1 ) â lies in ðð¢ð ð ( Ì Î) . Since ð IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 29 acts trivially on ð ( Ì Î) its conjugate will also act trivially on ð ( Ì Î) , that is, ð â â Î ð â Î ( ð â1 ) â lies in ðð¢ð ð ð ( Ì Î) . This proves the ï¬rst part of the assertion.The kernel of the map ðð«ðð¢ð ð (Î) ⶠðð«ðð¢ð ð ( Ì Î) is contained in ðð¢ð ð (Î) (see [RZ03, Lemma 3.4]).Since, by the assumption (B2) , the group ðð¢ð ð (Î) acts faithfully on ð (Î) , and îŽ â© ðð¢ð ð (Î) â ðð¢ð ð (Î) ,it follows that îŽ embeds into ðð«ðð¢ð ð ( Ì Î) , that is, îŽ â Ì îŽ . By the assumption (B3) it follows that ðð«ðð¢ð ð ( Ì Î) =
ðð¢ð ð ð ( Ì Î) â îŽ , and by faithfulness of îŽ â© ðð¢ð ð (Î) on ð ( Ì Î) also ðð¢ð ð ð ( Ì Î) â© îŽ = {1} . Thisproves ðð«ðð¢ð ð ( Ì Î) =
ðð¢ð ð ð ( Ì Î) â îŽ . If ðð¢ððð§ð ( Ì Î) â©
ðð¢ð ð ð ( Ì Î) = 1 then an element of
ðð¢ð ð ð ( Ì Î) is determinedby the automorphism of ð ( Ì Î) it induces, which implies the last bit of the assertion of the second part.Let us now prove the third part. Assumption (B5) implies that ðð«ðð¢ððð§ð ( Ì Î) = (
ðð¢ð ð ð ( Ì Î) â©
ðð«ðð¢ððð§ð ( Ì Î)) â ( Ì îŽ â© ðð«ðð¢ððð§ð ( Ì Î)) , and assumption (B4) ensures that ðð«ðð¢ððð§ð ( Ì Î) â©
ðð¢ð ð ð ( Ì Î) â Ì îŽ . It follows that ðð«ðð¢ððð§ð ( Ì Î) â Ì îŽ , fromwhich one infers (17). (cid:3) Proposition 8.4.
Assume ð is an algebraically closed ï¬eld, and let Ì Î be a ð -algebra of dihedral, semi-dihedral or quaternion type in the sense of Erdmann [Erd90] . Then Ì Î is silting-discrete.Proof. The class of algebras of dihedral, semi-dihedral or quaternion type is, by deï¬nition, closed underderived (and even stable) equivalences. All of these algebras are symmetric as well, which implies thattilting and silting complexes coincide. By [AM17, Theorem 2.4] it suï¬ces to show that all algebras inthe derived equivalence class of Ì Î are -tilting ï¬nite, and this was veriï¬ed in [EJR18, Theorem 16]. (cid:3) We will now verify the assumptions of Lemmas 8.2 and 8.3 for the lifts of the algebras îœ (3 ðŸ ) ð ,ð ,ð in characteristic two constructed in Proposition 5.2, as well as the orders lifting twisted Brauer graphalgebras from Proposition 4.4 (subject to mild assumptions on the Brauer graph). This is straight-forward in principle, but requires us to describe these ð [[ ð ]] -orders more explicitly. A large chunk ofthe veriï¬cation for the lifts of îœ (3 ðŸ ) ð ,ð ,ð is already contained in Remark 8.5 below. This remarkalso highlights that these ð [[ ð ]] -orders are equicharacteristic analogues of the lifts over an extension î» of the -adic integers coming from blocks of quaternion defect over î» (e.g. they share the samedecomposition matrix). The same is true for Brauer tree algebras. However, for other Brauer graphalgebras like î° (3 ðŸ ) ð ,ð ,ð our lifts over ð [[ ð ]] do have a diï¬erent decomposition matrix and a centre ofa diï¬erent dimension than the lifts over î» coming from modular representation theory. Remark îœ (3 ðŸ ) ð ,ð ,ð ) . Assume ð is an algebraically closed ï¬eld ofcharacteristic two. Let ð , ð , ð â , ð¡ â be as in Proposition 3.5, set ð ðŒ = 1 for all ðŒ â ð , and let ð â²â ⶠð â âš ð ⩠ⶠ†> be arbitrary. Consider the ð [[ ð ]] -order Î = Î(
ð, ð , ð â , ð â , ð¡ â , â ; ð â²â ) constructedin Proposition 5.2. Let ðŽ denote the ð (( ð )) -span of Î . By deï¬nition, Î is a pullback Ìð [ ð ] ð ( ðŒð ( ðŒ )â ð ÌðŒ, ðŒð ( ðŒ )â ÌðŒð ( ÌðŒ ) | ðŒ â ð ) ð / / / / Ìð [ ð ] ð ( ðŒð ( ðŒ ) , ðŒð ( ðŒ )â ÌðŒð ( ÌðŒ ) , ð + ð§ | ðŒ â ð ) Î O O / / Ìð [ ð ] ð ( ðŒð ( ðŒ ) , ð + ð§ | ðŒ â ð ) ð O O O O (18)where ð§ = â ð =1 ( ðŒ ð ð ( ðŒ ð ) + ð ( ðŒ ð ) ðŒ ð ) ð â² ðŒð and ð, ð are the natural surjections. Let us denote the orders in thetop left and bottom right corners of this diagram by Î and Î , respectively, and the ð [[ ð ]] -algebra inthe top right corner by Ì Î . We will view Î as a subset of Î â Î . Note that Î and Î can be described pictorially as ð [[ ð ]] ð [[ ð ]] ð [[ ð ]] ð [[ ð ]] ( ð ) ( ð )( ð ) ð [[ ð ]] ( ð )( ð ) ( ð ) ð [[ ð ]] âââââââ âââââââ ð [[ ð ]]â ð ð [[ ð ]]â ð ð [[ ð ]]â ð and ð [[ ð ð¢ ]] ð [[ ð ð¢ ]]( ð ð¢ ) ð [[ ð ð¢ ]] ââââââ ââââââ ð [[ ð ð£ ]] ð [[ ð ð£ ]]( ð ð£ ) ð [[ ð ð£ ]] ââââââ ââââââ ð [[ ð ð€ ]] ð [[ ð ð€ ]]( ð ð€ ) ð [[ ð ð€ ]] ââââââ ââââââ ð ðð , where ð¢, ð£, ð€ are the multiplicities ð â² ðŒ , ð â² ðŒ and ð â² ðŒ . The arcs indicate that the entries linked by themmust have the same image in the ring labelling the arc (either ð [[ ð ]]â( ð ) or ð ). This information canbe extracted from Proposition 3.5 and Proposition 4.4, respectively. Of course, the description abovemainly serves to illustrate, the facts we are actually going to use are the following (easily obtained fromPropositions 3.5 and 4.4 and the fact that Ì Î = Îâ ð Î is isomorphic to Î( ð, ð , ð â + ð â²â , ð â , ð¡ â , â ) ):(1) ( ð , ð ) , ( ð , ð ) and ( ð , ð ) form a full set of orthogonal primitive idempotents in Î â©œ Î â Î .When it is unambiguous we will write â ð ð â instead of ( ð ð , ð ð ) .(2) Î and Î are both orders in semisimple ð (( ð )) -algebras which are Morita equivalent to theircentres (i.e. no division algebras occur), and therefore so is Î . It follows that Î satisï¬esassumption (A2) of Lemma 8.2 with ð = ð [[ ð ]] .(3) The central primitive idempotents in the ð (( ð )) -span of Î are ð ð = ð ð â ð â2 ðŒ ð ðœ ð +1 for ð â {1 , , (set ðœ = ðœ ) and ð = 1 â ð â ð â ð . Those in the ð (( ð )) -span of Î are ð ð = ð â1 ( ðŒ ð ðœ ð +1 + ðœ ð +1 ðŒ ð ) ð â² ðŒð for ð â {1 , , .(4) The elements ð ð = ( ð ð , for â©œ ð â©œ and ð ð = (0 , ð ð ) for â©œ ð â©œ form a full set of primitiveidempotents in ðŽ . We have ð ð ð ( ðŽ ) â ð (( ð )) for â©œ ð â©œ , ð ð ð ( ðŽ ) â ð (( ð ð â² ðŒð )) for â©œ ð â©œ as ð (( ð )) -algebras, where ð ð ð ð ðŒð has preimage ðŒ ð ðœ ð +1 + ðœ ð +1 ðŒ ð in ð (Î ) . In particular, dim ð (( ð )) ( ð ( ðŽ )) = 4 + ð â² ðŒ + ð â² ðŒ + ð â² ðŒ = dim ð ( ð ( Ì Î)) , where the second equality is obtained by counting a basis of ð ( Ì Î) (alternatively see the appendixof [Erd90]). It follows that ð (Î) surjects onto ð ( Ì Î) , proving assumption ( ð² of Lemma 8.1.