Support varieties over skew complete intersections via derived braided Hochschild cohomology
aa r X i v : . [ m a t h . R A ] J a n SUPPORT VARIETIES OVER SKEW COMPLETEINTERSECTIONS VIA DERIVED BRAIDED HOCHSCHILDCOHOMOLOGY
LUIGI FERRARO, W. FRANK MOORE, AND JOSH POLLITZ
Abstract.
In this article we study a theory of support varieties over a skewcomplete intersection R , i.e. a skew polynomial ring modulo an ideal generatedby a sequence of regular normal elements. We compute the derived braidedHochschild cohomology of R relative to the skew polynomial ring and showits action on Ext R ( M, N ) is noetherian for finitely generated R -modules M and N respecting the braiding of R . When the parameters defining the skewpolynomial ring are roots of unity we use this action to define a support theory.In this setting applications include a proof of the Generalized Auslander-ReitenConjecture and that R possesses symmetric complexity. Introduction
The use of the cohomological spectrum has had a tremendous impact on modularrepresentation theory; two notable works are those of Carlson [17] and Quillen [32].Inspired by these successes, similar theories of support have been developed andsuccessfully applied for different kinds of algebras. For example, for restricted Liealgebras [23], finite-dimensional graded connected cocommutative Hopf algebras[28], commutative complete intersections [4], and quantum complete intersections[12].In this paper we define and study a support variety theory for skew completeintersections, a class of rings first studied in [22] by the first and second author;skew complete intersections contain the class of commutative graded complete in-tersections and the class of quantum complete intersections. Let k be a field and let Q = k q [ x , . . . , x n ] be a skew polynomial ring, i.e., a polynomial ring such that thevariables skew commute x i x j = q i,j x j x i , with q i,j ∈ k ∗ , q i,i = 1 and q i,j = q − j,i forall i, j . A skew complete intersection R is quotient of Q by an ideal generated by aregular sequence of normal elements. To define these support varieties we use colorDG homological algebra, a theory that was developed in [22]. The support theoryitself is inspired by ideas present in [3] and further developed in [30], to transfer thewell behaved homological properties over Q to homological properties over R . Inparticular, we prescribe the graded Ext modules over R with a ring of cohomologyoperators for which the action is noetherian.We arrive at this ring of cohomological operators in two ways. First, we realizethese operators as the derived braided Hochschild cohomology of R over Q . It isworth contrasting this with the approach of Bergh and Erdmann in [12] who use Mathematics Subject Classification.
Key words and phrases. skew complete intersections, Hochschild cohomology, support vari-eties, Auslander-Reiten conjecture, complexity, color DG algebras.The third author was supported by the National Science Foundation under Grant No. 2002173. classical Hochschild cohomology to define support varieties over quantum completeintersections; classical Hochschild cohomology can be difficult to calculate and typi-cally possesses a complicated structure. This is even the case for quantum completeintersections (cf. [11, 14, 19, 29]). The calculation of derived braided Hochschildcohomology of R over Q , found in Section 3, leads to an easier calculation of a ringof cohomology operators that has a particularly simple structure, see Section 3.Second, these operators are realized at the chain level by following the strategiesin [3, 30]. This is the perspective that allows us to define a support theory wheneach q i,j is a root of unity. Moreover, this approach is in line with the support theoryfor commutative complete intersections (cf. [1, 4, 5, 16, 25, 30]), and generalizesthe support variety theory for graded complete intersections.Our main applications are in Section 8 and Section 9. First, as the Ext modulesover R are noetherian over this ring of cohomology operators we generalize