The singularity category of an algebra with radical square zero
aa r X i v : . [ m a t h . R T ] A p r THE SINGULARITY CATEGORY OF AN ALGEBRA WITHRADICAL SQUARE ZERO
XIAO-WU CHEN
Abstract.
To an artin algebra with radical square zero, a regular algebrain the sense of von Neumann and a family of invertible bimodules over theregular algebra are associated. These data describe completely, as a triangu-lated category, the singularity category of the artin algebra. A criterion onthe Hom-finiteness of the singularity category is given in terms of the valuedquiver of the artin algebra. Introduction
Let R be a commutative artinian ring. All algebras, categories and functors are R -linear. We recall that an R -linear category is Hom-finite provided that all theHom sets are finitely generated R -modules.Let A be an artin R -algebra. Denote by A -mod the category of finitely generatedleft A -modules, and by D b ( A -mod) the bounded derived category. Following [14],the singularity category D sg ( A ) is the quotient triangulated category of D b ( A -mod)with respect to the full subcategory formed by perfect complexes; see also [3, 12, 10,15, 2] and [13]. Here, we recall that a complex in D b ( A -mod) is perfect provided thatit is isomorphic to a bounded complex consisting of finitely generated projectivemodules.The singularity category measures the homological singularity of an algebra inthe sense that an algebra A has finite global dimension if and only if its singular-ity category D sg ( A ) vanishes. In the meantime, the singularity category capturesthe stable homological features of an algebra ([3]). A fundamental result of Buch-weitz and Happel states that for a Gorenstein algebra A , the singularity category D sg ( A ) is triangle equivalent to the stable category of (maximal) Cohen-Macaulay A -modules ([3, 10]). This implies in particular that the singularity category of aGorenstein algebra is Hom-finite and has Auslander-Reiten triangles. We point outthat Buchweitz and Happel’s result specializes to Rickard’s result ([15]) on self-injective algebras. However, for non-Gorenstein algebras, not much is known abouttheir singularity categories ([4]).Our aim is to describe the singularity category of an algebra with radical squarezero. We point out that such algebras are usually non-Gorenstein ([5]). In whatfollows, we describe the results in this paper.We denote by r the Jacobson radical of A . The algebra A is said to be with radicalsquare zero provided that r = 0. In this case, r has a natural A/ r - A/ r -bimodulestructure. Set r ⊗ = A/ r and r ⊗ i +1 = r ⊗ A/ r ( r ⊗ i ) for i ≥
0. Then there are obviousalgebra homomorphisms End A/ r ( r ⊗ i ) → End A/ r ( r ⊗ i +1 ) induced by r ⊗ A/ r − . We Date : April 20, 2011.1991
Mathematics Subject Classification.
Key words and phrases. singularity category, von Neumann regular algebra, invertible bimod-ule, valued quiver.The author is supported by Special Foundation of President of The Chinese Academy of Sci-ences (No.1731112304061) and National Natural Science Foundation of China (No.10971206).E-mail: [email protected]. denote by Γ( A ) the direct limit of this chain of algebra homomorphisms. It is aregular algebra ([7, 8]) in the sense of von Neumann. We call Γ( A ) the associatedregular algebra of A . In most cases, the algebra Γ( A ) is not semisimple.For n ∈ Z and i ≥ max { , n } , Hom A/ r ( r ⊗ i , r ⊗ i − n ) has a natural End A/ r ( r ⊗ i − n )-End A/ r ( r ⊗ i )-bimodule structure. Set K n ( A ) to be the direct limit of the chainof maps Hom A/ r ( r ⊗ i , r ⊗ i − n ) → Hom A/ r ( r ⊗ i +1 , r ⊗ i +1 − n ), which are induced by r ⊗ A/ r − . Then K n ( A ) is naturally a Γ( A )-Γ( A )-bimodule for each n ∈ Z . Observethat K ( A ) = Γ( A ) Γ( A ) Γ( A ) as bimodules, and that composition of maps inducesΓ( A )-Γ( A )-bimodule morphisms φ n,m : K n ( A ) ⊗ Γ( A ) K m ( A ) → K n + m ( A ) for all n, m ∈ Z . These bimodules K n ( A ) are called the associated bimodules of A .Recall that for an algebra Γ, a Γ-Γ-bimodule K is invertible provided that thefunctor K ⊗ Γ − induces an auto-equivalence on the category of left Γ-modules. Theorem A.
Let A be an artin algebra with radical square zero. Use the notationas above. Then the associated Γ( A ) - Γ( A ) -bimodules K n ( A ) are invertible and themaps φ n,m are isomorphisms of bimodules. Since the algebra Γ( A ) is regular, the category proj Γ( A ) of finitely generatedright projective Γ( A )-module is a semisimple abelian category. The invertible bi-module K ( A ) induces an auto-equivalence Σ A = − ⊗ Γ( A ) K ( A ) : proj Γ( A ) → proj Γ( A ). We observe that the category proj Γ( A ) has a unique triangulatedstructure with Σ A its shift functor; see Lemma 3.4. This unique triangulated cate-gory is denoted by (proj Γ( A ) , Σ A ).The following result describes the singularity category of an artin algebra withradical square zero, which is based on a result by Keller and Vossieck ([12]). Theorem B.