(5) The bilinear form on ð ( ðŽ ) has Gram-matrix diag(1 , , , , ð â² ðŒ , ð â² ðŒ , ð â² ðŒ ) and the map ð· Πⶠð ( Ì Î) ⶠð ( ðŽ ) has matrix ð· = ââââââââââ ââââââââââ †(acting on row vectors) (19)with respect to the distinguished bases (note that the ordering of the idempotents ð , ⊠, ð induces an order on the basis of ð ( ðŽ ) as well). In fact, ð ð ðŽ is the direct sum of ð (( ð )) â ð [[ ð ]] ð ð Î and ð (( ð )) â ð [[ ð ]] ð ð Î , so this follows from the description of these two algebras.(6) Each ð ð Ì Î ð ð for ð â ð is generated by the unique arrow ðŒ â ð pointing from ð ð to ð ð . In fact, allother non-zero paths in Ì Î with the same source and target are ( ðŒð ( ðŒ )) ð ðŒ = ( ðŒð ( ðŒ ) + ÌðŒð ( ÌðŒ )) ð ðŒ forsome ð â â , and the element ðŒð ( ðŒ ) + ÌðŒð ( ÌðŒ ) is central. That is, Ì Î satisï¬es assumption ( ð² ofLemma 8.1. IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 31 (7) If ð, ð, ð â {1 , , are pair-wise distinct, the dim ð ð ð Ì Î ð ð Ì Î ð ð = 1 (by considering a basis), whichis equal to dim ð (( ð )) ( ð ð ðŽð ð ðŽð ð ) by (19). It also follows that Î satisï¬es assumption ( ð² if we setall ð ð,ð equal to the simple ðŽ -module corresponding to ð .Moreover, if ð â ð â {1 , , , then ð ð Ì Î ð ð Ì Î ð ð has the same dimension as ð ð Ì Î ð ð (just bychecking that multiplying this by the unique arrow from ð ð to ð ð does not annihilate any elements),which shows that dim ð ð ð Ì Î ð ð Ì Î ð ð = dim ð (( ð )) ð ð ðŽð ð ðŽð ð . In particular, Î satisï¬es assumption ( ð² of Lemma 8.1.(8) Assumption ( ð² of Lemma 8.1 is trivially satisï¬ed for Î since there is even an automorphismof Î (instead of ðŽ ) inducing the desired permutation of idempotents.(9) The diagram (18) restricts to a pullback diagram on the centres, from which one sees that ð (Î) âð ( ðŽ ) is spanned as a ð [[ ð ]] -lattice by (1 , , ( ð ( ð + ð ) , ð ð ð â² ðŒ ) , ( ð ( ð + ð ) , ð ð ð â² ðŒ ) , ( ð ( ð + ð ) , ð ð ð â² ðŒ ) , ( ð ð , ) , ( ð ð , ) , ( ð ð , ) , ( , ð ð ð â ð â² ðŒ + ð ð ð â ð â² ðŒ + ð ð ð â ð â² ðŒ ) , ( , ð ð ( ð +1)â ð â² ðŒ ) , ( , ð ð ( ð +1)â ð â² ðŒ ) for â©œ ð ð â©œ ð â² ðŒ ð . One then veriï¬es that the ð (( ð )) -algebra automorphism ðŒ ð of âš ð =1 ð (( ð )) ð ð sending ð ð to ð ð ( ð ) restricts to ð (Î) for all ð â âš (1 , , , , , (1 , , , (1 , , â© (1 , , , ,ð â² ðŒ ,ð â² ðŒ ,ð â² ðŒ ) , (20)where the subscript indicates a stabiliser ( ð acting on †simply by permutation).The only part of this that is not immediate from the deï¬nition is that ðŒ (1 , , maps ð (Î) into itself, speciï¬cally the third and fourth given generators (the others just get permuted). Toverify this one just needs the fact that ð ( ð + ð ) = ð ( ð + ð ) + ð , and and ð ( ð + ð ) = ð ( ð + ð ) + ð .(10) In the case where ð â² ðŒ ð = 1 for all ð â {1 , , the group of self-isometries of ð ( ðŽ ) which map Im ð· Î into itself was determined in [HKL07, Proposition 1.1] to be âš (1 , , , , , (1 , , , (3 , â4)(6 , â7) , â id â© (this is adapted to our labelling of simple ðŽ -modules) . The corresponding group of self-isometries of ð ( ðŽ ) for an arbitrary choice of ð â² ðŒ ð âsmust be contained in this group, and it is characterised by the fact that its image in ð stabilises (1 , , , , ð â² ðŒ , ð â² ðŒ , ð â² ðŒ ) â †. Therefore the group of self-isometries of ð ( ðŽ ) which preserve Im ð· Î maps onto the group of which ð is an element in equation (20). It follows that (A4) ofLemma 8.2 is satisï¬ed for this choice of Î and ð = ð [[ ð ]] .(11) Assume Î is a ð [[ ð ]] -order in a ð (( ð )) -algebra ðµ Morita-equivalent to ðŽ such that Ì Î = Îâ ð Î â Ì Î .Since dim ð (( ð )) ð ( ðµ ) = dim ð (( ð )) ð ( ðŽ ) = dim ð ð ( Ì Î) it follows that ð (Î) surjects onto ð ( Ì Î) ,and since ð Ì Î ð = ðð ( Ì Î) for any primitive idempotent ð â Ì Î it follows that ð Î ð = ð (Î) ð and ð ðµð = ð ð ( ðµ ) for any primitive idempotent ð â Î . What this means is that the correspondingprojective indecomposable module ð Î spans a multiplicity-free ðµ -module. That is, the entriesof the matrix ð· â ð ( †) representing ð· Î (with respect to the distinguished bases) has entriesbounded by one. Now consider the equation ð· â diag(1 , , , , ð â² ðŒ , ð â² ðŒ , ð â² ðŒ ) â ð· †= ð¶ Ì Î (21)where ð¶ Ì Î denotes the Cartan matrix of Ì Î (aï¬ording the bilinear form on ð ( Ì Î) ). If we deï¬ne a set ðŒ = ðŒ â ⊠âðŒ , where | ðŒ | = ⊠= | ðŒ | = 1 and | ðŒ ð | = ð â² ðŒ ð for ð â {1 , , , then we can deï¬nesets ð ð = â { ðŒ ð | â©œ ð â©œ and ð· ð,ð = 1} for ð â {1 , , . Clearly these sets determine ð· , and we can interpret the entries on the left hand side of equation (21) as the cardinalities | ð ð â© ð ð | for ð, ð â {1 , , . Note that ð· has no zero-columns, which implies that | ð ⪠ð ⪠ð | = | ðŒ | .By the inclusion-exclusion principle the cardinalities of all possible intersections of ð ð âs andcomplements of ð ð âs are now determined. In particular, by comparing to the matrix in (19)(which also satisï¬es (21)) we easily see that | ðŒ â© ðŒ â© ðŒ | = 1 and | ðŒ ð â© ( ðŒ ⧵ â ð â ð ðŒ ð ) | = 1 , whichshows that, up to a permutation ð â ð such that | ðŒ ð | = | ðŒ ð ( ð ) | applied to the columns of ð· (which corresponds to composing with an isometry of ð ( ðµ ) stabilising the distinguished basis),the ï¬rst four columns of ð· are as in the matrix given in (19), that is, ð ð â ðŒ ð ⪠ðŒ ⪠ðŒ ⪠ðŒ ⪠ðŒ for all ð .Now, for ( ð, ð ) â {(1 , , (2 , , (3 , we have | ð ð â© ð ð | = 1 + ð â² ðŒ ð . If we choose ð such that ð â² ðŒ ð is minimal amongst the values of ð â²â , then, after potentially applying a permutation whichï¬xes the cardinalities of the ðŒ ð âs, ð ð â© ð ð must be equal to ðŒ ⪠ðŒ ð . But then, again up to anadmissible permutation, ð ð = ðŒ ⪠ðŒ ⪠ðŒ ð ⪠ðŒ ð ( ð ) and ð ð = ðŒ ⪠ðŒ ⪠ðŒ ð ⪠ðŒ ð ( ð ) (where ð = (3 , , ), since that is again the only union of ðŒ ð âs with the right cardinality. If ð â² is theunique element of {1 , , ⧵ { ð, ð } , then we just have to consider the intersections ð ð â© ð ð â² and ð ð â© ð ð â² (whose cardinalities we know) to infer that ð ð â² = ðŒ ⪠ðŒ ⪠ðŒ ð Ⲡ⪠ðŒ ð ( ð â² ) , showingthat ð· is the same matrix as the one given in (19), up to the isometries of ð ( ðµ ) preserving thedistinguished bases which we applied. Hence Î satisï¬es assumption (A3) of Lemma 8.2. Proposition 8.6.
Assume ð is an algebraically closed ï¬eld of characteristic two. Let ð , ð , ð â , ð¡ â be asin Proposition 3.5, set ð ðŒ = 1 for all ðŒ â ð , and let ð â²â ⶠð â âš ð ⩠ⶠ†> be arbitrary. Then ðð¢ððð§ð (Î(
ð, ð , ð â , ð â , ð¡ â ; ð â²â )) = 1 . Proof.
