DG Structure on the Length 4 Big From Small Construction
aa r X i v : . [ m a t h . A C ] A p r DG STRUCTURE ON THE LENGTH 4 BIG FROM SMALLCONSTRUCTION
KELLER VANDEBOGERT
Abstract.
The big from small construction was introduced by Kustin andMiller in [13] and can be used to construct resolutions of tightly double linkedGorenstein ideals. In this paper, we expand on the DG-algebra techniquesintroduced in [11] and construct a DG-algebra structure on the length 4 bigfrom small construction. The techniques employed involve the constructionof a morphism from a Tate-like complex to an acyclic DG-algebra exhibitingPoincar´e duality. This induces homomorphisms which, after suitable modi-fications, satisfy a list of identities that end up perfectly encapsulating therequired associativity and DG axioms of the desired product structure for thebig from small construction. Introduction
Let ( R, m ) denote a local commutative Noetherian ring. Any quotient R/I withprojective dimension 2 has minimal free resolution that admits the structure ofa unique commutative associative differential graded (DG) algebra. A result ofBuchsbaum and Eisenbud (see [5]) establishes a similar result for quotients
R/I of R with projective dimension 3, where the DG structure is no longer unique.For quotients R/I with projective dimension at least 4, it is no longer guaranteedthat there exists an associative
DG structure on the minimal free resolution of
R/I .A standard counterexample is given by the homogeneous minimal free resolution ofthe ideal ( x , x x , x x , x x , x ) ⊂ k [ x , x , x , x ] (see Theorem 2 . . R/I is a Gorenstein ring with projective dimension 4, it is proved byKustin and Miller (see [12] when characteristic = 2) and Kustin (see [7] when char-acteristic = 3, [10] for a more general version) that the minimal free resolution does admit the structure of a commutative associative DG algebra exhibiting Poincar´eduality (see Definition 2.2). For Gorenstein rings R/I with projective dimension atleast 5, it is no longer guaranteed that an associative DG structure exists on theminimal free resolution (see [17]).The existence of DG-algebra structures on resolutions implies many desirableproperties of the module being resolved. Indeed, the aforementioned fact thatevery length 3 resolution of a cyclic module is a DG-algebra is then used to deducethe well-known structure theorem for the resolution of a grade 3 Gorenstein ideal(see [5], Theorem 2 . R, m ) is a regular localring and the minimal free resolution of a quotient R/I is a DG-algebra, then thePoincar´e series of
R/I may be written in terms of the Poincar´e series of the Koszulhomology algebra H ( K R/I ), which is a finite dimensional vector space.
Date : April 16, 2020.
There are many examples of “canonical” resolutions admitting DG-algebra struc-tures. The Koszul complex is a DG-algebra exhibiting Poincar´e duality with prod-uct given by exterior multiplication. The standard Taylor resolution for monomialideals, though not always minimal, is also a DG-algebra (see [1], 2 . L / K complexes introduced by Buchsbaumand Eisenbud (see [4]) and Buchsbaum (see [3]) are DG-algebras. A collection ofresults of a more combinatorial flavor relating to the existence of DG structures onhull/Lyubeznik resolutions and resolutions of monomial ideals in general may befound in [6].In practice, there are 3 common techniques of constructing DG structures onresolutions. The first two techniques use the fact that every resolution admits thestructure of a DG algebra that is associative up to homotopy by using induced com-parison maps from the so-called symmetric square complex. For sufficiently shortresolutions, these homotopies may be 0 maps (see [5]). If not, then a modificationprocedure may be used to construct products that are associative (see [10]).The third approach is to record an explicit multiplication and check by handthat all of the relevant DG axioms are satisfied. This can be done easily in thecase of the Koszul complex. This is also the approach used to show that the Taylorresolution is a DG algebra. Indeed, this is the approach that we will use to provethat the length 4 big from small construction is a DG algebra.The purpose of the current paper is to expand on techniques introduced in [11]and explore applications related to the construction of DG structures on particulartypes of complexes. More precisely, we construct a DG structure on the length 4“Big From Small Construction” (see Definition 3.1) introduced in [13]. In the casewhere the characteristic of R is not 2, this was already proved in [9], heavily relyingon the construction of a “complete higher order multiplication” as given by Palmerin [15].In the present case, the DG structure is built from the ground up, and the con-struction is totally characteristic free. It is the intent of the author to illustrate theuse of Tate-like complexes to construct interesting homomorphisms whose proper-ties are often ideal for inducing DG structures on complexes. We construct the DGstructure on the big from small construction since its structure is particularly wellsuited to quotients by DG ideals; indeed, in the case where R has characteristic = 2, it is shown in [9] that the minimal resolution of a grade 4 almost completeintersection admits the structure of a commutative associative DG algebra by suc-cessively taking DG quotients of the big from small construction applied to theappropriate Gorenstein ideal.The paper is organized as follows: in Section 2, basic facts on DG algebras withdivided powers and exhibiting Poincar´e duality are presented. The basic setup tobe used throughout the paper, Setup 2.7, is also introduced. In Section 3, we givea brief review of the big from small construction and its relation to the notion oftight double linkage (see Definition 3.2) of Gorenstein ideals.Section 4 is where the aforementioned techniques are introduced and employed.To be precise, we construct a Tate-like complex B along with a morphism of com-plexes c : B → K [ − K is a length 3 Koszul complex. By acyclicity, thereis an induced chain homotopy h ; moreover, this homotopy may be modified in such G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 3 a way that certain homomorphisms induced by Poincar´e duality will satisfy the re-quired associativity relations for the contended DG structure on the big from smallconstruction.In Section 5, we prove that the length 4 big from small construction admits thestructure of a commutative associative DG algebra exhibiting Poincar´e duality. Aspreviously mentioned, this amounts to writing down an explicit multiplication andchecking that all of the relevant DG axioms are satisfied.2.
DG Algebras
Definition 2.1.
Let R denote a commutative Noetherian ring. The grade of aproper ideal I is the length of the longest regular sequence on R in I . An ideal I is perfect if grade( I ) = pd R ( I ) (the projective dimension). An ideal of grade g iscalled Gorenstein if it is perfect and Ext gR ( R/I, R ) =
R/I .A complex F • : · · · → F → F → F → acyclic if the only nonzerohomology occurs at the 0th position. We say F • is a free resolution of R/I if H ( F • ) = R/I and all F i are free.Let F denote a free R -module. Then D ( F ) denotes the degree 2 piece of thedivided power algebra on F . Recall that by the divided power algebra structure,given x, x ′ ∈ F , ( x + x ′ ) (2) = x (2) + x · x ′ + x ′ (2) whence it suffices to determine the action of a homomorphism with domain D ( F )on elements of the form x (2) .We say that a pairing A ⊗ R B → C is perfect if the induced maps A → Hom R ( B, C ) B → Hom R ( A, C )are isomorphisms.