Let A be an artin algebra with radical square zero. Use the notationas above. Then we have a triangle equivalence D sg ( A ) ≃ (proj Γ( A ) , Σ A ) . We are interested in the Hom-finiteness of singularity categories. For this, werecall the notion of valued quiver of an artin algebra A . Choose a complete setof representatives of pairwise non-isomorphic simple A -modules { S , S , · · · , S n } .Set ∆ i = End A ( S i ); they are division algebras. Observe that Ext A ( S i , S j ) has anatural ∆ j -∆ i -bimodule structure. The valued quiver Q A of A is defined as follows:its vertex set is { S , S , · · · , S n } , here we identify each simple module S i with itsisoclass; there is an arrow from S i to S j whenever Ext A ( S i , S j ) = 0, in which casethe arrow is endowed with a valuation (dim ∆ j Ext A ( S i , S j ) , dim ∆ i op Ext A ( S i , S j ));here ∆ i op denotes the opposite algebra of ∆ i . We say that the valuation of Q A is trivial provided that all the valuations are (1 , resp. sink) provided that there is no arrow ending ( resp. starting)at it. For a valued quiver, to adjoin a (new) source ( resp. sink) is to add a vertextogether with some valued arrows starting ( resp. ending) at this vertex. For details,we refer to [1, III.1].The following result characterizes the Hom-finiteness of the singularity categoryin terms of valued quivers. Theorem C.
Let A be an artin algebra with radical square zero. Then the followingstatements are equivalent: (1) the singularity category D sg ( A ) is Hom-finite; (2) the associated regular algebra Γ( A ) is semisimple; (3) the valued quiver Q A is obtained from a disjoint union of oriented cycleswith the trivial valuation by repeatedly adjoining sources or sinks. HE SINGULARITY CATEGORY OF AN ALGEBRA WITH RADICAL SQUARE ZERO 3
The paper is structured as follows. In Section 2, we collect some basic facts onsingularity categories and recall a basic result of Keller and Vossieck. We proveTheorem A and B in Section 3, where an explicit example is presented. In Section4, we prove that one-point extensions and coextensions of algebras preserve theirsingularity categories. We introduce the notion of cyclicization of an algebra, whichis used in the proof of Theorem C in Section 5.For artin algebras, we refer to [1]. For triangulated categories, we refer to [11]and [9]. 2.
Preliminaries
In this section, we collect some facts on singularity categories of artin algebras.We recall a basic result due to Keller and Vossieck ([12]), which is applied to Ω ∞ -finite algebras.Let A be an artin algebra over a commutative artinian ring R . Recall that A -moddenotes the category of finitely generated left A -modules. We denote by A -proj thefull subcategory formed by projective modules, and by A -mod the stable category of A -mod modulo projective modules ([1, p.104]). The morphism space Hom A ( M, N )of two modules M and N in A -mod is defined to be Hom A ( M, N ) / p ( M, N ), where p ( M, N ) denotes the R -submodule formed by morphisms that factor through pro-jective modules.Recall that for an A -module M , its syzygy Ω( M ) is the kernel of its projec-tive cover P → M . This gives rise to the syzygy functor Ω : A -mod → A -mod([1, p.124]). Set Ω ( M ) = M and Ω i +1 ( M ) = Ω i (Ω( M )) for i ≥
0. Denote byΩ i ( A -mod) the full subcategory of A -mod formed by modules of the form P ⊕ Ω i ( M )for some module M and projective module P . Then an A -module X belongs toΩ i ( A -mod) if and only if there is an exact sequence 0 → X → P − i → · · · → P − → P with each P j projective.Recall that D b ( A -mod) denotes the bounded derived category of A -mod, whoseshift functor is denoted by [1]. For n ∈ Z , [ n ] denotes the n -th power of [1]. Themodule category A -mod is viewed as a full subcategory of D b ( A -mod) by identi-fying an A -module with the corresponding stalk complex concentrated at degreezero ([11, Proposition I.4.3]). Recall that a complex in D b ( A -mod) is perfect pro-vided that it is isomorphic to a bounded complex consisting of projective modules;these complexes form a full triangulated subcategory perf( A ). Recall that, via anobvious functor, perf( A ) is triangle equivalent to the bounded homotopy category K b ( A -proj); compare [3, 1.1-1.2].Following [14], we call the quotient triangulated category D sg ( A ) = D b ( A -mod) / perf( A )the singularity category of A . Denote by q : D b ( A -mod) → D sg ( A ) the quotientfunctor.The following two results are known; compare [14, Lemma 1.11] and [3, Lemma2.2.2]. Lemma 2.1.
Let X • be a complex in D sg ( A ) and n > . Then for any n largeenough, there exists a module M in Ω n ( A - mod) such that X • ≃ q ( M )[ n ] .Proof. Take a quasi-isomorphism P • → X • with P • a bounded above complex ofprojective modules ([11, Lemma I.4.6]). Take n ≥ n such that H i ( X • ) = 0 for all i ≤ n − n . Consider the good truncation σ ≥− n P • = · · · → → M → P − n → P − n → · · · of P • , which is quasi-isomorphic to P • . Then the cone of the obviouschain map σ ≥− n P • → M [ n ] is perfect, which becomes an isomorphism in D sg ( A ).This shows that X • ≃ q ( M )[ n ]. We observe that M lies in Ω n ( A -mod). (cid:3) XIAO-WU CHEN
Lemma 2.2.