By Proposition 5.3 the isomorphism type of Î( ð, ð , ð â , ð â , ð¡ â ; ð â²â ) as a ring is independent of ð â²â .Since ðð¢ððð§ð depends exclusively on the ring structure, we can assume without loss of generality that ð â² ðŒ = 1 for all ðŒ â ð . Set Î = Î(
ð, ð , ð â , ð â , ð¡ â ; ð â²â ) and ðŽ = ð (( ð )) â ð [[ ð ]] Î . Let ð , ð , ð bethe orthogonal primitive idempotents in Î as in Remark 8.5, and let ð , ⊠, ð denote the primitiveidempotents in ð ( ðŽ ) . By the shape of the matrix ð· given in equation (19) of Remark 8.5 it is clearthat ð ð ð ð is either zero or a primitive idempotent in ðŽ for all ð, ð . To be speciï¬c, ð ð ð ð is non-zero if andonly if the ( ð, ð ) -entry of the matrix ð· in (19) is non-zero. Moreover, by Remark 8.5 (4) it also followsthat ð ð ð ð ðŽð ð ð ð â ð (( ð )) as ð (( ð )) -algebras whenever ð ð ð ð â (using the assumption that ð â² ðŒ = 1 for all ðŒ â ð ).Now let us consider an arbitrary ðŸ â ðð®ðððð§ð (Î) (note that
ðð¢ððð§ð (Î) =
ðð®ðððð§ð (Î) , since thesimple Î -modules are the reduction modulo ð of irreducible lattices). The idea of the remainder of theproof is to modify ðŸ be inner automorphisms until we reach the identity automorphism. After modifying ðŸ by an inner automorphism we can assume ðŸ ( ð ð ) = ð ð for all ð â {1 , , , and therefore ðŸ ( ð ð ð ð ) = ð ð ð ð for all ð, ð . Since ðð®ðððð§ð ( ðŽ ) = 1 it follows that ðŸ ( ð¥ ) = ð¢ð¥ð¢ â1 for some unit ð¢ â ðŽ , and ðŸ ( ð ð ð ð ) = ð ð ð ð implies that ð¢ = â ð,ð ð¢ ðð ð ð ð ð for certain ð¢ ðð â ð (( ð )) à , where ð, ð run over all tuples indexing non-zeroentries of the matrix given in equation (19). Multiplying ð¢ by â ð =1 ð ð ð ð , where the ð ð are units in ð (( ð )) ,only changes ð¢ by an inner automorphism. We can therefore assume that ð¢ ð = 1 for all ð â {1 , , .Multiplying ð¢ by a unit in ð ( ðŽ ) does not change the automorphism it induces. Therefore we can alsoassume without loss of generality that ð¢ ð = 1 for all â©œ ð â©œ , and ð¢ = 1 , ð¢ = 1 as well as ð¢ = 1 .The only potentially non-trivial entries of ð¢ are therefore ð¢ , ð¢ and ð¢ .Now we must have ð¢ð Î ð ð¢ â1 = ( ð¢ ð + ð ) â ð Î ð â ð Î ð . Hence ð¢ ð + ð lies in theendomorphism ring of ð Î ð as a ð (Î) -module. Since ð Î ð is generated by a single element as a ð (Î) -module, this endomorphism ring is a quotient of ð (Î) , given by âš ð + ð , ðð â© ð [[ ð ]] (seen by projectingthe generators given in Remark 8.5 (9) to the fourth and ï¬fth component). We have ð Î ð = ð ð (Î) ,and by projecting the generators given in Remark 8.5 (9) we see that ( ð ð + ðð ) ð â ð Î ð . Hencewe can ï¬nd a unit ð£ â ð Î ð such that ð ð£ = ð¢ and ð ð ð£ = 1 for all ð â {2 , . We can then modify ð¢ by IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 33 ð + ð£ + ð to obtain a new ð¢ where ð¢ is equal to one, and all other ð¢ ðð are unchanged except possibly ð¢ . We can multiply by an appropriate unit in ð ( ðŽ ) to get ð¢ = 1 as well.We can repeat the process above to modify ð¢ in such a way that ð¢ and ð¢ also become equal toone, which means ð¢ = 1 . Hence we can modify an arbitrary central automorphism of Î by innerautomorphisms to obtain the identity automorphism, which implies ðð¢ððð§ð (Î) =
ðð®ðððð§ð (Î) = 1 . (cid:3) Theorem 8.7 (Quaternion-type algebras) . Assume ð is an algebraically closed ï¬eld of characteristic two.Deï¬ne an equivalence relation â ⌠â on †⩟ where ( ð , ð , ð ) ⌠( ð , ð , ð ) precisely when the followingholds: ð ð = ð ð if and only if ð ð = ð ð and ð ð = 2 if and only if ð ð = 2 , for all â©œ ð, ð â©œ .(1) For every ðŒ â †⩟ â ⌠there is a group îŽ ðŒ equipped with a homomorphism îŽ ðŒ ⶠð suchthat ðð«ðð¢ð ð ( îœ (3 ðŸ ) ð ,ð ,ð ) â ðð¢ð ð ð ( îœ (3 ðŸ ) ð ,ð ,ð ) â îŽ ðŒ for any choice of ( ð , ð , ð ) â ðŒ . The action of îŽ ðŒ on ðð¢ð ð ð ( îœ (3 ðŸ ) ð ,ð ,ð ) factors through ð .(2) There is a group î³ such that ðð«ðð¢ððð§ð ( îœ (3 ðŸ ) ð ,ð ,ð ) â î³ for all ð , ð , ð â©Ÿ .Proof. Let ð , ð , ð â , ð¡ â be as in Proposition 3.5, set ð ðŒ = 1 for all ðŒ â ð , and let ð â²â ⶠð â âš ð ⩠ⶠ†> take values ð â 1 , ð â 1 , and ð â 1 on the ð -orbits of ðŒ , ðŒ and ðŒ . Consider the ð [[ ð ]] -order Î = Î(
ð, ð , ð â , ð â , ð¡ â , â ; ð â²â ) constructed in Proposition 5.2. We know that Ì Î = Îâ ð Î â îœ (3 ðŸ ) ð ,ð ,ð .On top of that, deï¬ne ðŒ = {1 â©œ ð â©œ | ð ð = 2} and a function ð â²â²â ⶠð â âš ð ⩠ⶠ†> such that ð â²â² ðŒ ð = 1 if ð â ðŒ and ð â²â² ðŒ ð = 2 if ð â ðŒ . We set Î â² = Î( ð, ð , ð â , ð â , ð¡ â , â ; ð â²â²â ) . Note that by Proposition 5.3 wehave Î â Î â² as rings. What is even more, we are in the case of Proposition 5.3 (1), which means thatboth Î and Î â² are isomorphic as rings to the pullback Ìð [ ð ] ð ( ðŒð ( ðŒ )â ð ÌðŒ, ðŒð ( ðŒ )â ÌðŒð ( ÌðŒ ) | ðŒ â ð ) ð ⊠â â ð â ðŒ ðŒ ð ð ( ðŒ ð )+ ð ( ðŒ ð ) ðŒ ð / / / / Ìðð ( ðŒð ( ðŒ ) , ðŒð ( ðŒ )â ÌðŒð ( ÌðŒ ) | ðŒ â ð ) Î O O / / Ìðð ( ðŒð ( ðŒ ) | ðŒ â ð ) ð O O O O and both for Î and Î â² the pullback diagram of ð [[ ð ]] -algebras given in equation (18) is isomorphic tothe pullback diagram above by applying the identity to the top left entry, and mapping paths in Ìðð tothemselves in the top and bottom right corners of the diagram. That ï¬xes a ring isomorphism between Î â² and Î . If we set ðŽ = ð (( ð )) â ð [[ ð ]] Î , we can identify (as rings only) ð ( ðŽ ) â ð (( ð )) â (22)similar to Remark 8.5 (4). Then the image of ð â ð (Î) in the right hand side of equation (22) is ð Î = ( ð, ð, ð, ð, ð ð â² ðŒ , ð ð â² ðŒ , ð ð â² ðŒ ) , and that of ð â ð (Î â² ) is ð â²Î = ( ð, ð, ð, ð, ð ð â²â² ðŒ , ð ð â²â² ðŒ , ð ð â²â² ðŒ ) .Set ð = ð [[ ð Î ]] and ð = ð [[ ð Î â² ]] , both contained in the centre of Î , and deï¬ne ðŸ and ð¿ as theirrespective ï¬elds of fractions. Instead of working with Î and Î â² , we can now simply consider Î either asan ð -order or as an ð -order. We will now check that the assumptions of Lemma 8.2 are satisï¬ed for Î , ð and ð . Note that we have already checked the assumptions (A1) â (A4) in Remark 8.5.By the deï¬nition of ð â²â²â we have ðð®ð ð¿ ( ð ( ðŽ )) = ðð®ð ð (( ð )) ( ð (( ð )) â (4+ | ðŒ | ) ) à ðð®ð ð (( ð )) ( ð (( ð )) â (3â | ðŒ | ) ) â ð | ðŒ | à ð | ðŒ | , where we are using the fact that char( ð ) = 2 and therefore ðð®ð ð (( ð )) ( ð (( ð ))) = 1 . In particular,assumption (A7) of Lemma 8.2 is satisï¬ed.Any ðŸ â ðð®ð ðŸ ( ð ( ðŽ )) induces a permutation ð â ð which ï¬xes ð Î . This is because any two entriesof ð Î â ð ( ðŽ ) = ð (( ð )) â are either equal or they generate (complete) subï¬elds of diï¬erent index in ð (( ð )) . In particular, ð acting by permutation on the components of ð ( ðŽ ) also induces an element of ðð®ð ðŸ ( ð ( ðŽ )) , which in turn induces the same automorphism of ð ( ðŽ ) as ðŸ . Now, by deï¬nition of ð â²â²â , any ð â ð which ï¬xes ð Î also ï¬xes ð Î â² , and therefore the ð from before acting on ð ( ðŽ ) by permutation isalso an ð¿ -linear automorphism. We have thus found an element of ðð®ð ð¿ ( ð ( ðŽ )) â© ðð®ð ðŸ ( ð ( ðŽ )) inducingthe same automorphism of ð ( ðŽ ) as ðŸ , which proves that assumption (A6) of Lemma 8.2 is satisï¬ed.Since any self-isometry of ð ( ðŽ ) , where ðŽ is