Definition 2.2. A differential graded algebra ( F, d ) (DG-algebra) over a commu-tative Noetherian ring R is a complex of finitely generated free R -modules withdifferential d and with a unitary, associative multiplication F ⊗ R F → F satisfying(a) F i F j ⊆ F i + j ,(b) d i + j ( x i x j ) = d i ( x i ) x j + ( − i x i d j ( x j ),(c) x i x j = ( − ij x j x i , and(d) x i = 0 if i is odd,where x k ∈ F k . A DG-algebra F is a DGΓ-algebra if for each positive even index i and each x i ∈ F i , there is a family of elements { x ( k ) i } satisfying the divided poweraxioms in the module D k F i :(a) x (0) i = 1, x (1) i = x i , and x ( k ) i ∈ D k F i for x i ∈ F i .(b) x ( p ) i x ( q ) i = (cid:0) p + qq (cid:1) x ( p + q ) i for x i ∈ F i ,(c) ( x i + y i ) ( p ) = P pk =0 x ( p − k ) i y ( k ) i for x i , y i ∈ F i .(d) ( rx i ) ( p ) = r p x ( p ) i for r ∈ R , x i ∈ F i .(e) ( x ( p ) i ) ( q ) = ( pq )! q !( p q )! x ( pq ) i along with the extra condition d ik ( x ( k ) i ) = d i ( x i ) x ( k − i .A DG-algebra F exhibits Poincar´e duality if there is an integer m such that F i = 0 for i > m , F m ∼ = R , and for each i the multiplication map F i ⊗ R F m − i → F m KELLER VANDEBOGERT is a perfect pairing of R -modules. Given a DG-algebra F such that F i = 0 for i > m , an orientation isomorphism is a choice of isomorphism[ − ] F : F m → R, given that such an isomorphism exists. Remark . Let F be a DG-algebra exhibiting Poincar´e duality with m such that F i = 0 for i > m . Given any R -module M , observe that in order to specify an R -module homomorphism M → F i for any i m , it suffices to construct a morphism M ⊗ R F m − i → F m . Any such map induces a morphism M → Hom R ( F m − i , F m ),and by Poincar´e duality, Hom R ( F m − i , F m ) ∼ = F i . Lemma 2.4.
Let c : ( B, b ) → ( A, a ) be a morphism of complexes of projectivemodules with ( A, a ) acyclic. Assume B i = B ′ i ⊕ B ′′ i , b i ( B ′ i ) ⊆ B ′ i − and c i ( B ′ i ) = 0 for each i , with B = B ′ . Then there exists a homotopy h : B → A [1] such that h i ( B ′ i ) = 0 for each i .Proof. Build the sequence of homotopies inductively, setting h = 0. Let i > h i − has been defined. Take any h i such that a i +1 ◦ h i = c i − h i − ◦ b i and notice that a i ◦ ( c i − h i − ◦ b i ) = 0 since A is a complex. By acyclicity of A , this means c i − h i − ◦ b i has image contained in the image of a i +1 . Similarly,( c i − h i − ◦ b i )( B ′ i ) = 0 by our assumptions on c and b , and induction. Thus h i maybe taken to satisfy the desired properties. (cid:3) Theorem 2.5 ([10], Theorem 4 . . Let R be a commutative Noetherian ring and M a length resolution of a cyclic R -module by finitely generated free R -moduleswith rank( M ) = 1 and M ∼ = Hom R ( M, M ) . Then M has the structure of a DG Γ -algebra which exhibits Poincar´e duality.Remark . In particular, Theorem 2.5 tells us that in the case that R is local orstandard graded, the (homogeneous) minimal free resolution of a grade 4 Goren-stein ideal admits the structure of an associative DGΓ-algebra exhibiting Poincar´eduality.The following setup will be used for the rest of the paper. Setup 2.7.
Let R be a commutative Noetherian ring, I a grade Gorenstein ideal.Let ( M, m ) : 0 → M → · · · → M → M = R be a length resolution of R/I .Assume that M is also a DG Γ -algebra exhibiting Poincar´e duality.Let ( K, k ) denote the Koszul complex on a length regular sequence. Let R/ a denote the complete intersection that K is resolving; assume a ⊆ I . Moreover,assume that M = M , ⊕ M , , where rank( M , ) = 3 and m ( M , ) = a .Define α : K = R → M = R as the identity. Define α : K → M bythe condition that the composition K → M → M , makes the following diagramcommute: K k / / (cid:15) (cid:15) a M , m > > ⑤⑤⑤⑤⑤⑤⑤⑤ G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 5 and that K → M → M , is the map. Define α : K → M as the map whichextends α and α as described above. Given this data, define β : M → K [ − via (cid:2) β i ( θ i ) · φ − i (cid:3) K = ( − i +1 (cid:2) θ i α − i ( φ − i ) (cid:3) M Remark . The assumption that α ( K ) is a direct summand of M means thatthe resolution of M is not necessarily minimal. For our purposes, this will not besignificant since Proposition 3.3 applies regardless of the minimality of M .3. The Big From Small Construction
In this section we review the big from small construction of [13] as applied toSetup 2.7.
Definition 3.1.
Adopt notations and hypotheses as in Setup 2.7; let r ∈ R . The big from small construction applied to the data ( α, r ) yields the complex( F ( α, r ) , f ) : 0 → F → F → F → F → F with F = R , F = K ⊕ M , F = K ⊕ M ⊕ K , F = M ⊕ K , F = M , and f = ( m β + rk ) f = (cid:18) m β r − k − α (cid:19) , f = β − r − k − α m f = (cid:18) α β − rm − k β (cid:19) Definition 3.2.
Let I and I ′ be two grade g Gorenstein ideals in a commutativeNoetherian ring R . Suppose a is a grade g − tight double link between I and I ′ over K if there exists and ACI J = ( a , y, y ′ ) with( a , y ) and ( a , y ′ ) both complete intersections, and( a , y ) : I = ( a , y ′ ) : I The relation between Definitions 3.1 and 3.2 is the following:
Proposition 3.3 ([9], Proposition 4 . . Let a ⊆ I ideals in a commutative Noe-therian ring R with a a grade complete intersection and I a grade Gorensteinideal. Let K be a length Koszul complex resolving a and M a length Poincar´eDG algebra resolving I , with α : K → M the induced comparison map.If there is a tight double link between I and a grade Gorenstein ideal I ′ over a , then there exists r ∈ R such that F ( α, r ) resolves I ′ . Using Tate-Like Complexes To Induce DG Structures
We begin this section by examining additional properties of the maps α and β as in Setup 2.7. Proposition 4.1.