Let → M → P − n → · · · → P → N → be an exact sequencewith each P i projective. Then we have an isomorphism q ( N ) ≃ q ( M )[ n ] in D sg ( A ) .In particular, for an A -module M , we have a natural isomorphism q (Ω n ( M )) ≃ q ( M )[ − n ] .Proof. The stalk complex N is quasi-isomorphic to · · · → → M → P − n → · · · → P → → · · · . This gives rise to a morphism N → M [ n ] in D b ( A -mod), whosecone is perfect. Then this morphism becomes an isomorphism in D sg ( A ). (cid:3) Consider the composite q ′ : A -mod ֒ → D b ( A -mod) q → D sg ( A ); it vanishes onprojective modules. Then it induces uniquely a functor A -mod → D sg ( A ), whichis still denoted by q ′ . Then Lemma 2.2 yields, for each n ≥
0, the followingcommutative diagram A -mod q ′ (cid:15) (cid:15) Ω n / / A -mod q ′ (cid:15) (cid:15) D sg ( A ) [ − n ] / / D sg ( A ) . We refer to [3, Lemma 2.2.2] for a similar statement.The functor q ′ induces a natural mapΦ : Hom A ( M, N ) → Hom D sg ( A ) ( q ( M ) , q ( N ))for any modules M, N . Let n ≥
1. Lemma 2.2 yields a natural isomorphism θ M : q ( M ) ∼ −→ q (Ω n ( M ))[ n ]. Then we have a mapΦ n : Hom A (Ω n ( M ) , Ω n ( N )) → Hom D sg ( A ) ( q ( M ) , q ( N ))given by Φ n ( f ) = ( θ nN ) − ◦ (Φ ( f )[ n ]) ◦ θ nM .Consider the chain of maps Hom A (Ω n ( M ) , Ω n ( N )) → Hom A (Ω n +1 ( M ) , Ω n +1 ( N ))induced by the syzygy functor. It is routine to verify that Φ n are compatible withthis chain of maps. Then we have an induced mapΦ : lim −→ Hom A (Ω n ( M ) , Ω n ( N )) −→ Hom D sg ( A ) ( q ( M ) , q ( N )) . We recall the following basic result.
Proposition 2.3. (Keller-Vossieck)
Let
M, N be A -modules as above. Then themap Φ is an isomorphism.Proof. The statement follows from [12, Exemple 2.3]. We refer to [2, Corollary3.9(1)] for a detailed proof. (cid:3)
Recall that an additive category A is idempotent split provided that each idem-potent e : X → X splits, that is, it admits a factorization X u → Y v → X with u ◦ v = Id Y . For example, a Krull-Schmidt category is idempotent split ([6, Ap-pendix A]). In particular, for an artin algebra A , the stable category A -mod isidempotent split. Corollary 2.4.
The singularity category D sg ( A ) of an artin algebra A is idempo-tent split.Proof. By Lemma 2.1 it suffices to show that for each module M , an idempotent e : q ( M ) → q ( M ) splits. The above proposition implies that for a large n , there isan idempotent e n : Ω n ( M ) → Ω n ( M ) in A -mod which is mapped by Φ to e . Theidempotent e n splits as Ω n ( M ) u → Y v → Ω n ( M ) with u ◦ v = Id Y in A -mod. Thenthe idempotent e factors as q ( M ) ( q ( u )[ n ]) ◦ θ nM −→ q ( Y )[ n ] ( θ nM ) − ◦ ( q ( v )[ n ]) −→ q ( M ). (cid:3) HE SINGULARITY CATEGORY OF AN ALGEBRA WITH RADICAL SQUARE ZERO 5
Let A be an additive category. For a subcategory C , denote by add C the fullsubcategory of A formed by direct summands of finite direct sums of objects in C . For any algebra Γ, denote by proj Γ the category of finitely generated rightprojective Γ-modules. We observe that proj Γ = add Γ Γ .An artin algebra A is called Ω ∞ -finite provided that there exists a module E and n ≥ n ( A -mod) ⊆ add ( A ⊕ E ). In this case, we call E an Ω ∞ -generator of A . Proposition 2.5.
Let A be an Ω ∞ -finite algebra with an Ω ∞ -generator E . Thenwe have D sg ( A ) = add q ( E ) . Consequently, we have an equivalence of categories D sg ( A ) ≃ proj End D sg ( A ) ( q ( E )) , which sends q ( E ) to End D sg ( A ) ( q ( E )) .Proof. Observe that Ω n +1 ( A -mod) ⊆ Ω n ( A -mod). Then we may assume thatadd ( A ⊕ E ) ⊇ add Ω n ( A -mod) = add Ω n +1 ( A -mod) = · · · for n large enough.For the first statement, it suffices to show that each object X • in D sg ( A ) belongsto add q ( E ). By Lemma 2.1, X • ≃ q ( M )[ n ] for a module M ∈ Ω n ( A -mod) and n >
0. Since add Ω n ( A -mod) = add Ω n + n ( A -mod), we may assume that M ⊕ N ∈ Ω n + n ( A -mod) for some module N . Take an exact sequence 0 → M ⊕ N → P − n → · · · → P → L → P i projective and L ∈ Ω n ( A -mod).By Lemma 2.2, q ( L ) ≃ q ( M ⊕ N )[ n ] and then X • is a direct summand of q ( L ).Observing that L ∈ add ( A ⊕ E ), we are done with the first statement.The second statement follows from the projectivization; see [1, PropositionII.2.1]. The functor is given by Hom D sg ( A ) ( q ( E ) , − ). We point out that Corol-lary 2.4 is needed here. (cid:3) Algebras with radical square zero
In this section, we study the singularity category of an algebra with radicalsquare zero, and prove Theorem A and B. An explicit example is given at the end.Let A be an artin algebra. Denote by r the Jacobson radical of A . The algebra A is said to be with radical square zero provided that r = 0. In this case, r has an A/ r - A/ r -bimodule structure, which is induced from the multiplication of A .Denote by A -ssmod the full subcategory of A -mod formed by semisimple mod-ules. We observe that r ⊗ A/ r S = 0 for a simple projective module S . Then thefunctor r ⊗ A/ r − : A -ssmod → A -ssmod is well defined. We observe that the syzygyfunctor Ω sends semisimple modules to semisimple modules, and then we have therestricted functor Ω : A -ssmod → A -ssmod.The following result is implicitly contained in the proof of [1, Lemma X.2.1]. Lemma 3.1.