considered as a ðŸ -algebra, remains a self-isometry whenwe consider ðŽ as an ð¿ -algebra, and since in Remark 8.5 we have eï¬ectively also veriï¬ed assumption (A4) for Î considered as an ð -algebra, it follows that in the situation of assumption (A5) there alwaysexists a ðŸ â² â ðð®ð ð¿ ( ð ( ðŽ )) inducing the desired permutation of central primitive idempotents such that ðŸ â² ( ð (Î)) â ð (Î) . Now assumption (A5) asks for any ðŸ â ðð®ð ð¿ ( ð ( ðŽ )) inducing the same permutationof central primitive idempotents as ðŸ â² to also satisfy ðŸ ( ð (Î)) â ð (Î) . But from our veriï¬cation ofassumption (A7) it follows that ðŸ is fully determined by the permutation it induces on the central primitiveidempotents, which means that ðŸ = ðŸ â² . This shows that assumption (A5) of Lemma 8.2 is satisï¬ed.Let us now also verify assumptions (B1) â (B5) of Lemma 8.3. The algebra Ì Î â îœ (3 ðŸ ) ð ,ð ,ð is silting-discrete by Proposition 8.4, which implies assumption (B1) . If an element of ðð¢ð ð (Î) acts trivially on ð (Î) , then it must also act trivially on ð ( ðŽ ) by Remark 8.5 (10), since no self-isometry ï¬xes theï¬rst three simple ðŽ -modules and induces a non-trivial permutation on the others. Hence an element of ðð¢ð ð (Î) acting trivially on ð (Î) lies in ðð¢ððð§ð (Î) , which is trivial by Proposition 8.6. Assumption (B2) follows. By deï¬nition of Î there is an automorphism (both ð -linear and ð -linear) inducing the samepermutation on ð ( Ì Î) as some ð â ð if and only if ð stabilises ( ð , ð , ð ) , and the same is true forautomorphisms of Ì Î . This implies assumption (B3) .Since the ð -algebra structure on Î depends only on ðŒ as deï¬ned at the beginning of the proof,the group îŽ â©œ ðð«ðð¢ð ð (Î) from Lemma 8.2 depends only on ðŒ (as ðŒ determines which elements of ðð®ð ( ð ( ðŽ )) are isometries, and it determines ðŒ ). It therefore makes sense to denote îŽ by îŽ ðŒ . Now,any element of ðð¢ð ð ð ( Ì Î) can be represented by an automorphism ðŸ of Ì Î which ï¬xes ð , ð and ð . Sinceeach ð ð Ì Î ð ð is spanned by the paths along the ð -orbit of the unique arrow from ð ð to ð ð , it followsthat each ðŒ â ð gets mapped to ðŸ ( ðŒ ) = ð ( ðŒ ) ðŒ for some unit ð ( ðŒ ) â ð [ ðŒð ( ðŒ )] . Conjugation by ð¢ = ð ( ðŒ ) ð + ð ( ðŒ ) ð + ð ( ðŒ ) ð maps the arrow ðŒ ð to ð ð ð ( ðŒ ð ) ðŒ ð for ð â {1 , , , where ð , ð , ð â ð à areconstants. By further conjugation we can ï¬nd an inner automorphism which maps ðŒ and ðŒ to ðŸ ( ðŒ ) and ðŸ ( ðŒ ) , respectively, and ðŒ to ððŸ ( ðŒ ) , where ð = ð ð ð . That is, we may assume without loss of generalitythat ðŸ ( ðŒ ) = ðŒ , ðŸ ( ðŒ ) = ðŒ and ðŸ ( ðŒ ) = ð â1 ðŒ for some constant ð â ð à . If we now assume that ðŸ restrictsto the trivial automorphism of ð ( Ì Î) , then ðŸ ( ðŒ ðŒ ðŒ ) = ðŒ ðŒ ðŒ , and therefore ð = 1 . That is, ðŸ ( ðŒ ð ) = ðŒ ð for all ð â {1 , , . Moreover, again assuming that ðŸ is trivial on the centre, for an ðŒ â { ðŒ , ðŒ , ðŒ } we have ðŸ ( ðŒð ( ðŒ ) + ð ( ðŒ ) ðŒ ) = ðŒð ( ðŒ ) + ð ( ðŒ ) ðŒ and therefore ðŸ ( ðŒð ( ðŒ )) = ðŒð ( ðŒ ) . Given that ðŸ ( ðŒ ) = ðŒ , itfollows that ð ( ðŒ ) â ðŸ ( ð ( ðŒ )) is a linear combination of paths of the form ( ð ( ðŒ ) ðŒ ) ð ð ( ðŒ ) for ð â †⩟ whichannihilates ðŒ . This is only possible if ð ( ðŒ ) â ðŸ ( ð ( ðŒ )) is zero, since otherwise there would be an ð suchthat ( ð ( ðŒ ) ðŒ ) ð ð ( ðŒ ) â but ( ðŒð ( ðŒ )) ð +1 = 0 , which we know is not the case in generalised weighted surfacealgebras. Hence we have seen that if ðŸ â ðð¢ð ð ð ( Ì Î) restricts to the identity automorphism on ð ( Ì Î) , then ðŸ = 1 . That is, the map ðð¢ð ð ð ( Ì Î) ⶠðð®ð ð ( ð ( Ì Î)) is injective, so in particular condition (B4) holds.To verify condition (B5) ï¬rst recall that by Remark 8.5 (9) and (10) the image of îŽ ðŒ in ðð®ð ð ( ð (Î)) â ðð®ð ð¿ ( ð ( ðŽ )) â ð is contained in âš (1 , , , , , (1 , , , (1 , , â© (1 , , , ,ð â² ðŒ ,ð â² ðŒ ,ð â² ðŒ ) . One can also compute the images of the elements of îŽ ðŒ in ðð®ð ( ðœ â †ð ( Ì Î)) , since these are determinedby the self-isometries of ð ( ðŽ ) these elements induce (and Remark 8.5 (10) gives a list of possible self-isometries). One can check that an element of îŽ ðŒ acts trivially on ð (Î) if and only if it acts trivially on ðœ â †ð ( Ì Î) . The assumptions of Lemma 2.8 are satisï¬ed, and therefore it follows that if an element of îŽ ðŒ acts trivially on ðœ â †ð ( Ì Î) then it stabilises the subspaces ð ð soc( Ì Î) ð ð â ð ( Ì Î) for all ð â {1 , , .Conversely, if an element of îŽ ðŒ stabilises these subspaces, then it has no choice but to act trivially on IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 35 ðœ â †ð ( Ì Î) , and therefore also on ð (Î) . This is again by Lemma 2.8, but note that in characteristic ð it would only follow that each distinguished basis element of ðœ ð â †ð ( Ì Î) gets mapped to a non-zeromultiple of itself, which in characteristic ð = 2 happens to be suï¬cient.An element of ðð¢ð ð ð ( Ì Î) is induced by an automorphism ðŸ â ðð®ð ð ( Ì Î) which ï¬xes the idempotents ð , ð , and ð , which shows that ðŸ ( ð ð soc( Ì Î) ð ð ) = ð ð soc( Ì Î) ð ð for all ð â {1 , , . It follows that if an elementof ðð¢ð ð ð ( Ì Î) , represented by some ðŸ â ðð®ð ð ( Ì Î) which ï¬xes all ð ð , induces the same automorphism of ð ( Ì Î) as an element of îŽ ðŒ , then that element of îŽ ðŒ induces the identity on ð ( Ì Î) by the discussion inthe previous paragraph. It follows that condition (B5) holds.We now know by Lemma 8.3 that ðð«ðð¢ð ð ( Ì Î) =
ðð¢ð ð ð ( Ì Î) â îŽ ðŒ , where the action of îŽ ðŒ on ðð¢ð ð ð ( Ì Î) has ðð«ðð¢ððð§ð (Î) in its kernel and therefore factors through thenatural map îŽ ðŒ ⶠðð®ð ð ( ð (Î)) â©œ ðð®ð ð¿ ( ð ( ðŽ )) â©œ ð . We also know by Lemma 8.3 that ðð«ðð¢ððð§ð ( Ì Î) = Ker( îŽ ðŒ ⶠðð®ð ð ( ð ( Ì Î))) , and by the preceding discussion regarding (B5) this kernelis equal to
ðð«ðð¢ððð§ð (Î) . So we can set î³ = ðð«ðð¢ððð§ð (Î) . This completes the proof. (cid:3)
Remark . In the situation of the preceding theorem, the action of îŽ ðŒ on ðð¢ð ð ð ( Ì Î) can bedetermined explicitly. Namely, the image of the map îŽ ðŒ ⶠðð®ð ð ( ð (Î)) â©œ ð is contained in âš (1 , , , , , (1 , , , (1 , , â© by Remark 8.5 (10). Clearly ðð¢ð ð (Î) â© ðð¢ð ð (Î) is containedin îŽ ðŒ , which acts on Ì Î by permutation of vertices and arrows. This explains how the image of îŽ ðŒ in âš (1 , , , , , (1 , , â© â ð acts on ðð¢ð ð ð ( Ì Î) . But then one has to ï¬gure out how the permutation (12)(34) acts on ð ( Ì Î) . Of course Remark 8.5 (9) helps with that, but this is less straight-forward. Theorem 8.9 (Twisted Brauer graph algebras) . Let ð be an algebraically closed ï¬eld, and let ð and ð be as in §3. Assume that the Brauer graph of ( ð, ð ) in the sense of Deï¬nition 4.2 is a connectedsimple graph (i.e. it has no loops and no double edges). Deï¬ne an equivalence relation â ⌠â on the set îµ = { ð â ⶠð â âš ð ⩠ⶠ†> } , where ð â ⌠ð â²â means that ð ðŒ = ð ðœ if and only if ð â² ðŒ = ð â² ðœ for all ðŒ, ðœ â ð .(1) For each ðŒ â îµ â ⌠there is a group îŽ ðŒ such that ðð«ðð¢ð ð (Î tw ( ð, ð , ð â )) â ðð¢ð ð ð (Î tw ( ð, ð , ð â )) â îŽ ðŒ for all ð â â ðŒ .