Adopt notation and hypotheses of Setup 2.7. Then:(1) β ◦ α = 0 ,(2) β i + j ( α i ( φ i ) θ j ) = φ i β j ( θ j ) , and(3) β is an isomorphism. KELLER VANDEBOGERT
Proof. (1): Let φ i ∈ K i , φ − i ∈ K − i . Then, (cid:2) β i ( α i ( φ i )) φ − i (cid:3) K = ( − i +1 (cid:2) α i ( φ i ) α − i ( φ − i ) (cid:3) M = ( − i +1 (cid:2) α ( φ i ∧ φ − i ) (cid:3) M = 0 , since K i = 0 for i > φ − i − j ∈ K − i − j . (cid:2) β i + j (cid:0) θ j · α i ( φ i ) (cid:1) · φ − i − j (cid:3) K = ( − i + j +1 (cid:2) θ j · α i ( φ i ) · α − i − j ( φ − i − j ) (cid:3) M = ( − i + j +1 (cid:2) θ j · α − j ( φ i ∧ φ − i − j ) (cid:3) M = ( − i (cid:2)(cid:0) β j ( θ j ) · φ i (cid:1) · φ − i − j (cid:3) K , so that β i + j ( θ j α i ( φ i )) = ( − i β j ( θ j ) φ i . Using skew-commutativity, the result fol-lows.(3): Notice that α is the identity map, which is an isomorphism. By Poincar´eduality, β must also be an isomorphism. (cid:3) Proposition 4.2.
Adopt notation and hypotheses of Setup 2.7. Let B : B → B → B → B → B → B be the complex with B = M , B = M , B = ( ^ M ) ⊕ M B = ^ M ⊕ ( M ⊗ M ) ⊕ M , B = ( ^ M ⊗ M ) ⊕ D M ⊕ ( M ⊗ M ) B = ( M ⊗ D M ) ⊕ ( ^ M ⊗ M ) and the differential d being the induced Tate differential (that is, just use the gradedLeibniz rule). Define a map c : B → K [ − via: c = c = c = 0 c θ ∧ θ ′ ∧ θ ′′ θ ′′′ ⊗ θ θ ! = β ( θ ∧ θ ′ ) β ( θ ′′ ) − β ( θ ∧ θ ′′ ) β ( θ ′ )+ β ( θ ′ ∧ θ ′′ ) β ( θ ) ,c θ ∧ θ ′ ⊗ θ ′ θ (2)2 θ ′′′ ⊗ θ ! = β ( θ ′ ) β ( θ θ ′ ) − β ( θ ) β ( θ ′ θ ′ ) − β ( θ ∧ θ ′ ) β ( θ ′ ) ,c (cid:18) θ ⊗ θ (2)2 θ ′ ∧ θ ′′ ⊗ θ (cid:19) ! = β ( θ ) β ( θ (2)2 ) − β ( θ θ ) β ( θ ) G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 7 − β ( θ ′′ ) β ( θ ′ θ ) + β ( θ ′ ) β ( θ ′′ θ ) − β ( θ ′ ∧ θ ′′ ) β ( θ ) . Then c is a morphism of complexes.Proof. We verify that all of the appropriate maps commute. The first nontrivialplace to check is c : B → K . k ◦ c θ ∧ θ ′ ∧ θ ′′ θ ′′′ ⊗ θ θ ! = β ( m ( θ ∧ θ ′ )) β ( θ ′′ ) − β ( m ( θ ∧ θ ′′ )) β ( θ ′ )+ β ( m ( θ ′ ∧ θ ′′ )) β ( θ )= β ( m ( θ ) θ ′ ) β ( θ ′′ ) − β ( θ m ( θ ′ )) β ( θ ′′ ) − β ( m ( θ ) θ ′′ ) β ( θ ′ ) + β ( θ m ( θ ′′ )) β ( θ ′ )+ β ( m ( θ ′ ) θ ′′ ) β ( θ ) − β ( θ ′ m ( θ ′′ )) β ( θ ) . Since β i is a map of R -modules, we see that β ( m ( θ ) θ ′ ) β ( θ ′′ ) = β ( θ ′ ) β ( m ( θ ) θ ′′ ),so this term cancels with the first term of the second line. Similarly for the otherterms, so this composition is 0 as needed. Next: k ◦ c θ ∧ θ ′ ⊗ θ ′ θ (2)2 θ ′′′ ⊗ θ ! = β ( θ ′ ) β ( m ( θ θ ′ )) − β ( θ ) β ( m ( θ ′ θ ′ )) − β ( m ( θ ∧ θ ′ )) β ( θ ′ ) + β ( θ ∧ θ ′ ) β ( m ( θ ′ ))= β ( θ ′ ) β ( m ( θ ) θ ′ ) − β ( θ ′ ) β ( θ m ( θ ′ )) − β ( θ ) β ( m ( θ ′ ) θ ′ ) + β ( θ ) β ( θ ′ m ( θ ′ )) − β ( m ( θ ) ∧ θ ′ ) β ( θ ′ ) + β ( θ ∧ m ( θ ′ )) β ( θ ′ ) β ( θ ∧ θ ′ ) β ( m ( θ ′ ))= β ( θ ∧ θ ′ ) β ( m ( θ ′ )) + β ( θ ) β ( θ ′ ∧ m ( θ ′ ) − β ( θ ′ ) β ( θ ′ ∧ m ( θ ′ ))= c θ ∧ θ ′ ∧ m ( θ ′ ) m ( θ ) ⊗ θ + m ( θ ∧ θ ′ ) ⊗ θ ′ − θ ′′ ⊗ m ( θ ) m ( θ ′′ ) θ ! = c ◦ d θ ∧ θ ′ ⊗ θ ′ θ (2)2 θ ′′′ ⊗ θ ! . And, finally: k ◦ c (cid:18) θ ⊗ θ (2)2 θ ′ θ ′′ ⊗ θ (cid:19) ! = β ( θ ) β ( m ( θ (2)2 )) − β ( m ( θ θ )) β ( θ ) − β ( θ θ ) β ( m ( θ )) − β ( θ ′′ ) β ( m ( θ ′ θ )) + β ( θ ′ ) β ( m ( θ ′′ θ )) KELLER VANDEBOGERT − β ( m ( θ ′ ∧ θ ′′ )) β ( θ ) + β ( θ ′ ∧ θ ′′ ) β ( m ( θ ))= β ( θ ) β ( m ( θ ) θ ) − β ( m ( θ ) θ ) β ( θ ) + β ( θ m ( θ )) β ( θ ) − β ( θ θ ) β ( m ( θ )) − β ( θ ′′ ) β ( m ( θ ′ ) θ ) + β ( θ ′′ ) β ( θ ′ m ( θ ))+ β ( θ ′ ) β ( m ( θ ′′ ) θ ) − β ( θ ′ ) β ( θ ′′ m ( θ ))+ β ( m ( θ ′ ) θ ′′ ) β ( θ ) − β ( θ ′ m ( θ ′′ )) β ( θ )+ β ( θ ′ ∧ θ ′′ ) β ( m ( θ )) . Notice that β ( m ( θ ) θ ) β ( θ ) = m ( θ ) β ( θ ) β ( θ ) = 0, since β ( θ ) ∈ K .Moreover, β ( θ ′′ ) β ( m ( θ ′ ) θ ) = β ( m ( θ ′ ) θ ′′ ) β ( θ ), so this term cancels with thefirst term on the second line from the bottom. The same goes for β ( m ( θ ′ ) θ ′′ ) β ( θ ).Thus we are left with:= β ( θ ) β ( m ( θ ) θ ) + β ( θ m ( θ )) β ( θ ) − β ( θ θ ) β ( m ( θ )) + β ( θ ′ ∧ θ ′′ ) β ( m ( θ )) − β ( θ ′ ) β ( θ ′′ m ( θ )) + β ( θ ′′ ) β ( θ ′ m ( θ ))= c − θ ∧ m ( θ ) ⊗ θ + θ ′ ∧ θ ′′ ⊗ m ( θ ) m ( θ ) θ (2)2 m ( θ ′ ∧ θ ′′ ) ⊗ θ ! = c ◦ d (cid:18) θ ⊗ θ (2)2 θ ′ θ ′′ ⊗ θ (cid:19) ! . This shows that we have a morphism of complexes. (cid:3)
Proposition 4.3.
Adopt notation and hypotheses of Proposition 4.2. Then themap c satisfies: c (cid:16) M ∧ α ( K ) ⊗ M (cid:17) = 0 ,c (cid:16) ^ M ⊗ α ( K ) (cid:17) = 0 ,c (cid:0) θ ∧ θ ′ ⊗ α ( φ ) · θ (cid:1) = 0 ,c (cid:16) θ ∧ θ ′ ⊗ α ( φ ) θ ′′ + θ ′′ ∧ θ ′ ⊗ α ( φ ) θ (cid:17) = 0 . Additionally, the map c satisfies: c ( ^ M ∧ α ( K )) = 0 . Proof.
For the first formula, c ( θ ∧ α ( φ ) ⊗ θ ) = β ( α ( φ )) β ( θ θ ) − β ( θ ) β ( α ( φ ) θ ) − β ( θ ∧ α ( φ )) β ( θ ) . G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 9
Since β ◦ α = 0, the first term vanishes. For the second terms, β ( α ( φ ) θ ) = φ β ( θ ) ,β ( θ ∧ α ( φ )) = − β ( θ ) φ , so we see that these cancel out and we obtain 0. In the next case, c ( θ ∧ θ ′ ⊗ α ( φ )) = β ( θ ′ ) β ( θ α ( φ )) − β ( θ ) β ( θ ′ α ( φ )) − β ( θ ∧ θ ′ ) β ( α ( φ )) . In this case, the last term is 0. For the first two terms, β ( θ α ( φ )) = β ( θ ) φ ,β ( θ ′ α ( φ )) = β ( θ ′ ) φ , so these terms again cancel. For the last property of c : c (cid:16) θ ∧ θ ′ ⊗ α ( φ ) θ ′′ (cid:17) = β ( θ ′ ) β ( θ α ( φ ) θ ′′ ) − β ( θ ) β ( θ ′ α ( φ ) θ ′′ ) − β ( θ ∧ θ ′ ) β ( α ( φ ) θ ′′ )= − β ( θ ′ ) β ( θ ′′ α ( φ ) θ ) + β ( θ ) φ β ( θ ′ θ ′′ ) − β ( θ ∧ θ ′ ) φ β ( θ ′′ )= − β ( θ ′ ) β ( θ ′′ α ( φ ) θ ) + β ( θ ′′ ∧ θ ′ ) β ( α ( φ ) θ )+ β ( θ ′′ ) β ( θ α ( φ ) θ ′ )= − c (cid:16) θ ′′ ∧ θ ′ ⊗ α ( φ ) θ (cid:17) . Moreover, if we set θ = θ ′′ in the above, notice that the second equality in theabove computation becomes 0, so the penultimate property for c also holds.For the c property, we compute in similar fashion: c ( θ ∧ θ ′ ∧ α ( φ )) = β ( θ ∧ θ ′ ) β ( α ( φ )) − β ( θ ∧ α ( φ )) β ( θ ′ )+ β ( θ ′ ∧ α ( φ )) β ( θ )= − β ( θ ) φ β ( θ ′ ) + β ( θ ′ ) φ β ( θ )= 0 , where in the above we have used that β ◦ α = 0 and β i + j ( α i ( φ i ) θ j ) = φ i β j ( θ j ). (cid:3) Corollary 4.4.
Adopt notation and hypotheses of Proposition 4.3. Then thereexists a homotopy h : B → K [ − with c = kh + hd . Moreover, h may be chosen tosatisfy the following:(1) h restricted to any summand of each B i with fewer than terms in theproduct is identically .(2) h (cid:16) V M ∧ α ( K )) = 0 (3) h (cid:16) M ∧ α ( K ) ⊗ M (cid:17) = 0 (4) h (cid:16) V M ⊗ α ( K ) (cid:17) = 0 (5) h ( θ ∧ θ ′ ⊗ α ( φ ) · θ ) = 0 (6) h (cid:16) θ ∧ θ ′ ⊗ α ( φ ) θ ′′ + θ ′′ ∧ θ ′ ⊗ α ( φ ) θ (cid:17) = 0 Proof.