resultsfrom [22] which pertain to the Poincar´e series of color R -modules. Further assumingeach of the q i,j defining R is a root of unity, we can apply our theory of supportsand obtain other interesting results. For example, in Section 8, we prove that colormodules over such skew complete intersections satisfy the Generalized Auslander-Reiten Conjecture. Also, in Section 9, we prove complexity for pairs of finitelygenerated color R -modules is symmetric in the module arguments for such skewcomplete intersections; therefore we deduce that if M and N are finitely generatedcolor modules over a skew complete intersection R , then Ext ≫ R ( M, N ) = 0 if andonly if Ext ≫ R ( N, M ) = 0. The latter result should be compared with the analogousresult from [10] for modules over quantum complete intersections, with the sameassumption on the q i,j ’s.The paper is organized as follows. In Section 1 we recall background informa-tion on color commutative rings and DG algebras that can be found in [22], and weprovide the definition of the derived braided Hochschild cohomology of a ring. InSection 2 we recall the definition of skew complete intersection and make the neces-sary constructions needed to compute the derived braided Hochschild cohomologyof a skew complete intersection in Section 3. In Section 4 we show that two naturalproducts on the derived braided Hochschild cohomology coincide. Section 5 andSection 6 are dedicated to defining our ring of cohomological operators at the chainlevel and showing that Ext modules are finitely generated modules over this ring.In Section 7 we assume that the parameters q i,j are roots of unity, this allows us towork with a subring of cohomological operators that is a commutative polynomialring. We use this smaller ring to define support varieties. The rest of Section 7is dedicated to the study of the properties of these support varieties and to sev-eral consequences. As discussed above, the main applications are in Section 8 andSection 9. 1. Background and conventions
Color commutative rings.
Let G be an abelian group and k a field. An alter-nating bicharacter on G is a function κ : G × G → k ∗ such that for all α, β, σ ∈ Gκ ( σ, αβ ) = κ ( σ, α ) κ ( σ, β ) ,κ ( σ, σ ) = 1 . Let A be a G -graded k -algebra with decomposition A = L σ ∈ G A σ , and let κ bean alternating bicharacter defined on G . We say that A is κ -color commutative (or UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 3 simply color commutative if κ is understood) if for every x ∈ A σ and y ∈ A τ , onehas xy = κ ( σ, τ ) yx . An element x ∈ A σ is said to be G - homogeneous . We callthe G -degree of a G -homogeneous element x the color of x , and we denote this by G ( x ). We also refer to G as the group of colors of A . If x and y are G -homogeneouswe abuse notation and use κ ( x, y ) to denote κ ( G ( x ) , G ( y )). Color DG algebras.
Let Q be a color commutative connected graded k -algebraand denote by G its group of colors. Let A be a DG Q -algebra. We say that A is a κ -color DG Q -algebra provided A is G -graded with a grading compatible with thehomological and internal grading of A , and the differential on A is G -homogeneousof color e G .We also assume that a color DG Q -algebra A is graded color commutative . Thatis, for all x, y ∈ A , homogeneous with respect to all gradings, we assume that xy = ( − | x || y | κ ( x, y ) yx , and that x = 0 when x is of odd homological degree.1.1 . The color opposite of A is the DG algebra A op with the same underlyingcomplex as A and product given by a · op b = ( − | a || b | κ ( a, b ) ba, where a, b are elements of A homogeneous with respect to all the gradings of A .By the color commutativity of A it follows that a · op b = ab , therefore A op is just A . The enveloping algebra of A , denoted by A e , is the color DG Q -algebra withunderlying complex A ⊗ Q A and product given by( a ⊗ b )( c ⊗ d ) = ( − | b || c | κ ( b, c ) ac ⊗ bd. where a, b, c, d are elements of A homogeneous with respect to all the gradings of A . Color DG modules.