(2) There is a group î³ such that ðð«ðð¢ððð§ð (Î tw ( ð, ð , ð â )) â î³ for all ð â â îµ for which Î tw ( ð, ð , ð â ) is symmetric.Proof. Deï¬ne ð â²â ⶠð â âš ð ⩠ⶠ†> by setting ð â² ðŒ = 1 for all ðŒ â ð . Then Î tw ( ð, ð , ð â ) and Î tw ( ð, ð , ð â²â ) are isomorphic as rings, but they carry diï¬erent ð [[ ð ]] -algebra structures (thiscomes directly from Proposition 4.4). Set Î = Î tw ( ð, ð , ð â ) , where ð = ð [[ ð ]] acts as deï¬ned inProposition 4.4. This becomes an ð = ð [[ ð ]] -order by letting ð act as â ðŒ â ð ðŒð ( ðŒ ) ⯠ð ð ðŒ â1 ( ðŒ ) , and Î is isomorphic to Î tw ( ð, ð , ð â²â ) as an ð -order. Let us also deï¬ne ðŸ and ð¿ as in Lemma 8.2, and set ðŽ = ðŸ â ð Î . We should mention that we will exclude the case where the Brauer graph of ( ð, ð ) consists of only a single edge whenever necessary (in that case the assertion follows easily from [RZ03,Proposition 3.3]).Let us ï¬rst verify that Î satisï¬es the assumptions of Lemma 8.1 as an ð -order. Note that Ì Î = Îâ ð Î â Î tw ( ð, ð , ð â ) . Condition ( ð² is clear from the presentation of Î tw ( ð, ð , ð â ) . For condition ( ð² we canuse the second option: if ð is a primitive idempotent in Î , and ðŒ and ÌðŒ are the two arrows whose sourceis the corresponding vertex in ð , then ðŒ and ÌðŒ lie in diï¬erent ð -orbits by our âno loopsâ-assumption onthe Brauer graph, which implies that ð Î ð is isomorphic to {( ð, ð ) â ð [[ ð ]] â ð [[ ð ]] | ð (0) = ð (0)} as an ð -algebra (directly from the presentation of Î tw ( ð, ð , ð â²â ) ). This is a subspace of ð -codimension one in ð [[ ð ]] â ð [[ ð ]] , which is the unique maximal order in ð (( ð )) â ð (( ð )) , irrespective of whether we view itas an ð -order or an ð -order. This shows that there is no local ð -order in ððŽð properly containing ð Î ð .For condition ( ð² pick vertices ð , ð , ð â ð such that ð â ð and ð â ð (these correspond toedges in the Brauer graph and idempotents in Î ). If there is no vertex in the Brauer graph of ( ð, ð ) such that the three edges associated with ð , ð and ð are incident to that vertex, then ð Î ð Î ð = 0 and condition ( ð² holds for these three idempotents. If there is such a vertex, then it is unique by ourassumption on the Brauer graph. By swapping ð and ð if necessary we can assume that ð is between ð and ð in the cyclic order around that vertex. Let ðŒ â ð be the arrow whose ð -orbit corresponds to theaforementioned vertex in the Brauer graph and whose source is ð . It then follows from the presentationof Î that ð Î ð Î ð is the completed span of all paths of positive length from ð to ð along the ð -orbitof ðŒ , since any such path passes through ð anyway. This is a pure sublattice of Î , since if ð§ð€ lies in itfor some ð€ â Î (with ð§ as in Proposition 4.4), then ð€ lies in it too. To see this one just has to note thatif ð€ involves any paths along other ð -orbits, even of length zero, then so does ð§ð€ . Condition ( ð² nowfollows from purity.To check condition ( ð² , consider ð â ð â ð . If the corresponding edges in the Brauer graphdo not meet in a vertex, then ð ðŽð is zero and there is nothing to show. If these edges do meet in avertex, then this vertex is unique by our assumptions on the Brauer graph, and therefore we can assignto ð and ð the simple ð ( ðŽ ) -module corresponding to that vertex in the Brauer graph (recall that weparametrised the simple ðŽ -modules in Proposition 4.4). If we now take pair-wise distinct ð , ð , ð â ð ,then ð ðŽð ðŽð is non-zero if and only if the edges in the Brauer graph corresponding to ð , ð and ð meet in a single vertex, and the simple ð ( ðŽ ) -module belonging to that vertex is by deï¬nition the simple ð ( ðŽ ) -module we attached to any pair selected from ð , ð and ð , which veriï¬es ( ð² .We will deal with ( ð² further below, but ï¬rst we need to have a closer look at the centre of ðŽ .The assumption (A2) is trivially satisï¬ed, since no matrix rings over (proper) skew-ï¬elds occur in theWedderburn decomposition of ðŽ . The centre of ðŽ can be described as ð ( ðŽ ) = âš ðŒ âš ð â© â ð â âš ð â© ð (( ð ð ðŒ )) (23)where ð â ð acts as ð on each component, and ð â ð acts as ð ð ðŒ on the component labelled by ðŒ âš ð â© .So clearly ðð®ð ð¿ ( ð ( ðŽ )) is the whole symmetric group on the components of the direct sum above, whichare labelled by ð â âš ð â© . If, for some ð¿ -linear automorphism of ð ( ðŽ ) , the corresponding permutation of ð â âš ð â© ï¬xes ð â , then this ð -linear automorphism is also ð -linear. This already implies (A7) . Similarly,an element of the group ðð®ð ðŸ ( ð ( ðŽ )) induces a permutation of those components of (23) which share thesame multiplicity ð ðŒ , possibly followed by automorphisms of ð (( ð ð ðŒ )) as a ð (( ð )) -algebra. In particularcondition (A6) holds.Now note that the centre of Î is embedded in ð ( ðŽ ) as described in (23) as follows: ð (Î) = { ð = ( ð ðŒ ) ðŒ â âš ðŒ âš ð â© â ð â âš ð â© ð [[ ð ð ðŒ ]] â ð ( ðŽ ) |||| ð ðŒ (0) = ð ðœ (0) for all ðŒ, ðœ â ð } . (24)Clearly this order is ï¬xed by all elements of ðð®ð ðŸ ( ð ( ðŽ )) and ðð®ð ð ( ð ( ðŽ )) , which shows thatconditions (A4) and (A5) hold.A map ð as in ( ð² can be interpreted as a permutation of the edges of the Brauer graph of ( ð, ð ) ,ignoring multiplicities for the time being. To simplify notation, let us regard such a ð as a map from ð into itself. If ð , ⊠, ð ð â ð (for ð â©Ÿ ) are pair-wise distinct and correspond to ð edges in the Brauergraph incident to some vertex ð£ , and without loss of generality are ordered with respect to the cyclicorder around ð£ , then ð Ì Î ð Ì Î â¯ Ì Î ð ð â . By assumption on ð we then have ð ( ð ) ðŽð ( ð ) ðŽ ⯠ðŽð ( ð ð ) â , which is only possible if the edges corresponding to ð ( ð ) , ⊠, ð ( ð ð ) are all incident to some vertex ð£ â² in the Brauer graph. That is, ð induces an automorphism of the line graph of the Brauer graph,which moreover preserves stars (that is, collections of edges that meet in a single vertex). Since ð is an IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 37 automorphism of a line graph, the map ð â1 needs to preserve stars as well (as the number of stars inthe range of ð is equal to the number of stars in the domain). By [Hem72, Theorem 1] (a variation ofWhitneyâs line graph theorem) ð is induced by a graph automorphism of the Brauer graph. The vertices ofthe Brauer graph correspond to the distinguished basis of ð ( ðŽ ) , and therefore there is a ÌðŸ â ðð®ð ( ð ( ðŽ )) permuting the distinguished basis such that ÌðŸ ([ ððŽ ]) = [ ð ( ð ) ðŽ ] for all ð â ð . In particular ÌðŸ ([ ðŽ ]) = [ ðŽ ] ,so if ÌðŸ is induced by a Morita auto-equivalence, then it is induced by an automorphism. To check that ÌðŸ comes from a ðŸ â ðð®ð ðŸ ( ðŽ ) one only needs to check that there is a corresponding automorphism of ð ( ðŽ ) (since ðŽ is Morita equivalent to its centre), which reduces to showing that ÌðŸ is an isometry. Thatis, we need to show that ð preserves the multiplicities of vertices in the Brauer graph of ( ð, ð ) . If avertex in the Brauer graph is incident to at least two edges, corresponding to idempotents ð , ð â Î ,then the