The existence of the homotopy follows from the fact that c is a morphismof complexes and K is acyclic. The fact that we may arrange h to have property(1) follows from the definition of c and Lemma 2.4. It is clear that V M ∧ α ( K )is a direct summand of B and M ∧ α ( K ) ⊗ M is a direct summand of B bythe splitting assumption M = α ( K ) ⊕ M , . This yields properties (1), (2), and(3). Assume now that h has been chosen to satisfy these properties.For property (4), applying the Tate differential yields: d ( θ ∧ θ ′ ⊗ α ( φ )) = m ( θ ) θ ′ ⊗ α ( φ ) − m ( θ ′ ) θ ⊗ α ( φ )+ θ ∧ θ ′ ⊗ α ( k ( φ )) ∈ (cid:16) M ⊗ M (cid:17) ⊕ ^ M ∧ α ( K ) . By our selection of h , we see that h ( d ( θ ∧ θ ′ ⊗ α ( φ )) = 0. Since c = hd + kh ,Proposition 4.3 combined with the previous sentence yields k ( h ( θ ∧ θ ′ ⊗ α ( φ ))) = 0and since k is injective, property (4) follows.For property (5), we again apply the Tate differential: d ( θ ∧ θ ′ ⊗ α ( φ ) · θ ) = m ( θ ) θ ′ ⊗ α ( φ ) θ − m ( θ ′ ) θ ⊗ α ( φ ) θ + k ( φ ) θ ∧ θ ′ ∧ θ − m ( θ ) θ ∧ θ ′ ∧ α ( φ ) ∈ (cid:16) M ⊗ M (cid:17) ⊕ ^ M ∧ α ( K ) , so that in an identical manner to property (4), we obtain (5). Finally, for property(6), assume that h has been chosen to satisfy property (5) as well. We simply let θ θ + θ ′ in (5) to obtain (6). (cid:3) Definition 4.5.
Adopt notation and hypotheses as in Corollary 4.4. Define h : V M ⊗ M → K by composing with the inclusion V M ⊗ M → B . Then,define X : V M → M , X t : M ⊗ M → M via X ( θ ∧ θ ′ ) · θ = ( β − ◦ h )( θ ∧ θ ′ ⊗ θ ) = θ ′ · X t ( θ ⊗ θ ) Proposition 4.6.
The maps X and X t of Definition 4.5 have the following prop-erties:(1) β X ( θ ∧ θ ′ ) = 0 , (2) β X t ( θ ⊗ θ ) = 0 , (3) X t ( θ ⊗ α ( φ )) = 0 and α ( φ ) · X ( θ ∧ θ ′ ) = 0 , (4) α ( φ ) θ ′′ · X ( θ ∧ θ ′ ) + α ( φ ) θ · X ( θ ′′ ∧ θ ′ ) = 0 . In particular, this implies θ ′′ · X ( θ ∧ θ ′ ) + θ · X ( θ ′′ ∧ θ ′ ) = 0 , (5) m X ( θ ∧ θ ′ ) = β ( θ ′ ) θ − β ( θ ) θ ′ − α β ( θ θ ′ ) , (6) X t ( θ ⊗ m ( θ )) = θ α β ( θ ) − β ( θ ) θ − α β ( θ θ ) , (7) X ( θ ∧ m ( θ )) + m X t ( θ ⊗ θ ) = α β ( θ θ ) − θ α β ( θ ) − β ( θ ) θ , G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 11 (8) X t ( θ ′ ⊗ m ( θ )) + X t ( θ ⊗ m ( θ ′ )) = α β ( θ θ ′ ) − α β ( θ ) θ ′ − α β ( θ ′ ) θ , (9) X t ( θ ⊗ X ( θ ′′ ∧ θ ′ )) + X t ( θ ′′ ⊗ X ( θ ∧ θ ′ )) = 0 . Proof. (1): β X ( θ ∧ θ ′ ) · φ = X ( θ ∧ θ ′ ) · α ( φ )= ( β − ◦ h )( θ θ ′ ⊗ α ( φ ))= 0 (by Corollary 4.4) . (2): φ · β X t ( θ ⊗ θ ) = − α ( φ ) · X t ( θ ⊗ θ )= − ( β − ◦ h )( α ( φ ) ∧ θ ⊗ θ )= 0 (by Corollary 4.4) . (3): For the first equality, θ ′ · X t ( θ ⊗ α ( φ )) = ( β − ◦ h )( θ ′ ∧ θ ⊗ α ( φ ))= 0 (by Corollary 4.4) . For the second equality, multiply by an arbitrary θ ∈ M : θα ( φ ) · X ( θ ∧ θ ′ ) = β − ◦ h ( θ ∧ θ ′ ⊗ θα ( φ ))= 0 (by Corollary 4.4) . (4): α ( φ ) θ ′′ · X ( θ ∧ θ ′ ) + α ( φ ) θ · X ( θ ′′ ∧ θ ′ )=( β − ◦ h )( θ ∧ θ ′ ⊗ α ( φ ) θ ′′ + θ ′′ ∧ θ ′ ⊗ α ( φ ) θ )=0 (by Corollary 4.4) . To prove the additional claim, apply m to the equality α ( φ ) θ · X ( θ ∧ θ ′ ) = 0Recalling that α is the identity, we obtain: k ( φ ) θ X ( θ ∧ θ ′ ) + α ( φ ) m ( θ ) X ( θ ∧ θ ′ )+ α ( φ ) θ m ( X ( θ ∧ θ ′ )) . Observe that m ( θ ) α ( φ ) X ( θ ∧ θ ′ ) = 0 by Property (3).We want to show that α ( φ ) θ m ( X ( θ ∧ θ ′ )) = 0. If we multiply by any α ( φ ′ ), then we must obtain 0 by property (3). Applying m and expanding usingthe Leibniz rule, m (cid:16) α ( φ ′ ) α ( φ ) θ m ( X ( θ ∧ θ ′ )) (cid:17) = k ( φ ′ ) α ( φ ) θ m ( X ( θ ∧ θ ′ )) − k ( φ ) α ( φ ′ ) θ m ( X ( θ ∧ θ ′ ))+ m ( θ ) α ( φ ′ ) α ( φ ) m ( X ( θ ∧ θ ′ )) . The last term is 0 by property (3) combined with the Leibniz rule, whence theabove shows that for all φ , φ ′ ∈ K , k ( φ ′ ) α ( φ ) θ m ( X ( θ ∧ θ ′ )) = k ( φ ) α ( φ ′ ) θ m ( X ( θ ∧ θ ′ ))Since φ and φ ′ are totally arbitrary and k ( K ) has grade >