There is a natural isomorphism Ω ≃ r ⊗ A/ r − of functors on A - ssmod .Proof. Let X be a semisimple module with a projective cover P → X . Tensoring P with the natural exact sequence of A - A -bimodules 0 → r → A → A/ r → X ) ≃ r ⊗ A P . Using isomorphisms r ⊗ A P ≃ r ⊗ A/ r P/ r P and P/ r P ≃ X , weget an isomorphism Ω( X ) ≃ r ⊗ A/ r X . It is routine to verify that this isomorphismis natural in X . (cid:3) Recall that an algebra Γ is regular in the sense of von Neumann provided thatfor each element a there exists a ′ such that aa ′ a = a . For example, a semisimplealgebra is regular. Then a direct limit of semisimple algebras is regular. For details,we refer to [7, Theorem and Definition 11.24].Recall that for an artin algebra A with radical square zero, there is a chain ofalgebra homomorphisms End A/ r ( r ⊗ i ) → End A/ r ( r ⊗ i +1 ) induced by r ⊗ A/ r − . Here, r ⊗ = A/ r and r ⊗ i +1 = r ⊗ A/ r ( r ⊗ i ). We set Γ( A ) to be the direct limit of this XIAO-WU CHEN chain. Since each algebra End A/ r ( r ⊗ i ) is semisimple, the algebra Γ( A ) is regular.It is called the associated regular algebra of A . We refer to [8, 19.26B, Example] fora related construction.We recall the associated Γ( A ) - Γ( A ) -bimodules K n ( A ) of A , n ∈ Z . For i ≥ max { , n } , Hom A/ r ( r ⊗ i , r ⊗ i − n ) has a natural End A/ r ( r ⊗ i − n )-End A/ r ( r ⊗ i )-bimodulestructure. Consider a chain of maps Hom A/ r ( r ⊗ i , r ⊗ i − n ) → Hom A/ r ( r ⊗ i +1 , r ⊗ i +1 − n ),which are induced by r ⊗ A/ r − , and define K n ( A ) to be its direct limit. Then K n ( A ) is naturally a Γ( A )-Γ( A )-bimodule for each n ∈ Z . Observe that K ( A ) = Γ( A ) Γ( A ) Γ( A ) as Γ( A )-Γ( A )-bimodules. Proposition 3.2.
Let A be an artin algebra with radical square zero. Then thereis a natural isomorphism K n ( A ) ≃ Hom D sg ( A ) ( q ( A/ r ) , q ( A/ r )[ n ]) for each n ∈ Z .Proof. Consider the case n ≤ q ( A/ r )[ n ] ≃ q (Ω − n ( A/ r )) ≃ q ( r ⊗− n ). Then Proposition 2.3 yields an isomor-phism Hom D sg ( A ) ( q ( A/ r ) , q ( A/ r )[ n ]) ≃ lim −→ Hom A (Ω i ( A/ r ) , Ω i ( r ⊗− n )). By Lemma3.1 again we have Ω i ( A/ r ) ≃ r ⊗ i and Ω i ( r ⊗− n ) = r ⊗ i − n . Then we have a surjec-tive map ψ : K n ( A ) → Hom D sg ( A ) ( q ( A/ r ) , q ( A/ r )[ n ]). On the other hand, everymorphism f : r ⊗ i → r ⊗ i − n that is zero in A -mod necessarily factors through asemisimple projective module. However, the functor r ⊗ A/ r − vanishes on semisim-ple projective modules. Then r ⊗ A/ r f is zero. This forces that ψ is injective. Weare done in this case.For the case n >
0, we observe that Hom D sg ( A ) ( q ( A/ r ) , q ( A/ r )[ n ]) is isomor-phic to Hom D sg ( A ) ( q ( A/ r )[ − n ] , q ( A/ r )), and by the same argument as above, it isisomorphic to lim −→ Hom A/ r ( r ⊗ i + n , r ⊗ i ). Then we get a surjective map K n ( A ) → Hom D sg ( A ) ( q ( A/ r ) , q ( A/ r )[ n ]). Similarly as above, we have that this map is injec-tive. (cid:3) Remark 3.3.
In the case n = 0, the above isomorphism is an isomorphism Γ( A ) ≃ End D sg ( A ) ( q ( A/ r )) of algebras. Then for an arbitrary n , the above isomorphismbecomes an isomorphism of Γ( A )-Γ( A )-bimodules.Recall that an abelian category A is semisimple provided that each short exactsequence splits. For example, for a regular algebra Γ, the category proj Γ of finitelygenerated right projective Γ-modules is a semisimple abelian category. Here, weuse the fact that all finitely presented Γ-modules are projective; see [7, Theoremand Definition 11.24(a)].The following observation is well known. Lemma 3.4.
Let A be a semisimple abelian category, and let Σ be an auto-equivalence on A . Then there is a unique triangulated structure on A with Σ theshift functor. The obtained triangulated category in this lemma will be denoted by ( A , Σ).
Proof.
We use the fact that each morphism in A is isomorphic to a direct sum ofmorphisms of the forms K → I Id I → I and 0 → C . Then all possible trianglesare a direct sum of the following trivial triangles K → → Σ( K ) Id Σ( K ) → Σ( K ), I Id I → I → → Σ( I ) and 0 → C Id C → C → Σ(0). (cid:3)
Proposition 3.5.