multiplicity of the vertex is dim ðŸ ð ðŽð , which is preserved by ð . If the vertex is incidentonly to a single edge, corresponding to an idempotent ð , then this edge must be incident to another edge,corresponding to some idempotent ð (unless the Brauer graph has only one edge, a case we have excludedearlier). In this case the multiplicity of the vertex in question is dim ðŸ ð ðŽð â dim ðŸ ð ðŽð , which againis preserved by ð . It follows that ð preserves multiplicities, which gives us an automorphism ðŸ inducing ÌðŸ . Since ðŸ ( ð (Î)) â ð (Î) is trivially satisï¬ed, by the description of ð (Î) given in equation (24) and thesubsequent remarks, the condition ( ð² holds.Let us now show condition (A3) . To this end, ï¬x an ð -order Î derived equivalent to Î in a ðŸ -algebra ðµ Morita equivalent to ðŽ , and assume Ì Î = Îâ ð Î â Ì Î . In particular, ð (Î) â ð (Î) . Fix an isomorphism ð â¶ Ì Î â¶ Ì Î . If ð , ⊠, ð ð â ð (for some ð â©Ÿ ) are pair-wise distinct, and ð = ð + ⊠+ ð ð , then ð Ì Î ð is again a twisted Brauer graph algebra whose Brauer graph is the subgraph of the Brauer graph of ( ð, ð ) retaining only the edges corresponding to ð , ⊠, ð ð (removing all orphaned vertices and keepingthe multiplicities of all other vertices unchanged). In the same vein, ð Î ð is the corresponding order ofthe form Î tw (âŠ) as deï¬ned in Proposition 4.4. This can be veriï¬ed using the presentations given inDeï¬nition 4.1 and Proposition 4.4 (refer to [Gne19, Lemma 1.12] for a proof).Let us choose ð , ⊠, ð ð â ð corresponding to a spanning tree of the Brauer graph of ( ð, ð ) . Let ð â²1 , ⊠, ð â² ð â Î be pair-wise orthogonal lifts of ð ( ð ) , ⊠, ð ( ð ð ) , and ð â² = ð â²1 + ⊠+ ð â² ð . Then ð Ì Î ð and ð â² Ì Î ð â² are isomorphic Brauer graph algebras whose graph is a tree with ð edges. We know furthermore that rank †ð ( ððŽð ) â©œ rank †ð ( ðŽ ) = ð + 1 = rank †ð ( ðµ ) â©Ÿ rank †ð ( ð â² ðµð â² ) . We also know that the leftmost â â©œ â is actually an equality, and we have a description of the map ð· ð Î ð from Proposition 4.4. The problem is that we do not have the corresponding information for ð â² Î ð â² .By [MH97, Theorem 7.4] the algebra ð Ì Î ð is derived equivalent to a Brauer graph algebra Ω whoseBrauer graph is a star, with the same multiplicities as those occurring in ð Ì Î ð , up to permutation. By[MH97, Theorem 8.3] we may actually assume that the multiplicity of the central vertex is minimalamong all multiplicities, that is, it is equal to ð = min{ ð ðŒ | ðŒ â ð } . Now we can choose a one-sided tilting complex Ìð â over ð Ì Î ð such that ðð§ð î° ð ( ð Ì Î ð ) ( Ìð â ) op â Ω . Then Ìð â²â = ð â² Ì Î ð â² ð â ð Ì Î ð Ìð â is aone-sided tilting complex over ð â² Ì Î ð â² also with endomorphism ring Ω . We can choose lifts ð â and ð â²â of Ìð â and Ìð â²â to ð Î ð and ð â² Î ð â² , whose endomorphism rings are ð -orders Ω and Ω reducing to Ω . We canfurthermore pick two-sided tilting complexes ð â and ð â²â whose restrictions to the left are ð â and ð â²â . If Ìð â and Ìð â²â denote the reductions of ð â and ð â²â modulo ð , then their restrictions to the right diï¬er onlyby an automorphism of Ω , which acts on ð (Ω) as some permutation ð of the distinguished basis. Hencewe get a diagram ð ( ð Ì Î ð ) ð Ìð / / ð· ð Î ð (cid:15) (cid:15) ð (Ω) ð· Ω1 (cid:15) (cid:15) ð / / ð (Ω) ð· Ω2 (cid:15) (cid:15) ð ( ð â² Ì Î ð â² ) ð· ð â²Î ð â² (cid:15) (cid:15) ð Ìð â² o o ð ( ððŽð ) ð ðŸð / / ð ( ðŸ â ð Ω ) â ð ? / / âŽâŽâŽ ð ( ðŸ â ð Ω ) ð ( ð â² ðµð â² ) ð ðŸð â² o o (25) where the leftmost and the rightmost squares are commutative. The composition ð â1 Ìð Ⲡ⊠ð ⊠ð Ìð correspondsto an isomorphism between ð Ì Î ð and ð â² Ì Î ð â² , and therefore maps the distinguished basis to the distinguishedbasis. We will now show that we can ï¬nd an isometry ð ⶠð ( ðŸ â ð Ω ) ⶠð ( ðŸ â ð Ω ) making themiddle square commute. If such a map exists, then ð â1 ðŸð Ⲡ⊠ð ⊠ð ðŸð must actually map the distinguishedbasis of ð ( ððŽð ) to that of ð ( ð â² ðµð â² ) , since ð· ð Î ð , ð· ð â² Î ð â² and ð â1 Ìð Ⲡ⊠ð ⊠ð Ìð all preserve the †⩟ -span of thedistinguished bases (and together these form a commutative square). It then follows that for a projectiveindecomposable ð â² Ì Î ð â² -module ð the image ð· ð â² Î ð â² ([ ð ]) is the sum of exactly two distinguished basiselements of ð ( ð â² ðµð â² ) , since the analogous statement is true for ð· ð Î ð .To ï¬nd ð ï¬rst note that if ð , ⊠, ð ð are a full set of orthogonal primitive idempotents in Ω such thatthe corresponding edges in the star-shaped Brauer graph follow the cyclic order around the central vertex,then ð Ω ð Ω ⯠Ω ð ð â . If ð ( ð ) , ⊠, ð ( ð ) ð (for ð â {1 , ) denote mutually orthogonal lifts of ð , ⊠, ð ð to Ω ð , then ð ( ð ) Ω ð ð ( ð ) Ω ð ⯠Ω ð ð ( ð ) ð â . It follows that for each ð â {1 , there is a distinguished basiselement [ ð ð ] in ð ( ðŸ â ð Ω ð ) such that for each â©œ ð â©œ ð we have ð· Ω ð ([ ð ð Ω]) = [ ð ð ] + [ ð ð,ð ] for some ðŸ â ð Ω ð -module ð ð,ð . We know that ( ð· Ω ð ([ ð ð Ω]) , ð· Ω ð ([ ð ð Ω])) = ([ ð ð Ω] , [ ð ð Ω]) = dim ð ð ð Ω ð ð = ð for all ð â ð , and ð happens to be the minimal length of an element of ð ( ðŸ â ð Ω ð ) . Hence [ ð ] and [ ð ] are actually of length ð , and the [ ð ð,ð ] must be both mutually orthogonal and orthogonal to [ ð ð ] . Moreover, the [ ð ð,ð ] are all non-zero, since ( ð· Ω ð ([ ð ð Ω]) , ð· Ω ð ([ ð ð Ω])) = dim ð ð ð Ω ð ð > ð for all ð .Since rank †( ð ( ðŸ â ð Ω ð )) â©œ ð + 1 it follows (essentially by the âpigeonhole principleâ) that all [ ð ð,ð ] must be distinguished basis elements, pair-wise distinct and distinct from [ ð ð ] (and rank †( ð ( ðŸ â ð Ω ð )) = ð + 1 also follows). To deï¬ne ð we can now simply map [ ð ] to [ ð ] and [ ð ,ð ] to [ ð ,ð ] for â©œ ð â©œ ð .Let us now assign a graph ðº Ω to any ð -order Ω in a semisimple ðŸ -algebra which has the property thatfor each primitive idempotent ð â Ω the ðŸ â ð Ω -module ðŸ â ð ð Ω has exactly two simple constituents,non-isomorphic to each other. Write Ì Î© = Ωâ ð Ω , as usual. We deï¬ne the vertices of ðº Ω to be inbijection with the elements of the distinguished basis of ð ( ðŸ â ð Ω) , and the edges of ðº Ω to be inbijection with the elements of the distinguished basis of ð ( Ì Î©) . We want the edge labelled by [ ð Ì Î©] , fora primitive idempotent ð â Ω , to link the two vertices for which the sum of the corresponding basiselements is equal to ð· Ω ([ ð Ì Î©]) . By this deï¬nition, two edges labelled by [ ð Ì Î©] and [ ð Ì Î©] are adjacentif and only if ð Ω ð â , which happens if and only if ð Ì Î© ð â . If dim ð ð Ì Î© ð > dim ð ð Ì Î© ð for all [ ð Ì Î©] â [ ð Ì Î©] â ð ( Ì Î©) , then ðº Ω is a simple graph. By the discussion above (applied to all spanningtrees of the Brauer graph) we get simple graphs ðº Î and ðº Î , and ðº Î is just the Brauer graph of ( ð, ð ) .The isomorphism ð â¶ Ì Î â¶ Ì Î induces a bijection between the edges of ðº Î and ðº Î by sending [ ð Ì Î] ,for a primitive idempotent ð â Î , to [ ð ( ð ) Ì Î] . By the discussion in the previous paragraph, this assignmentpreserves adjacency of edges. That is, ð induces an