Let A be a graded color commutative DG Q -algebra, where Q is a color commutative connected graded k -algebra. In this section we reviewconventions and terminology regarding color DG A -(bi)modules; see [22, Section4] for details. Throughout this article, a left (right) color DG A -module is a left(right) DG A -module equipped with a G -grading, where G is the group of colors of A , compatible with respect to the homological and internal gradings.1.2 . As usual a color DG A -bimodule is a left and right color DG A -module wherethe two actions are compatible; equivalently, it is a left color DG A e -module. Wesay that a DG A -bimodule M is symmetric provided am = ( − | a || m | κ ( a, m ) ma for all m ∈ M, a ∈ A homogeneous with respect to all gradings. Each right colorDG A -module M can be canonically made into a symmetric color bimodule byprescribing a left DG A -module with the following action: a · m := ( − | a || m | κ ( a, m ) ma for each a ∈ A and m ∈ M . Similarly, we can switch from left color DG A -modulesto right color DG A -modules according to the following rule: m · a := ( − | a || m | κ ( m, a ) am for each a ∈ A and m ∈ M . Hence, each left color (or right) DG A -modulecan naturally be viewed as a symmetric color DG A -bimodule according to thisconvention; therefore, we will not specify a side for color DG modules over colorcommutative algebras. L. FERRARO, W. F. MOORE, AND J. POLLITZ . Let
M, N be color DG A -modules. The color DG A -modules M ⊗ A N andHom A ( M, N ) are defined in the standard way, see [22, Definition 4.6 and Definition4.8]. The suspension of M , denoted by Σ M , is the color DG A -module with( Σ M ) i := M i − , ∂ Σ M := − ∂ M , and a · m := ( − | a | am. The homology of M , denoted by H( M ), is a color H( A )-module over the colorcommutative H ( A )-algebra H( A ) . We say M is finite provided that the color mod-ule M ♮ is finitely generated over A ♮ where ( − ) ♮ is the forgetful functor to thecategory of color A -modules. In particular, any bounded complex of finitely gen-erated color H ( A )-modules is a finite color DG A -module via restricting scalarsalong the augmentation A → H ( A ).1.4 . The diagonal is the multiplication map A e → A , which is a morphism of colorDG Q -algebras. In the sequel, it is through the diagonal that A will always beregarded as a color DG A e -module. Color DG homological algebra.
Let A be a graded color commutative DGalgebra.1.5 . Let F be a color DG A -module. We say that F is semifree if there exists achain of color DG A -modules0 = F ( − ⊆ F (0) ⊆ F (1) ⊆ F (2) ⊆ . . . satisfying S F ( n ) = F and for each n ≥ A -modules F ( n ) /F ( n − ∼ = M i ∈ Z Σ i A ( X i,n ) for some (possibly infinite) sets X i,n .1.6 . A color DG A -module P is said to be semiprojective provided Hom A ( P, − )preserves surjections and quasi-isomorphisms. Below are standard properties when A has trivial color, so we leave the proofs to the reader (cf. [20, Chapter 6] or [2,Chapter 1]).From the definition it follows easily that a quasi-isomorphism between semipro-jectives is in fact a homotopy equivalence. Also, any semifree DG A module issemiprojective. Finally, if P is semiprojective P ⊗ A − preserves injections andquasi-isomorphisms.1.7 . Let P be a color DG A -module. We say that P is perfect over Q providedthat P is quasi-isomorphic to a bounded complex of finitely generated projective Q -modules; here we are regarding P as a color DG Q -module via restriction ofscalars along the structure map Q → A. We say P is strongly perfect over Q if it isitself a bounded complex of finitely generated projective Q -modules.1.8 . For each DG A -module M there exists a surjective quasi-isomorphism of DG A -modules F ≃ −→ M where F is semifree. We call F ≃ −→ M a semifree resolutionof M over A , and any two semifree resolutions of M are homotopy equivalent.Furthermore, when A is a nonnegatively graded DG algebra with H( A ) noetherianand H( M ) a noetherian graded H( A )-module, there exists a semifree resolution F ≃ −→ M such that F ♮ ∼ = ∞ M j = i Σ j ( A β j ) ♮ UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 5 for a some fixed integer