Let A be an artin algebra with radical square zero and let Γ( A ) be its associated regular algebra. Then there is a triangle equivalence Ψ : D sg ( A ) ≃ (proj Γ( A ) , Σ) HE SINGULARITY CATEGORY OF AN ALGEBRA WITH RADICAL SQUARE ZERO 7 for some auto-equivalence Σ on proj Γ( A ) , which sends q ( A/ r ) to Γ( A ) .Proof. We observe that for any A -module M , its syzygy Ω( M ) is semisimple. Hencewe have Ω ( A -mod) ⊆ add ( A ⊕ A/ r ). We apply Proposition 2.5 to obtain anequivalence of categories D sg ( A ) ≃ proj End D sg ( A ) ( q ( A/ r )). By Proposition 3.2this yields an equivalence of categories D sg ( A ) ≃ proj Γ( A ).By transport of structures, the shift functor [1] on D sg ( A ) corresponds to anauto-equivalence Σ on proj Γ( A ), and then proj Γ( A ) becomes a triangulated cat-egory. However, by Lemma 3.4 the semisimple abelian category proj Γ( A ) has aunique triangulated structure with Σ the shift functor. Then this structure neces-sarily coincides with the transported one. Then we are done. (cid:3) We are interested in the auto-equivalence Σ above. The following result charac-terizes it using the bimodules K n ( A ). Lemma 3.6.
Use the notation as above. Then for each n ∈ Z , the auto-equivalence Σ n is isomorphic to − ⊗ Γ( A ) K n ( A ) : proj Γ( A ) → proj Γ( A ) .Proof. Recall that the above equivalence Ψ is given by Hom D sg ( A ) ( q ( A/ r ) , − ),which sends q ( A/ r ) to Γ( A ). The auto-equivalence Σ n corresponds, via Ψ, to [ n ] on D sg ( A ). Then by Proposition 3.2 we have an isomorphism φ : K n ( A ) ∼ −→ Σ n (Γ( A ))of right Γ( A )-modules. Recall that Σ n (Γ( A )) has a natural Γ( A )-Γ( A )-bimodulestructure such that Σ n is isomorphic to − ⊗ Γ( A ) Σ n (Γ( A )). Thanks to Remark 3.3,the isomorphism φ is an isomorphism of bimodules. This proves the lemma. (cid:3) Recall that for an algebra Γ, a Γ-Γ-bimodule K is invertible provided that thefunctor − ⊗ Γ K induces an auto-equivalence on the category of right Γ-modules.For details, we refer to [7, Definition and Proposition 12.13].We recall that for an artin algebra A with radical square zero, the associ-ated Γ( A )-Γ( A )-bimodules K n ( A ) are defined to be lim −→ Hom A/ r ( r ⊗ i , r ⊗ i − n ), where i ≥ max { , n } . Then composition of maps between the A/ r -modules r ⊗ j yieldsmorphisms φ n,m : K n ( A ) ⊗ Γ( A ) K m ( A ) −→ K n + m ( A )of Γ( A )-Γ( A )-bimodules, for all n, m ∈ Z . More precisely, let f ∈ K n ( A ) and g ∈ K m ( A ) be represented by f ′ : r ⊗ j − m → r ⊗ j − m − n and g ′ : r ⊗ j → r ⊗ j − m forsome large j , respectively. Then φ n,m ( f ⊗ g ) is represented by the composite f ′ ◦ g ′ .The following result is Theorem A. Theorem 3.7.
Let A be an artin algebra with radical square zero. Use the notationas above. Then for all n, m ∈ Z , the Γ( A ) - Γ( A ) -bimodules K n ( A ) are invertibleand the morphisms φ n,m are isomorphisms.Proof. By Lemma 3.6 the functor − ⊗ Γ( A ) K n ( A ) : proj Γ( A ) → proj Γ( A ) is anauto-equivalence for each n . This functor extends naturally to an auto-equivalenceon the category of all right Γ( A )-modules. Then K n ( A ) is an invertible bimodule.The second statement follows from Lemma 3.6 and the fact that Σ m Σ n is isomorphicto Σ n + m . Here, we use [7, Proposition 12.9] implicitly. (cid:3) We now have Theorem B. Denote the functor − ⊗ Γ( A ) K ( A ) : proj Γ( A ) → proj Γ( A ) by Σ A . Theorem 3.8.
Let A be an artin algebra with radical square zero. Use the notationas above. Then we have a triangle equivalence D sg ( A ) ≃ (proj Γ( A ) , Σ A ) , which sends q ( A/ r ) to Γ( A ) .Proof. It follows from Proposition 3.5 and Lemma 3.6. (cid:3)
XIAO-WU CHEN
Let A be an artin algebra with radical square zero. For each n ≥
1, we considerthe artin algebra G n = A/ r ⊕ r ⊗ n , which is the trivial extension of the A/ r - A/ r -bimodule r ⊗ n ([1, p.78]). All these algebras G n have radical square zero.The following observation seems to be of independent interest. Proposition 3.9.
Use the notation as above. Then for each n ≥ , we have atriangle equivalence D sg ( G n ) ≃ (proj Γ( A ) , Σ nA ) . In particular, we have a triangle equivalence D sg ( A ) ≃ D sg ( G ) .Proof. Write G n = A ′ . Then from the very definition, we have a natural identi-fication Γ( A ′ ) = Γ( A ). Moreover, the Γ( A ′ )-Γ( A ′ )-bimodule K ( A ′ ) correspondsto the Γ( A )-Γ( A )-bimodule K n ( A ). Then by Lemma 3.6 Σ A ′ corresponds to Σ nA .Then the result follows from Theorem 3.8 immediately. (cid:3) Remark 3.10.