isomorphism Ìð between the line graphs of ðº Î and ðº Î . It follows from Whitneyâs line graph theorem [Hem72, Corollary] that either ðº Î is isomorphic to ðº Î , or one of them is the complete graph ðŸ and the other one is the complete bipartite graph ðŸ , . Thelatter case cannot occur since both ðº Î and ðº Î have the same number of vertices, as their number is equalto the rank of ð ( ðŽ ) â ð ( ðµ ) .The discussion following equation (25) implies that if ð = ð + ⊠+ ð ð is a sum of orthogonal primitiveidempotents in Î corresponding to a spanning tree of ðº Î , and ð â² is a lift to Î of ð ( ð ) , then ðº ð Î ð isisomorphic to ðº ð â² Î ð â² . Note that, by deï¬nition, ðº ð Î ð is a subgraph of ðº Î , ðº ð â² Î ð â² is a subgraph of ðº Î , and ðº ð â² Î ð â² is the image of ðº ð Î ð under Ìð . That is, Ìð maps spanning trees to trees. Since every star in ðº Î iscontained in a maximal subtree, which is the same as a spanning tree, the map Ìð maps stars to trees aswell. But if a line graph isomorphism maps a star to a tree, then that tree must again be a star (all otherpossible images of a star contain a triangle). It follows that Ìð maps stars to stars, and because it is a linegraph isomorphism between two isomorphic graphs (with the same number of stars), the inverse of Ìð preserves stars as well. By [Hem72, Theorem 1] the map Ìð is induced by a graph isomorphism.We can attach the multiplicity ([ ð ] , [ ð ]) ðµ to the vertex of ðº Î belonging to the distinguished basiselement [ ð ] â ð ( ðµ ) . The analogously deï¬ned multiplicities on ðº Î coincide with the multiplicities IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 39 already deï¬ned on it (as a Brauer graph). The Cartan matrices of Ì Î and Ì Î determine the multiplicities ofthe vertices of ðº Î and ðº Î by the same argument that was used in proving condition ( ð² . In particular, Ìð induces an isomorphism between ðº Î and ðº Î as graphs with multiplicities. By deï¬nition, thesegraphs determine the matrices of the maps ð· Î and ð· Î with respect to the respective distinguished bases(remember that both vertices and edges are labelled by such basis elements). The graph isomorphism Ìð induces bijections between the sets of vertices and the sets of edges, which produces maps ð and ð as required in assumption (A3) (the fact that these are isometries follows from the fact that Ìð preservesmultiplicities). This ï¬nishes the veriï¬cation of the conditions of Lemma 8.2, which implies that there isa group ðð«ðð¢ððð§ð (Î) â©œ îŽ ðŒ â©œ ðð«ðð¢ð ð (Î) (which is the same for any ð â â ðŒ ) such that ðð¢ð ð ( Ì Î) â Ì îŽ ðŒ isequal to ðð«ðð¢ð ð ( Ì Î) .To ï¬nish the proof we will need to look at automorphism groups. Note that all algebras we consider arebasic, so Picard groups and outer automorphism groups coincide. Let ðŸ â ðð®ð ð (Î) â© ðð®ð ð (Î) representan element which lies in the kernel of the natural map ðð®ð ð (Î) ⶠðð®ð ð ( Ì Î) . Then ðŸ must act triviallyon ð ( Ì Î) . So we can assume that ðŸ ( ð ) = ð for all ð â ð , and by further modifying ðŸ by an innerautomorphism we can assume that ðŸ is trivial on Ì Î . In particular, ðŸ induces an automorphism of theBrauer graph of ( ð, ð ) which ï¬xes all edges, and unless the Brauer graph consists of a single edge (whichwe assume it does not) such an automorphism must ï¬x all vertices of the Brauer graph as well. So ðŸ actstrivially on ð ( ðŽ ) and therefore becomes inner in ðŽ (since ðŸ is ð -linear and ðŽ is a split semisimple ð¿ -algebra). Hence we can assume that ðŸ is induced by conjugation by ð¢ = â ð â ð â ðŒ âš ð â© â ð â âš ð â© ð¢ ð,ðŒ âš ð â© â ð ðŒ âš ð â© ð, where ð ðŒ âš ð â© denotes the primitive idempotent in ðŽ belonging to the ð -orbit ðŒ âš ð â© , and ð¢ ð,ðŒ âš ð â© â ð (( ð ð ðŒ )) à .We can multiply ð¢ by a unit of ð ( ðŽ ) and assume without loss of generality that for each ðŒ â ð thereis an ð ( ðŒ ) â ð such that ð¢ ð ( ðŒ ) ,ðŒ âš ð â© = 1 . Fix an orbit ðŒ âš ð â© , and pick the representative ðŒ such that ð ( ðŒ ) is the source of ðŒ . Then conjugation by ð¢ maps ðŒð ( ðŒ ) ⯠ð ð ( ðŒ ) (for any ð ) to ð¢ ð¡ ( ð ð ( ðŒ )) ,ðŒ âš ð â© â ðŒð ( ðŒ ) ⯠ð ð ( ðŒ ) ,where ð¡ ( ð ð ( ðŒ )) denotes the target of ð ð ( ðŒ ) . At the same time we know that the image of this element in Ì Î must be equal to ðŒð ( ðŒ ) ⯠ð ð ( ðŒ ) , which implies that ð¢ ð¡ ( ð ð ( ðŒ )) ,ðŒ âš ð â© must lie in ð ð ðŒ ð [[ ð ð ðŒ ]] . Itfollows that ð¢ ð,ðŒ âš ð â© lies in ð ð ðŒ ð [[ ð ð ðŒ ]] for all ðŒ â ð and ð â ð . But from the descriptionof ð (Î) in equation (24) it is clear that then ð¢ â âš ð â ð ð (Î) ð â Î , that is, ðŸ is inner. It follows that ðð®ð ð (Î) â© ðð®ð ð (Î) embeds into ðð®ð ð (Î) , and in particular Ì îŽ ðŒ â îŽ ðŒ .A permutation of the distinguished basis of ð ( Ì Î) is induced by an automorphism of Ì Î if and only ifit corresponds to a multiplicity preserving automorphism of the Brauer graph of ( ð, ð ) . Similarly, thereis an element of ðð®ð ð (Î) acting isometrically on ð ( ðŽ ) (carrying the bilinear form coming from the ðŸ -algebra structure of ðŽ ) which induces a given permutation of the distinguished basis of ð (Î) â ð ( Ì Î) if and only if the permutation corresponds to a multiplicity preserving automorphism of the Brauer graph.Such an element of ðð®ð ð (Î) then gives rise to an element of îŽ ðŒ . It follows that ðð¢ð ð ( Ì Î) â îŽ ðŒ = ðð¢ð ð ð ( Ì Î) â îŽ ðŒ , proving the ï¬rst assertion.For the second assertion we will verify conditions (B4) and (B5) of Lemma 8.3. First let us show (B4) ,that is, ðð¢ððð§ð ( Ì Î) â©
ðð¢ð ð ð ( Ì Î) â îŽ ðŒ . Let ðŸ â ðð®ðððð§ð ( Ì Î) be an automorphism ï¬xing the all ð â ð . Foreach ðŒ â ð the element ð§ ðŒ âš ð â© = â ðœ â ðŒ âš ð â© ðœð ( ðœ ) ⯠ð ð ðŒ â1 ( ðœ ) â Ì Î is central and therefore ï¬xed by ðŸ , which implies ðŸ ( ðŒð ( ðŒ ) ⯠ð ð ðŒ â1 ( ðŒ )) = ðŒð ( ðŒ ) ⯠ð ð ðŒ â1 ( ðŒ ) . For each â©œ ð â©œ ð ðŒ â 1 we have ðŸ ( ð ð ( ðŒ )) = ð¢ ð â ð ð ( ðŒ ) for some unit ð¢ ð â ð [ ð§ ðŒ âš ð â© ] , and ð¢ ð¢ ⯠ð¢ ð ðŒ â1 ð§ ðŒ âš ð â© = ð§ ðŒ âš ð â© .Write ð¢ ð = (1 + ð ð ) â ð ð for some ð ð â ð§ ðŒ âš ð â© ð [ ð§ ðŒ âš ð â© ] and ð ð â ð à . Since ð§ ðŒ âš ð â© annihilates all arrows notcontained in the ð -orbit of ðŒ we can construct a unit ð£ â Ì Î such that ð£ðœð£ â1 = ðœ whenever ðœ â ðŒ âš ð â© and ð£ð ð ( ðŒ ) ð£ â1 = (1 + ð ð ) â1 ð ð ( ðŒ ) for all â©œ ð â©œ ð ðŒ â 1 . If we compose ðŸ and conjugation by ð£ we can assumewithout loss of generality that ð¢ ð â ð à for all â©œ ð â©œ ð ðŒ â 1 , which by ð¢ ð¢ ⯠ð¢ ð ðŒ â1 ð§ ðŒ âš ð â© = ð§ ðŒ âš ð â© implies that all ð¢ ð lie in ð à (to see this note that if an element in ð [ ð§ ðŒ âš ð â© ] does not annihilate the arrow ðŒ , then italso does not annihilate ð§ ðŒ âš ð â© ). Note that we did not alter the image of any ðœ â ðŒ âš ð â© . We can thereforeassume without loss of generality that there are ð¢ ðŒ â ð à such that ðŸ ( ðŒ ) = ð¢ ðŒ ðŒ for all ðŒ â ð , andmoreover â ðœ â ðŒ âš ð â© ð¢ ðœ = 1 for all ðŒ . Now deï¬ne a central automorphism of the order Î = Î tw ( ð, ð , ð â ) that maps any ðŒ â ð to ð¢ ðŒ ðŒ (this would not have been possible with the original ð¢ ð âs, since their productwas equal to one only in Ì Î but not necessarily in ðð ). Using the presentation given in Proposition 4.4one sees that this automorphism is well-deï¬ned and one checks that it is indeed trivial on ð (Î) . Thisshows that ðð¢ððð§ð ( Ì Î) â©