i and nonnegative integers β j (see [7, Proposition B.2]).Mimicking the proofs in [20, Proposition 6.6] one can show the existence anduniqueness, up to homotopy, of semifree resolutons; one only need to mind thecolors while adapting the arguments there.Let M, N be color DG A -modules, and let F be a semifree resolution of M . Itfollows from the previous properties that the homology of Hom A ( F, N ) and F ⊗ A N does not depend on the choice of the resolution F . We defineExt A ( M, N ) := H(Hom A ( F, N )) and Tor A ( M, N ) = H( F ⊗ A N )for each color DG A -module N . These are naturally graded color modules overH( A ) (see 1.3 for the H( A )-structures).1.9 . Let A be a color DG algebra over a color commutative ring. Let z be acycle of A homogeneous with respect to all gradings. There is a semifree extensionof A , denoted by A h y | ∂y = z i , where z becomes a boundary. This extension isconstructed by adjoining the variable y which is an skew exterior variable when z has even homological degree or a skew divided power variable when z has oddhomological degree; in either case, the variable y color commutes with A . It isstraightforward to see that A is a DG subalgebra of A h y i . Hence, we can inductivelyadjoin variables to kill cycles: A h y , . . . , y m | ∂y i = z i i := A h y , . . . , y m − | ∂y i = z i ih y m | ∂y m = z m i . See [22, Proposition 2.5 and Proposition 2.7] for further details.The final two subsections provide context for the the calculations in Section 3,and explain the choice of terminology regarding the operators discussed in Section 4.
Braided tensor categories, general case.
In this subsection we recall someconstructions from [8]. A monoidal category ( C , ⊗ ) is said to be braided if for allobjects U and V of C there exist functorial isomorphisms R U,V : U ⊗ V → V ⊗ U, satisfying the hexagon axioms , see [8, Section 2]. If, furthermore, R U,V R V,U = id V ⊗ U for all objects in C , then C is called a symmetric monoidal category .In [8], Baez defines braided tensor categories using commutative rings, we workwith the more general notion of color commutative rings; the proofs of the state-ments below remain unchanged under this more general hypothesis. Given a colorcommutative ring F , we define a braided (symmetric) tensor category of F - modules to be a braided (symmetric) monoidal category C equipped with a faithful functor F to the category of color F -modules, satisfying a list of axioms that the interestedreader can find in loc. cit. . Therefore for the remainder of this subsection, we willwork in a fixed braided tensor category V of F -modules, where F is some colorcommutative ring. As in [8], we will identify objects and morphisms in V withtheir images under F .Let A be an algebra object in V , with an associative multiplication given by m A .Let A op denote A with the multiplication map m A R A,A . It is proved in [8, Lemma1], that A op is an algebra in V . Let B be another algebra in V with product m B ,then A ⊗ B is an algebra in V with multiplication given by ( m A ⊗ m B )( id A ⊗ R B,A ⊗ id B ) (cf. [8, Lemma 2]). The enveloping algebra of A , denoted by A e , is A ⊗ A op .Assume that V is a symmetric tensor category. L. FERRARO, W. F. MOORE, AND J. POLLITZ (1) [8, Lemma 4] If A is an algebra object in V , then A is a left A e -module in V with action given by m A ( id A ⊗ m A R A,A ).(2) [8, Lemma 5] If A is an algebra object in V , then A is a right A e -modulein V with action given by m A R A,A ( m A ⊗ id A ).These facts are applied in [8, Section 3] to define the braided Hochschild homologyof an algebra object A in V , whenever A is flat over F . We define a dual notionbelow. Definition 1.10.
Let F be a color commutative ring, and let V be a symmetrictensor category of F -modules. Let A be an algebra object in V which is projectiveover F . The braided Hochschild cohomology of A with respect to F isHH( A | F ) = Ext A e ( A, A ) , where the right-hand side is the homology of the Hom complex of right linearmaps Hom A e ( B, B ); here B is the bar resolution constructed in [8, Theorem 1] andExt A e ( A, A ) has the composition product.