We point out that for n ≥ D sg ( G n ) might not be triangleequivalent to D sg ( A ), although the underlying categories are equivalent.We conclude this section with an example. Example 3.11.
Let k be a field and let n ≥ . Consider the algebra A = k [ x , x , · · · , x n ] / ( x i x j , ≤ i, j ≤ n ) , which is with radical square zero. We iden-tify A/ r with k , and r with the n -dimensional k -space V = kx ⊕ kx ⊕ · · · ⊕ kx n .Consequently, for each i ≥ , the algebra End A/ r ( r ⊗ i ) is isomorphic to End k ( V ⊗ i ) ,which is identified with the n i × n i total matrix algebra M n i ( k ) . Then the associ-ated regular algebra Γ( A ) is isomorphic to the direct limit of the following chain ofalgebra embeddings k −→ M n ( k ) −→ M n ( k ) −→ M n ( k ) −→ · · · Here, for each algebra B , B → M n ( B ) is the algebra embedding sending b to bI n with I n the n × n identity matrix.We observe that Γ( A ) is a simple algebra. We point out that this constructionis classical; see [8, 19.26 B, Example] .Let ≤ r, s ≤ n . Define E rs : V → V to be the linear map such that E rs ( x i ) = δ i,s x r , where δ is the Kronecker symbol. Consider, for all i ≥ , the linear maps −⊗ k E rs : End k ( V ⊗ i ) → End k ( V ⊗ i +1 ) . Taking the limit, we have the induced linearmap − ⊗ k E rs : Γ( A ) → Γ( A ) for each pair of r, s . Then we have an isomorphism σ : M n ( A ) → A of algebras, which sends an n × n matrix ( a ij ) to P ≤ i,j ≤ n a ij ⊗ k E ij .The associated Γ( A ) - Γ( A ) -bimodule K ( A ) is described as follows. As a k -space, K ( A ) = Γ( A ) ⊕ Γ( A ) ⊕ · · · Γ( A ) with n copies of Γ( A ) . The left actionis given by a ( a , a , · · · , a n ) = ( aa , aa , · · · , aa n ) , while the right action is givenby ( a , a , · · · , a n ) a = ( a , a , · · · , a n ) σ − ( a ) .We remark that the regular algebra Γ( A ) is related to a quotient abelian categorystudied in [16] , which might relate to the singularity category D sg ( A ) via a versionof Koszul duality. One-point (co)extensions and cyclicizations of algebras
In this section, we prove that one-point extensions and coextensions of algebraspreserve their singularity categories. We then introduce the notion of cyclicizationof an algebra, which is a repeated operation to remove sources and sinks on thevalued quiver. The obtained result will be used in the proof of Theorem C.Let A be an artin algebra. Let D be a simple artin algebra and let A M D bean A - D -bimodule, on which R acts centrally. The one-point extension of A by M is the upper triangular matrix algebra A [ M ] = (cid:18) A M D (cid:19) . A left A [ M ]-module HE SINGULARITY CATEGORY OF AN ALGEBRA WITH RADICAL SQUARE ZERO 9 is denoted by a column vector (cid:18) XV (cid:19) φ , where X and V are a left A -module and D -module, respectively, and that φ : M ⊗ D V → X is a morphism of A -modules.We sometimes suppress the morphism φ , when it is clearly understood. For details,we refer to [1, III.2].Recall from [1, III.1] the notion of valued quiver Q A for an artin algebra A . Weobserve that for the unique simple D -module S , the corresponding A [ M ]-module (cid:18) S (cid:19) is simple injective, which corresponds to a source in the valued quiver Q A [ M ] of the one-point extension A [ M ]. Indeed, this valued quiver is obtained from Q A by adding this source together with some valued arrows starting at it.One-point extensions of algebras preserve singularity categories. Observe thenatural exact embedding i : A -mod → A [ M ]-mod, which sends A X to i ( X ) = (cid:18) X (cid:19) . Proposition 4.1.
Let A [ M ] be the one-point extension of A as above. Then theexact embedding i : A - mod → A [ M ] - mod induces a triangle equivalence D sg ( A ) ≃ D sg ( A [ M ]) . Proof.