ðð¢ð ð ð ( Ì Î) â ðð¢ððð§ð (Î) â
ðð¢ððð§ð (Î) â îŽ ðŒ , proving (B4) .For condition (B5) we note that an element of îŽ ðŒ induces a permutation on the elements of the form ð§ ðŒ âš ð â© deï¬ned earlier, as these are the reductions modulo ð of the elements ð ð ðŒ ð ðŒ âš ð â© â ð (Î) . Anelement of ðð¢ð ð ð ( Ì Î) is induced by an automorphism ðŸ of Ì Î which ï¬xes all ð â ð , and therefore can onlymap ð§ ðŒ âš ð â© to ð§ ðœ âš ð â© for ðŒ âš ð â© â ðœ âš ð â© if the exact same vertices appear as sources of arrows in the orbits ðŒ âš ð â© and ðœ âš ð â© . If two distinct ð -orbits of arrows have more than one vertex in common (as a source ofan arrow), then by deï¬nition there is a double edge in the Brauer graph, which we do not allow. Hence ðŸ can only map ð§ ðŒ âš ð â© to ð§ ðœ âš ð â© if both ðŒ and ðœ are loops attached to the same vertex. In that case the Brauergraph has only a single edge, a case we exclude. So, if ðŸ induces the same automorphism of ð ( Ì Î) assome element of îŽ ðŒ , then ðŸ ï¬xes ð§ ðŒ âš ð â© for all ðŒ â ð . But then ðŸ also ï¬xes ðð§ ð ðŒ ðŒ âš ð â© for all ð â ð and ðŒ â ð . The latter elements together with the ð§ ðŒ âš ð â© generate ð ( Ì Î) . That is, ðŸ must be trivial on ð ( Ì Î) ,which implies (B5) .It follows by Lemma 8.3 that ðð«ðð¢ððð§ð ( Ì Î) = Ker( îŽ ðŒ ⶠðð®ð ð ( ð ( Ì Î))) . As already discussed above,an element of îŽ ðŒ induces a permutation of the ð§ ðŒ âš ð â© for ðŒ â ð , and since we are assuming that thereis more than one edge in the Brauer graph we have ð§ ðŒ âš ð â© â ð§ ðœ âš ð â© in Ì Î whenever ðŒ âš ð â© â ðœ âš ð â© . Inparticular ðð«ðð¢ððð§ð ( Ì Î) â ðð«ðð¢ððð§ð (Î) â
ðð«ðð¢ððð§ð (Î) , and we can therefore write
ðð«ðð¢ððð§ð ( Ì Î) =Ker(
ðð«ðð¢ððð§ð (Î) ⶠðð®ð ð ( ð ( Ì Î))) . Now ð ( Ì Î) is generated by the ð§ ðŒ âš ð â© for ðŒ â ð and the elementsof soc( Ì Î) = âš ðð§ ð ðŒ ðŒ âš ð â© | ð â ð , ðŒ â ð â© ð (one obtains this from the presentation of Ì Î ). In particular, soc( Ì Î) = soc( ð ( Ì Î)) , and an element of
ðð«ðð¢ððð§ð (Î) induces the identity on ð ( Ì Î) if and only if it mapsthe elements ðð§ ð ðŒ ðŒ âš ð â© to multiples of themselves, since it ï¬xes the elements ð§ ðŒ âš ð â© anyway. In this part ofthe proof we can also assume that Ì Î is symmetric, so Lemma 2.8 applies and it follows that ðð«ðð¢ððð§ð ( Ì Î) = Ker(
ðð«ðð¢ððð§ð (Î) ⶠðð®ð ð ( ð â †ð (Î))) , which is independent of the ð -algebra structure on Î . If we let î³ denote the right hand side of theexpression above then the second assertion follows. (cid:3) Remark . (1) While there may be some twisted Brauer graph algebras of independent interest,the main intended application of Theorem 8.9 are twisted Brauer graph algebras which areisomorphic to their ordinary counterparts (e.g. in characteristic two, or when the Brauer graphis bipartite). In those cases the symmetry condition in the second part of Theorem 8.9 isautomatically met.(2) One case we are particularly interested in are the algebras of dihedral type î° (3 ðŸ ) ð ,ð ,ð fromErdmannâs classiï¬cation [Erd90] in characteristic two, where ð , ð , ð â©Ÿ . These are Brauergraph algebras, where the graph is a triangle (i.e. a complete graph on three vertices). ObviouslyTheorem 8.9 applies, but these algebras are also silting-connected, which by Lemma 8.3 impliesthat ðð¢ð ð ð ( î° (3 ðŸ ) ð ,ð ,ð ) ⎠ðð«ðð¢ð ð ( î° (3 ðŸ ) ð ,ð ,ð ) . However, condition (B2) of Lemma 8.3 fails to hold, and we do not get a semi-direct productdecomposition as we did for îœ (3 ðŸ ) ð ,ð ,ð . Speciï¬cally, the quiver ð for this algebra is an inProposition 3.5, and for each ð â ð à there is an automorphism sending ðŒ to ððŒ , ðœ to ð â1 ðœ ,and ï¬xing all other arrows. These automorphisms are trivial on the centre of the algebra, and lift IJECTIONS OF SILTING COMPLEXES AND DERIVED PICARD GROUPS 41 to the ð [[ ð ]] -order Î used in the proof of Theorem 8.9, that is, they lie in îŽ ðŒ â© ðð¢ð ð ð ( Ì Î) (in fact,this intersection consists exactly of the automorphisms we just described).Of course Theorem 8.9 also applies to Brauer tree algebras, which are Brauer graph algebras whosegraph is a tree and only a single vertex may have multiplicity bigger than one. Their derived Picard groupswere already described in [Zvo15, VZ17] (which to some extent motivated Theorems 8.7 and 8.9). Wecan recover the fact that the derived Picard group decomposes as a direct product of ðð¢ð ð ð ( Ì Î) and a groupwhose isomorphism type is mostly independent of multiplicities. If more than one vertex has multiplicitybigger than one, this becomes a semidirect product. Proposition 8.11.
Let ð be algebraically closed and let ðŽ ( ð , ð â ) denote a Brauer graph algebra whosegraph ð is a star with multiplicities ð â . Deï¬ne îµ and â ⌠â as in Theorem 8.9. For any ðŒ â îµ â ⌠thereis a group îŽ ðŒ such that ðð«ðð¢ð ð ( ðŽ ( ð , ð â )) â ðð¢ð ð ð ( ðŽ ( ð , ð â )) â îŽ ðŒ . for all ð â â ðŒ .Given a multiplicity function ð â which assigns multiplicity one to all except the central vertex, there areexactly two possibilities for the equivalence class ðŒ â îµ â ⌠containing ð â (one in which the multiplicityof the central vertex is also equal to one, and one in which it is bigger than one), and ðð«ðð¢ð ð ( ðŽ ( ð , ð â )) â ðð¢ð ð ð ( ðŽ ( ð , ð â )) Ã îŽ ðŒ . Proof.
The ï¬rst part of the assertion follows from Lemma 8.3. Clearly (B1) holds since ðŽ ( ð , ð â ) is silting-discrete (see [AAC18, Theorem 6.7]). An argument like the one in Proposition 8.6 showsthat ðð¢ððð§ð (Î tw ( ð, ð )) = 1 , where Î tw ( ð, ð , ð â ) is the twisted Brauer graph algebra isomorphic to ðŽ ( ð , ð â ) (note that ð is not a circular quiver, but rather a circular quiver with a loop attached to eachvertex). This implies condition (B2) of Lemma 8.3. Condition (B3) is also satisï¬ed, as both images in ðð®ð ( ð ( ðŽ ( ð , ð â ))) correspond precisely to the automorphisms of the tree ð . By applying Lemma 8.3on top of Theorem 8.9 we get a semidirect product decomposition ðð¢ð ð ð ( ðŽ ( ð , ð â )) â îŽ ðŒ . For the secondpart of the assertion we should note that ðð¢ððð§ð ( ðŽ ( ð , ð â )) = 1 irrespective of ð â , and the centre of ðŽ ( ð , ð â ) â Î tw ( ð, ð , ð â ) is the reduction of the centre of Î tw ( ð, ð , ð â ) . In particular, the action of îŽ ðŒ on ðð¢ð ð ð ( ðŽ ( ð , ð â )) factors through ðð®ð ( ð ( ð (( ð )) â ð [[ ð ]] Î tw ( ð, ð , ð â ))) . Hence one only needs to checkthat the automorphisms of ð ( ðŽ ( ð , ð â )) coming from automorphisms of ð (Î tw ( ð, ð , ð â )) commute withthose which are induced by automorphisms of ðŽ ( ð , ð â ) . This is true if only the central vertex is allowedto have multiplicity bigger than one, and false otherwise (this requires a computation). (cid:3) R EFERENCES [AAC18] T. Adachi, T. Aihara, and A. Chan. Classiï¬cation of two-term tilting complexes over Brauer graph algebras.
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