Symmetric tensor categories, color commutative case.
Let Q be a colorcommutative ring, and let C be the category of color DG Q -modules. The tensorproduct ⊗ Q , defined in [22, Definition 4.6], gives C the structure of a monoidalcategory. Let κ be the bicharacter associated to Q . For every pair of objects U, V ∈ C , we define maps R U,V : U ⊗ Q V → V ⊗ Q U by u ⊗ v ( − | u || v | κ ( u, v ) v ⊗ u, for all u ∈ U and v ∈ V homogeneous elements with respect to all gradings.These maps give C the structure of a symmetric monoidal category. The naturalembedding of C into the category of color Q -modules gives C the structure of asymmetric tensor category, we denote it by V .Let A be a color commutative DG Q -algebra in V . The product on A op , definedby m A R A,A , simplifies to a · op b = ( − | a || b | κ ( a, b ) ba = ab, where a, b are homogeneous elements with respect to all the gradings. This showsthat A op is isomorphic to A as a color DG Q -algebra.Let A and B be color commutative algebras in V . The multiplication on thetensor product A ⊗ B simplifies to( a ⊗ b )( a ′ ⊗ b ′ ) = ( − | b || a ′ | κ ( b, a ′ )( aa ′ ) ⊗ ( bb ′ ) , where a, b, a ′ , b ′ are homogeneous elements with respect to all the gradings.The enveloping algebra A e is the tensor product A ⊗ Q A op = A ⊗ Q A . It followsfrom [8, Lemma 3], or from a straightforward check, that A e is a color commutativeDG Q -algebra.The left and right A e action on A , given by [8, Lemma 4 & Lemma 5], simplifyto ( a ⊗ b ) · c = ( − | b || c | κ ( b, c ) acb, c · ( a ⊗ b ) = ( − | a || c | κ ( c, a ) acb, for a, b, c ∈ A , homogeneous elements with respect to all gradings.Therefore, the braided Hochschild cohomology of A relative to Q isHH( A | Q ) = Ext A e ( A, A ) , provided that A is semiprojective over Q . UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 7 If Q is a skew polynomial ring and R a quotient of Q by an ideal generated bynormal elements, then there exists a color DG Q -algebra resolution of R , obtainedvia the process of killing cycles illustrated in 1.9, see also [22, Section 2]. We denotethis DG algebra resolution by E . As in [6, Remark 3.3] we make the following Definition 1.11.
Let Q be a skew polynomial ring and R a quotient of Q by anideal generated by normal elements. Let E be a color DG Q -algebra resolution of R . The derived braided Hochschild cohomology of R relative to Q is HH ( R | Q ) = HH( E | Q ) . Observe that HH ( R | Q ) = Ext R ⊗ L Q R ( R, R ) , which justifies the use of the adjective derived in the definition of HH ( R | Q ). Remark . Braided Hochschild homology was first introduced in [8] by Baez.Derived Hochschild cohomology was defined by MacLane in [26] for Z -algebras andlater generalized by Shukla in [34] for algebras over general commutative noetherianrings. Quillen, in [31], recognized it as a derived version of Hochschild cohomology;[6] notes that derived Hochschild cohomology is also known as Shukla cohomology .For a comparison of Hochschild cohomology and derived Hochschild cohomologysee [9]. 2.
Semifree resolution of the diagonal
Throughout this paper we fix the following notation. Let Q = k q [ x , . . . , x n ] bea skew polynomial ring, where k is a field and q = ( q i,j ) is a matrix with invertibleentries such that q i,j = q − j,i for all i, j and q i,i = 1 for all i . The ring Q can beregarded as a color commutative ring in the standard way, see [22, Example 3.2],denote by G its group of colors. We fix a sequence of normal elements f = f , . . . , f c in ( x , . . . , x n ) and set R to be the quotient R = Q/ ( f ) where we regard R as acolor commutative ring with the same group of colors as Q . Finally, let E = Q h e , . . . , e c | ∂ ( e i ) = f i i be the (skew) Koszul complex on f over Q (cf. 1.9).2.1 . Recall from 1.4, E is regarded as a color DG E e -algebra via the diagonal E e → E given by a ⊗ b ab. There is an isomorphism of color DG Q -algebras E e ∼ = Q h e , . . . , e c , e ′ , . . . , e ′ c | ∂ ( e i ) = ∂ ( e ′ i ) = f i i , that identifies 1 ⊗ e i and e i ⊗ e i and e ′ i , respectively. Hence, we will freelyidentify these models without further mention in the rest of the article. Construction 2.2.