The exact functor i extends naturally to a triangle functor i ∗ : D b ( A -mod) → D b ( A [ M ]-mod). We observe that i ( A ) is projective, and then i ∗ sends perfectcomplexes to perfect complexes. Then it induces a triangle functor ¯ i ∗ : D sg ( A ) → D sg ( A [ M ]). We claim that ¯ i ∗ is an equivalence.For the claim, recall that the functor i admits a right adjoint j : A [ M ]-mod → A -mod which sends (cid:18) XV (cid:19) φ to X/ Im φ . Observe that the corresponding counit ji ∼ −→ Id A -mod is an isomorphism . One checks that the cohomological di-mension ([11, p.57]) of j is at most one. In particular, the left derived functor L b j : D b ( A [ M ]-mod) → D b ( A -mod) is defined. Moreover, we have the adjoint pair( L b j, i ∗ ), and the counit is an isomorphism. Since the functor j sends projectivemodules to projective modules, the functor L b j preserves perfect complexes. Thenit induces a triangle functor L b ¯ j : D sg ( A [ M ]) → D sg ( A ). Moreover, we have the in-duced adjoint pair ( L b ¯ j, ¯ i ∗ ), whose counit ( L b ¯ j ) ¯ i ∗ ∼ −→ Id D sg ( A ) is an isomorphism;see [14, Lemma 1.2]. In particular, the functor ¯ i ∗ is fully faithful.It remains to show the denseness of ¯ i ∗ . We now view the essential image Im ¯ i ∗ of ¯ i ∗ as a full triangulated subcategory of D sg ( A [ M ]). It suffices to show that foreach A [ M ]-module (cid:18) XV (cid:19) , its image in D sg ( A [ M ]) lies in Im ¯ i ∗ ; see Lemma 2.1.Observe that Ω( (cid:18) V (cid:19) ) lies in Im i , and then by Lemma 2.2, q ( (cid:18) V (cid:19) ) lies in Im ¯ i ∗ .The following natural exact sequence induces a triangle in D sg ( A [ M ])0 −→ i ( X ) −→ (cid:18) XV (cid:19) −→ (cid:18) V (cid:19) −→ . This triangle implies that q ( (cid:18) XV (cid:19) ) lies in Im ¯ i ∗ . Then we are done. (cid:3) Let D be a simple artin algebra, and let D N A be a D - A -bimodule, on which R acts centrally. The one-point coextension of A by N is the upper triangular matrixalgebra [ N ] A = (cid:18) D N A (cid:19) . A left [ N ] A -module is written as (cid:18) VX (cid:19) φ , where D V and A X are a left D -module and A -module, respectively, and that φ : M ⊗ A X → V is amorphism of D -modules. The valued quiver Q [ N ] A is obtained from Q A by adding a sink together with some valued arrows ending at it, where the sink correspondsto the simple projective [ N ] A -module (cid:18) S (cid:19) for a simple D -module S .For the one-point coextension [ N ] A , we have an exact embedding i : A -mod → [ N ] A -mod, which sends A X to i ( X ) = (cid:18) X (cid:19) .The following result is similar to Proposition 4.1, while the proof is simpler. Thisresult is closely related to [4, Theorem 4.1(1)]. Proposition 4.2.
Let [ N ] A be the one-point coextension as above. Then the em-bedding i : A - mod → [ N ] A - mod induces a triangle equivalence D sg ( A ) ≃ D sg ([ N ] A ) . Proof.
We observe that i ( A ) has projective dimension at most one. Then the ob-viously induced functor i ∗ : D b ( A -mod) → D b ([ N ] A -mod) preserves perfect com-plexes, and it induces the required functor ¯ i ∗ : D sg ( A ) → D sg ([ N ] A ).The functor i admits an exact left adjoint j : [ N ] A -mod → A -mod, which sends (cid:18) VX (cid:19) to X ; moreover, j preserves projective modules. Then it induces a trianglefunctor ¯ j ∗ : D sg ([ N ] A ) → D sg ( A ), which is left adjoint to ¯ i ∗ . Then as in the proofof Proposition 4.1, we have that ¯ i ∗ is fully faithful. The denseness of ¯ i ∗ follows fromthe natural exact sequence0 −→ (cid:18) V (cid:19) −→ (cid:18) VX (cid:19) −→ i ( X ) −→ , for each [ N ] A -module (cid:18) VX (cid:19) , and that the module (cid:18) V (cid:19) is projective. We omit thedetails. (cid:3) We use the above two propositions to reduce the study of the singularity categoryof arbitrary artin algebras to cyclic-like ones.Let A be an artin algebra. Consider the valued quiver Q A . A vertex e is called cyclic provided that there is an oriented cycle containing it, and the correspondingsimple A -module is called cyclic . More generally, a vertex e is called cyclic-like provided that there is a path through e , which starts with a cyclic vertex and endsat a cyclic vertex, while the corresponding simple A -module is called cyclic-like .An artin algebra A is called cyclic-like provided that its valued quiver Q A is cyclic-like. This is equivalent to that A has neither simple projective nor simple injectivemodules.For an artin algebra A , its cyclicization is an artin algebra A c which is eithersimple or cyclic-like, such that there is a sequence A c = A , A , · · · , A r = A witheach A i +1 is a one-point (co)extension of A i .The following is an immediate consequence of the definition. Lemma 4.3.
Let A be an artin algebra with its cyclicization A c . Then we have atriangle equivalence D sg ( A c ) ≃ D sg ( A ) . Proof.
Apply Propositions 4.1 and 4.2, repeatedly. (cid:3)
The following result seems to be well known.
Lemma 4.4.
The following statements hold. (1)
Each artin algebra has a cyclicization. (2)
Let A c and A c ′ be two cyclicizations of A . Then if A c is simple, so is A c ′ .Otherwise, we have an isomorphism A c ≃ A c ′ of algebras. HE SINGULARITY CATEGORY OF AN ALGEBRA WITH RADICAL SQUARE ZERO 11
Proof. (1) It follows from the well-known fact that the existence of a simple injec-tive ( resp. projective) module of A implies that A is a one-point extension ( resp. coextension) of A ′ . Moreover, the valued quiver Q A ′ of A ′ is obtained from the oneof A by deleting the relevant source ( resp. sink).(2) The first statement follows from the observation that passing from A to A ′ in (1), the set of cyclic-like vertices stays the same.For the isomorphism of algebras, it suffices to observe that A c -mod is equiva-lent to the smallest Serre subcategory ([7, Chapter 15]) of A -mod containing thecyclic-like simple A -modules S ; moreover, the multiplicity of P A c ( S ) in the in-decomposable decomposition of A c equals the multiplicity of P ( S ) in the one of A . Here, P ( S ) and P A c ( S ) denote the projective cover of S as an A -module and A c -module, respectively. (cid:3) Hom-finiteness of singularity categories
In this section, we study the Hom-finiteness of the singularity category of anartin algebra with radical square zero, and prove Theorem C.Throughout, A is an artin R -algebra such that its Jacobson radical r satisfies r = 0. Recall that in this case, the syzygy Ω( X ) of any module X is semisimple. Lemma 5.1.