2, we must have that α ( φ ) θ m ( X ( θ ∧ θ ′ )) = 0. Combining this with the above, we find k ( φ ) θ X ( θ ∧ θ ′ ) = 0. Since k ( φ ) may be chosen to be regular, θ X ( θ ∧ θ ′ ) = 0. Now let θ θ + θ ′′ to obtain the desired equality.(5): m X ( θ ∧ θ ′ ) · θ = − X ( θ ∧ θ ′ ) · m ( θ )= − ( β − ◦ h )( θ ∧ θ ′ ⊗ m ( θ ))= − ( β − ◦ h )( d ( θ ∧ θ ′ ⊗ θ ))= − β − ◦ c ( θ ∧ θ ′ ⊗ θ )= β − (cid:16) β ( θ ′ ) β ( θ θ ) − β ( θ ) β ( θ ′ θ )+ β ( θ ∧ θ ′ ) β ( θ ) (cid:17) = (cid:16) β ( θ ′ ) θ − β ( θ ) θ ′ − α β ( θ θ ′ ) (cid:17) θ . (6): This follows from (5) since: m X ( θ ∧ θ ′ ) · θ = θ ′ · X t ( θ ⊗ m ( θ ))= β ( θ ′ ) θ θ − β ( θ ) θ ′ θ − α β ( θ θ ′ ) θ = − θ ′ α β ( θ θ ) − θ ′ β ( θ ) θ + θ ′ · θ α β ( θ )= θ ′ · (cid:16) θ α β ( θ ) − β ( θ ) θ − α β ( θ θ ) (cid:17) . (7): (cid:16) X ( θ ∧ m ( θ )) + m X t ( θ ⊗ θ ) (cid:17) · θ ′ = ( β − ◦ h )( θ ∧ m ( θ ) ⊗ θ ′ + θ ∧ m ( θ ′ ) ⊗ θ )= ( β − ◦ h )( d ( − θ ⊗ θ · θ ′ ))= β − ◦ c ( − θ ⊗ θ · θ ′ )= β − (cid:16) − β ( θ ) β ( θ θ ′ ) + β ( θ θ ) β ( θ ′ ) + β ( θ θ ′ ) β ( θ ) (cid:17) = − β ( θ ) θ θ ′ + α β ( θ θ ) θ ′ + θ θ ′ α β ( θ )= (cid:16) − β ( θ ) θ + α β ( θ θ ) − θ ′ α β ( θ ) (cid:17) θ ′ . G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 13 (8): This follows from (7), since this is just the adjoint version. θ (cid:16) X t ( θ ′ ⊗ m ( θ )) + X t ( θ ⊗ m ( θ ′ )) (cid:17) = − X ( θ ∧ m ( θ )) · θ ′ − m X t ( θ ∧ θ ) · θ ′ = − β ( θ ) θ θ ′ + α β ( θ θ ) θ ′ + θ θ ′ α β ( θ )= + θ α β ( θ θ ′ ) − θ θ α β ( θ ′ ) − θ θ ′ α β ( θ )= θ · (cid:16) α β ( θ θ ′ ) − θ α β ( θ ′ ) − θ ′ α β ( θ ) (cid:17) . (9): Observe that it suffices to show X t ( θ ⊗ X ( θ ∧ θ ′ )) = 0, since we may thensubstitute θ θ + θ ′′ to obtain the general case. Multiplying by an arbitrary θ ∈ M , we obtain X ( θ ∧ θ ) X ( θ ∧ θ ′ )so it suffices to show this product is 0. Since this is an element of M and m isinjective, it suffices to show that m applied to the above is 0. We compute: m ( X ( θ ∧ θ ) X ( θ ∧ θ ′ ))= m ( X ( θ ∧ θ )) X ( θ ∧ θ ′ ) + X ( θ ∧ θ ) m ( X ( θ ∧ θ ′ ))= (cid:0) β ( θ ) θ − β ( θ ) θ − α β ( θ θ ) (cid:1) X ( θ ∧ θ ′ )+ (cid:0) β ( θ ′ ) θ − β ( θ ) θ ′ − α β ( θ θ ′ ) (cid:1) X ( θ ∧ θ ) (by property (5))= (cid:0) β ( θ ) θ − β ( θ ) θ (cid:1) X ( θ ∧ θ ′ )+ ( β ( θ ′ ) θ − β ( θ ) θ ′ ) X ( θ ∧ θ ) (by property (2))= − β ( θ ) (cid:0) θX ( θ ∧ θ ′ ) + θ ′ X ( θ ∧ θ ) (cid:1) (by property (4))=0 (again, by property (4)) . (cid:3) The Length 4 Big From Small Construction is a DG Algebra
Theorem 5.1.
Adopt notation and hypotheses of Setup 2.7. Then the complex F ( α, r ) of Definition 3.1 admits the structure of a commutative associative DG-algebra exhibiting Poincar´e duality via the following multiplication: F ⊗ F → F (cid:18) φ θ (cid:19) (cid:18) φ ′ θ ′ (cid:19) = φ φ ′ − α ( φ ) θ ′ − θ α ( φ ′ ) − rθ θ ′ + X ( θ ∧ θ ′ ) α ( θ ) φ ′ − α ( φ ′ ) θ + β ( θ θ ′ ) F ⊗ F → F (cid:18) φ θ (cid:19) φ θ φ ′ = (cid:18) θ α ( φ ) − [ φ φ ] K α ( h ) − α ( φ ) θ − rθ θ + X t ( θ ⊗ θ ) φ φ ′ − m ( θ ) φ − β ( θ θ ) (cid:19) F ⊗ F → F (cid:18) φ θ (cid:19) (cid:18) θ φ (cid:19) = [ φ φ ] K h − θ θ F ⊗ F → F