Notice that the elements e ′ i − e i are G -homogeneous cycles inthe color DG Q -algebra E e . Moreover, they pairwise skew commute and square tozero and so by [22, Proposition 2.9], they are killable. Therefore, we introduce aset of (skew) divided power variables Y = { y , . . . , y c } of homological degree 2 anddefine E e h Y i := E e h Y | ∂y i = e ′ i − e i i . L. FERRARO, W. F. MOORE, AND J. POLLITZ
Note that E e h Y i is a color DG E e -algebra and is equipped with a morphism ofcolor DG Q -algebras ǫ : E e h Y i → E given by( a ⊗ b ) y ( h )1 y ( h )2 . . . y ( h c ) c ( h i > ab otherwise . Moreover, the morphism is compatible with the E e -actions meaning that we havethe following commutative diagram of color DG E e -algebras where the unlabeledmaps are the canonical ones E e h Y i E e E ǫ Theorem 2.3.
The morphism of DG E e -algebras ǫ : E e h Y i → E defined in Con-struction 2.2 is a semifree resolution of E over E e .Proof. Clearly E e h Y i is a semifree DG E e -algebra. So it suffices to check that ǫ isa quasi-isomorphism.First, let x = x , . . . , x c be (skew) exterior variables of homological degree 1 andsuch that G ( x i ) = G ( e i ). Set Q h x i = Q h x | ∂x i = 0 i ;that is, the (skew) exterior algebra on L ci =1 Qx i and let π : Q h x i → Q be thecanonical augmentation map. There is an evident isomorphism of color DG Q -algebras ϕ : E e → Q h x i ⊗ Q E determined by e i ⊗ ⊗ e i and 1 ⊗ e i ⊗ e i − x i ⊗ . Furthermore ϕ is compatible with the augmentation maps to E , meaning the fol-lowing diagram of color DG Q -algebras commutes E e Q h x i ⊗ Q EE E. ϕ π ⊗ By [22, Lemma 6.4], ϕ extends to an morphism of color DG Q -algebras˜ ϕ : E e h Y i → ( Q h x i ⊗ Q E ) h Y | ∂y i = ϕ ( e i ⊗ − ⊗ e i ) i ;as ˜ ϕ maps a graded Q -basis of E e to a graded Q -basis of Q h x i ⊗ Q E , we concludethat ˜ ϕ is an isomorphism of color DG Q -algebras. Now note ϕ ( e i ⊗ − ⊗ e i ) = x i ⊗ ψ ( Q h x i ⊗ Q E ) h Y | ∂y i = ϕ ( e i ⊗ − ⊗ e i ) i ∼ = E ⊗ Q Q h x , Y | ∂x i = 0 , ∂y i = x i i as augmented, to E , DG Q -algebras. Finally, it follows from [22, Theorem 2.15]that Q h x , Y | ∂x i = 0 , ∂y i = x i i ≃ −→ Q and so the commutative diagram UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 9 E e h Y i E ⊗ Q Q h x , Y i E E ψ ˜ ϕ ≃ = completes the proof. (cid:3) Notation 2.4.