Suppose that A is cyclic-like. Then we have (1) each simple A -module has infinite projective dimension; (2) for each i ≥ , the algebra homomorphism End A/ r ( r ⊗ i ) → End A/ r ( r ⊗ i +1 ) induced by r ⊗ A/ r − is injective.Proof. (1) Recall that a cyclic-like algebra does not have simple projective modules.Then the statement follows from the observation that for a simple module S withfinite projective dimension, we have that proj . dim Ω( S ) = proj . dim S − A -ssmod is the full subcategory of A -mod consisting of semisim-ple modules. Then by (1), A -ssmod is naturally equivalent to A -ssmod, and thesyzygy functor Ω : A -ssmod → A -ssmod is faithful. Now the result follows fromLemma 3.1. (cid:3) Recall that the singularity category D sg ( A ) is naturally R -linear. We are inter-ested in the problem when it is Hom-finite, that is, all the Hom sets are finitelygenerated R -modules. Theorem 5.2.
Let A be an artin algebra with radical square zero. Then the fol-lowing statements are equivalent: (1) the singularity category D sg ( A ) is Hom-finite; (2) the associated regular algebra Γ( A ) is semisimple; (3) the cyclicization A c of A is either simple or isomorphic to a finite productof self-injective algebras. We point out that the cyclicization A c of A is necessarily with radical squarezero. Recall that an indecomposable non-simple artin algebra with radical squarezero is self-injective if and only if its valued quiver is an oriented cycle with thetrivial valuation; see [1, Proposition IV.2.16] or the proof of [5, Corollary 1.3]. Thenthe statement (3) above is equivalent to the corresponding one in Theorem C. Proof.
Recall from Proposition 3.2 the isomorphism Γ( A ) ≃ End D sg ( A ) ( q ( A/ r )).Then we have the implication “(1) ⇒ (2)”, since an artin regular algebra is neces-sarily semisimple.For “(2) ⇒ (1)”, consider the cyclicization A c of A , whose Jacobson radical isdenoted by r c . Then by Lemma 4.3 we have an equivalence D sg ( A c ) ≃ D sg ( A ).Applying Proposition 3.5 we have an equivalence proj Γ( A c ) ≃ proj Γ( A ), that is, Γ( A c ) and Γ( A ) are Morita equivalent. Then Γ( A c ) is also semisimple. Recall thatΓ( A c ) = lim −→ End A c / r c ( r ⊗ ic ). By Lemma 5.1 all the canonical maps End A c / r c ( r ⊗ ic ) → Γ( A c ) are injective. Recall that for a semisimple algebra, the number of pair-wise orthogonal idempotents is bounded. Then the R -lengths of the algebrasEnd A c / r c ( r ⊗ ic ) are uniformly bounded. Consequently, the algebra Γ( A ) is an artin R -algebra. By Proposition 3.5 the singularity category D sg ( A c ) is Hom-finite. Thenwe are done by Lemma 4.3.Recall from [15, Theorem 2.1] that the singularity category of a self-injectivealgebra is equivalent to its stable category. In particular, it is Hom-finite. Thenthe implication “(3) ⇒ (1)” follows from Lemma 4.3.It remains to show “(1) ⇒ (3)”. Without loss of generality, we assume that thealgebra A is cyclic-like such that D sg ( A ) is Hom-finite. We will show that A isself-injective.We claim that the sysygy Ω( S ) of any cyclic simple A -module S is simple. Thenthere is only one arrow starting at S in Q A , which is valued by (1 , b ) for somenatural number b . Since Q A is cyclic-like, this forces that Q A is a disjoint union oforiented cycles. In each oriented cycle, every arrow has valuation (1 , b i ) for some b i . Then the symmetrization condition implies that all these b i ’s are necessarilyone; compare the proof of [1, Proposition VIII. 6.4]. As we point out above, thisimplies that A is self-injective.We prove the claim. Since by Corollary 2.4 D sg ( A ) is idempotent split, we havethat D sg ( A ) is a Krull-Schmidt category ([6, Appendix A]). In particular, eachobject is uniquely decomposed as a direct sum of finitely many indecomposableobjects. We observe that for each semisimple module X , lX ≤ l Ω( X ). Here, l denotes the composition length. Consider a cyclic simple A -module S , and takea path S = S → S → · · · → S r → S r +1 = S in Q A . Assume on the contrarythat l Ω( S ) ≥
2. Then we have Ω( S ) = S ⊕ X for some nonzero semisimplemodule X . Observe that S is a direct summand of Ω r − ( S ), and then we haveΩ r ( S ) = S ⊕ X ′ for a nonzero semisimple module X ′ . Consequently, we haveΩ nr ( S ) = S ⊕ X ′ ⊕ Ω r ( X ′ ) ⊕ · · · ⊕ Ω ( n − r ( X ′ ). Then the lengths of the semisimplemodules Ω nr ( S ) tend to the infinity, when n goes to the infinity. By Lemma5.1(1), q ( T ) is not zero for any simple A -module T . Recall from Lemma 2.2 that q ( S ) ≃ q (Ω nr ( S ))[ nr ]. This contradicts to the Krull-Schmidt property of q ( S ), andwe are done with the claim. (cid:3) Acknowledgements.
The author thanks Professor Zhaoyong Huang and Profes-sor Yu Ye for their helpful comments.
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