44 KELLER VANDEBOGERT φ θ φ φ ′ θ ′ φ ′ = [ φ φ ′ ] K h + [ φ φ ′ ] K h − θ θ ′ where h ∈ M is such that [ h ] M = 1 and the maps X and X t are defined inDefinition 4.5.Proof. We first show that associatvity holds for 3 elements of degree 1. Consider: (cid:18) φ θ (cid:19) (cid:18) φ ′ θ ′ (cid:19) (cid:18) φ ′′ θ ′′ (cid:19) ! − (cid:18) φ ′′ θ ′′ (cid:19) (cid:18) φ θ (cid:19) (cid:18) φ ′ θ ′ (cid:19) ! We first compute the top entry of the above: θ α ( φ ′ φ ′′ ) − ( θ ′′ α ( φ ′ ) − θ ′ α ( φ ′′ ) − rθ ′ θ ′′ + X ( θ ′ ∧ θ ′′ )) α ( φ ) − rθ ( θ ′′ α ( φ ′ ) − θ ′ α ( φ ′′ ) − rθ ′ θ ′′ + X ( θ ′ ∧ θ ′′ ))+ X t ( θ ⊗ ( θ ′′ α ( φ ′ ) − θ ′ α ( φ ′′ ) − rθ ′ θ ′′ + X ( θ ′ ∧ θ ′′ ))) − θ ′′ α ( φ φ ′ ) + ( θ ′ α ( φ ) − θ α ( φ ′ ) − rθ θ ′ + X ( θ ∧ θ ′ )) α ( φ ′′ )+ rθ ′′ ( θ ′ α ( φ ) − θ α ( φ ′ ) − rθ θ ′ + X ( θ ∧ θ ′ )) − X t ( θ ′′ ⊗ ( θ ′ α ( φ ) − θ α ( φ ′ ) − rθ θ ′ + X ( θ ∧ θ ′ ))) . After cancelling off the easy terms, we are left with: − X ( θ ′ ∧ θ ′′ ) α ( φ ) − rθ X ( θ ′ ∧ θ ′′ )+ X t ( θ ⊗ ( θ ′′ α ( φ ′ ) − θ ′ α ( φ ′′ ) − rθ ′ θ ′′ + X ( θ ′ ∧ θ ′′ )))+ X ( θ ∧ θ ′ ) α ( φ ′′ )+ rθ ′′ X ( θ ∧ θ ′ ) − X t ( θ ′′ ⊗ ( θ ′ α ( φ ) − θ α ( φ ′ ) − rθ θ ′ + X ( θ ∧ θ ′ ))) . Notice that X t ( θ ⊗ θ ′′ α ( φ ′ ) + θ ′′ ⊗ θ α ( φ ′ )) = 0 , since after taking the product with any other θ ∈ M , we obtain − β − ◦ h ( θ ∧ θ ⊗ θ ′′ α ( φ ′ ) + θ ′′ ∧ θ ⊗ θ α ( φ ′ )) , and this is 0 by Corollary 4.4. Two other terms like this cancel in a similar fashion.We are then left with: − X ( θ ′ ∧ θ ′′ ) α ( φ ) + X ( θ ∧ θ ′ ) α ( φ ′′ ) − rθ X ( θ ′ ∧ θ ′′ ) + rθ ′′ X ( θ ∧ θ ′ )+ X t ( θ ⊗ X ( θ ′ ∧ θ ′′ )) − X t ( θ ′′ ⊗ X ( θ ∧ θ ′ )) . After multiplying by θ ∈ M , the first term of the top line is: − β − ◦ h ( θ ′ ∧ θ ′′ ⊗ θα ( φ )) . G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 15
This is 0 by property (3) of Proposition 4.6. Similarly for the second term. For themiddle expression, this vanishes by Property (4) of Proposition 4.6, and the final 2terms vanish by Property (9) of 4.6.The expression in the bottom entry is then computed as: φ ( m ( θ ′ ) φ ′′ − m ( θ ′′ ) φ ′ + β ( θ ′ θ ′′ )) − m ( θ ) φ ′ φ ′′ − β ( θ ( θ ′′ α ( φ ′ ) − θ ′ α ( φ ′′ ) − rθ ′ θ ′′ + X ( θ ′ ∧ θ ′′ ))) − φ ′′ ( m ( θ ) φ ′ − m ( θ ′ ) φ + β ( θ θ ′ ))+ m ( θ ′′ ) φ φ ′ + β ( θ ′′ ( θ ′ α ( φ ) − θ α ( φ ′ ) − rθ θ ′ + X ( θ ∧ θ ′ ))) . Recall that β ◦ X = 0, so all X terms vanish. Using that β ( α ( φ i ) θ j ) = φ i β ( θ j ),more terms cancel in this way. The rest of the terms cancel without any specialrules.For the next associativity term, we take the product of 2 elements in degree 1and 1 element in degree 2 to obtain:[ φ ( φ ′ φ ′′ − m ( θ ′ ) φ − β ( θ ′ θ ))] h − θ (cid:0) θ ′ α ( φ ) − [ φ ′ φ ] m ( h ) − θ α ( φ ′ ) − rθ ′ θ + X t ( θ ′ θ ) (cid:1) − [ φ ( m ( θ ) φ ′ − m ( θ ′ ) φ + β ( θ θ ′ ))] h − [ φ ′′ φ φ ′ ] h + θ ( θ ′ α ( φ ) − θ α ( φ ′ ) − rθ θ ′ + X ( θ θ ′ )) . For the nontrivial cancellations, observe first that [ φ β ( θ ′ θ )] h = θ θ ′ α ( φ ); thiscancels with the first term on the bottom line. A second term cancels similarly.There are 4 terms leftover after all trivial cancellations. Firstly, there is the expres-sion − θ X t ( θ ′ ⊗ θ ) + θ X ( θ ∧ θ ′ ) , but this is zero by definition. Lastly, we have[ φ ′ φ ] θ m ( h ) − [ φ m ( θ ) φ ′ ] h. Since M = 0, θ m ( h ) = m ( θ ) h , so the above is 0. This proves associativity.For the Leibniz rule on two elements of degree 1, the top entry is k ( φ φ ′ ) + β ( θ ′ α ( φ )) − β ( θ α ( φ ′ )) − rβ ( θ θ ′ )+ β ( X ( θ θ ′ )) + rm ( θ ) φ ′ − rm ( θ ′ ) φ + rβ ( θ θ ′ ) − k ( φ ) φ ′ − β ( θ ) φ ′ − rm ( θ ) φ ′ + k ( φ ′ ) φ + β ( θ ′ ) φ + rm ( θ ′ ) φ . Recalling that β ◦ X = 0, all other terms cancel trivially. For the bottom entry,this is computed as: − m ( θ ′ α ( φ )) + m ( θ α ( φ ′ )) + rm ( θ θ ′ ) − m ( X ( θ θ ′ )) − α ( m ( θ ) φ ′ ) + α ( m ( θ ′ ) φ ) + α ( β ( θ θ ′ )) − k ( φ ) θ ′ − β ( θ ) θ ′ − rm ( θ ) θ ′ + k ( φ ′ ) θ + β ( θ ′ ) θ + rm ( θ ′ ) θ = − m ( X ( θ ∧ θ ′ ) + α ( β ( θ θ ′ )) − β ( θ ) θ ′ + β ( θ ′ ) θ . By property (5) of Proposition 4.6, this is 0. For the Leibniz rule on elements ofdegree 1 and 2, the top entry is computed as: β ( θ α ( φ )) − [ φ φ ] β ( m ( h )) − β ( θ α ( φ )) − rβ ( θ θ )+ β ( X t ( θ θ ) − rφ φ ′ + rm ( θ ) φ + rβ ( θ θ ) − k ( φ ) φ − β ( θ ) φ − rm ( θ ) φ + φ k ( φ ) + φ β ( θ ) + rφ φ ′ . Almost all terms cancel trivially. Recall that [ φ φ ] β ( m ( h )) = k ( φ φ ) to see thatthese terms cancel. The middle entry is − m ( θ α ( φ )) + [ φ φ ] m ( m ( h )) + m ( θ α ( φ )) + rm ( θ θ ) − m ( X t ( θ ⊗ θ )) − α ( φ φ ′ ) + α ( m ( θ ) φ ) + α ( β ( θ θ )) − k ( φ ) θ − β ( θ ) θ − rm ( θ ) θ − m ( θ ) α ( φ ) − α ( φ ′ ) α ( φ ) − θ α ( k ( φ )) − θ α ( β ( θ )) − rθ α ( φ ′ )+ rθ m ( θ ) + rθ α ( φ ′ ) − X ( θ ∧ m ( θ )) − X ( θ ∧ α ( φ ′ ))= − m ( X t ( θ ⊗ θ )) + α ( β ( θ θ )) − β ( θ ) θ − θ α ( β ( θ )) − X ( θ ∧ m ( θ )) . By Property (7) of Proposition 4.6, this is 0. The bottom entry is: k ( φ φ ′ ) − m ( θ ) k ( φ ) − k ( β ( θ θ ) − k ( φ ) φ ′ − β ( θ ) φ ′ − rm ( θ ) φ ′ m ( θ ) k ( φ ) + m ( θ ) β ( θ ) + rm ( θ ) φ ′ + m ( α ( φ ′ )) φ − β ( θ m ( θ )) − β ( θ α ( φ ′ )) , and these terms cancel without any additional properties. For the Leibniz rule onelements of degree 1 and 3, the top entry is:[ φ φ ] α ( β ( h )) − α ( β ( θ θ )) − r [ φ φ ] m ( h ) + rm ( θ θ ) − k ( φ ) θ − β ( θ ) θ − rm ( θ ) θ + θ α ( β ( θ )) − rθ α ( φ ) − [ φ β ( θ ) − rφ φ ] m ( h )+ m ( θ ) α ( φ ) + α ( φ ) α ( φ ) + rθ m ( θ ) + rθ α ( φ ) − X t ( θ ⊗ m ( θ )) − X t ( θ ⊗ α ( φ ))= − α ( β ( θ θ )) − β ( θ ) θ + θ α ( β ( θ )) − X t ( θ ⊗ m ( θ )) . By property (6) of Proposition 4.6, this is 0. The bottom entry is: − [ φ φ ] k ( β ( h )) + k ( β ( θ θ )) − k ( φ ) φ − β ( θ ) φ − rm ( θ ) φ
2G STRUCTURE ON THE LENGTH 4 BIG FROM SMALL CONSTRUCTION 17 + φ k ( φ ) − m ( θ ) β ( θ ) + rm ( θ ) φ + β ( θ m ( θ )) + β ( θ α ( φ )) . Again, recalling that β ( α ( φ ) θ ) = φ β ( θ ), the above terms cancel easily.For the Leibniz rule on elements of degree 1 and 4: − k ( φ ) θ − β ( θ ) θ − rm ( θ ) θ − [ φ k ( β ( θ ))] h − θ α ( β ( θ )) + rθ m ( θ ) . Notice [ φ kβ ( θ )] h = − k ( φ )[ θ ] h = − k ( φ ) θ and θ α ( β ( θ )) = β ( θ ) θ , so theseterms cancel. For the Leibniz rule on elements both of degree 2, the top entry is:[ φ ′ φ ] α ( β ( h )) + [ φ φ ′ ] α ( β ( h )) − α ( β ( θ θ ′ )) − r [ φ ′ φ ] m ( h ) − [ φ φ ′ ] rm ( h ) + rm ( θ θ ′ )+ m ( θ ) α ( φ ′ ) + α ( φ ) α ( φ ′ )+ [ rk ( φ ) φ ′ + β ( θ ) φ ′ + rφ φ ′ ] m ( h ) + θ ′ α ( k ( φ ))+ θ ′ α ( β ( θ )) + rθ ′ α ( φ ) − rm ( θ ) θ ′ − rα ( φ ) θ ′ + X t ( m ( θ ) ⊗ θ ′ ) + X t ( α ( φ ) ⊗ θ ′ )+ m ( θ ′ ) α ( φ ) + α ( φ ′ ) α ( φ )+ [ rk ( φ ′ ) φ + β ( θ ′ ) φ + rφ ′ φ ] m ( h ) + θ α ( k ( φ ′ ))+ θ α ( β ( θ ′ )) + rθ α ( φ ′ ) − rm ( θ ′ ) θ − rα ( φ ′ ) θ + X t ( m ( θ ′ ) ⊗ θ ) + X t ( α ( φ ′ ) ⊗ θ )= − α ( β ( θ θ ′ )) + θ ′ α ( β ( θ )) + X t ( m ( θ ) ⊗ θ ′ )+ θ α ( β ( θ ′ )) + X t ( m ( θ ′ ) ⊗ θ ) . By property (8) of Proposition 4.6, this final term is 0. The bottom entry is: − [ φ ′ φ ] k ( β ( h )) − [ φ φ ′ ] k ( β ( h )) + k ( β ( θ θ ′ )) − k ( φ ) φ ′ − β ( θ ) φ ′ − rφ φ ′ − m ( α ( φ )) φ ′ − β ( m ( θ ) θ ′ ) − β ( α ( φ ) θ ′ ) − k ( φ ′ ) φ − β ( θ ′ ) φ − rφ ′ φ − m ( α ( φ ′ )) φ − β ( m ( θ ′ ) θ ) − β ( α ( φ ′ ) θ ) , and everything cancels trivially, keeping in mind that [ φ φ ′ ] k ( β ( h )) = k ( φ φ ′ ).For the Leibniz rule on elements of degree 2 and 3:[ k ( φ ) φ ′ + β ( θ ) φ ′ + rφ φ ′ ] h + m ( θ ) θ + α ( φ ) θ + [ φ k ( φ ′ )] h + [ φ β ( θ ) − rφ φ ′ ] h + θ m ( θ ) + θ α ( φ ′ ) , and again, all of these terms cancel trivially. (cid:3) References [1] L. Avramov,
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