For the rest of the paper E e h Y i will denote the resolution of E from Construction 2.2.3. Derived braided Hochschild cohomology
Directly from Theorem 2.3,Tor E e ( E, E ) ∼ = H( E e h Y i ⊗ E e E ) . It is clear that there is an isomorphism of color DG Q -algebras E e h Y i ⊗ E e E ∼ = E h Y | ∂y i = 0 i . Therefore, the braided Hochschild homology of E over Q is the graded color Q -algebra H( E ) ⊗ Q Q h Y i . This section is devoted to the calculation of Hochschild co homology, HH( E | Q ) =Ext E e ( E, E ), as a graded color Q -algebra.3.1 . Let A be a DG R -algebra and let A h Y i be a semifree extension of A . Welet D := Der A ( A h Y i , A h Y i ) denote the subset of Hom A ( A h Y i , A h Y i ) consistingof A -linear color derivations, see [22, Definition 5.1]. It is straightforward to checkthat D is a subcomplex of Hom A ( A h Y i , A h Y i ). Equipping it with the followingbracket and squaring operations[ θ, ξ ] = θξ − ( − | θ || ξ | κ ( θ, ξ ) ξθζ [2] = ζ , where θ, ξ, ζ ∈ Der A ( A h Y i , A h Y i ) are homogeneous with respect to all the gradingsand ζ has odd homological degree, makes D a color DG Lie A ♮ -algebra, see [22,Definition 7.1 and 7.9] for the definition of color DG Lie algebra. The proof isessentially contained in [22, Lemma 7.10], where it is proved when A = R . The sameproof works in this more general case; we show that the bracket is A ♮ -bilinear. Let a ∈ A, θ, ξ ∈ Der A ( A h Y i , A h Y i ) be homogeneous with respect to all the gradings,then [ aθ, ξ ] = aθξ − ( − | aθ || ξ | κ ( aθ, ξ ) ξ ( aθ )= aθξ − ( − | a || ξ | + | θ || ξ | κ ( a, ξ ) κ ( θ, ξ )( − | ξ || a | κ ( ξ, a ) aξθ = a [ θ, ξ ] . The remaining checks are similar.3.2 . Let A be a color DG algebra. Let g , . . . , g c be elements of the group of colorsof A , let n , . . . , n c be integers, and let z , . . . , z c be indeterminates, we denote by A [ z , . . . , z c | G ( z i ) = g i , | z i | = n i , i = 1 , . . . , c ] the Ore extension obtained by adding the indeterminates z , . . . , z c to A , satisfyingthe following commuting relations: z i s = ( − n i | s | κ ( g i , s ) sz i , for all s ∈ A,z i z j = ( − n i n j κ ( g i , g j ) z j z i , for all i, j = 1 , . . . , c, where s is homogeneous with respect to all the gradings of A .We use E e h Y i to compute the derived braided Hochschild cohomology of E relative to Q (cf. Definition 1.11). Theorem 3.3.
There is an isomorphism of graded color H( E e ) -algebras HH( E | Q ) ∼ = H( E )[ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2 , i = 1 , . . . , c ] . Moreover the variables χ , . . . , χ c correspond to the homology classes of derivations θ , . . . , θ c ∈ Der E e ( E e h Y i , E e h Y i ) , satisfying θ i ( y j ) = δ i,j .Proof. We consider the following diagramDer E e ( E e h Y i , E e h Y i ) Hom E e ( E e h Y i , E e h Y i )Der E e ( E e h Y i , E ) Hom E e ( E e h Y i , E )Hom E e h Y i (Diff E e E e h Y i , E ) Hom E ( E e h Y i ⊗ E e E, E )Hom E ((Diff E e E e h Y i ) ⊗ E e h Y i E, E )Hom E ( EY, E ) Hom E ( E h Y | ∂y i = 0 ∀ i i , E ) Eχ ⊕ · · · ⊕ Eχ c E [ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2][22, Corollary 5.3] [22, Proposition 4.12][22, Proposition 5.2] [22, Proposition 4.9][22, Proposition 4.9]where the horizontal maps are inclusions, the vertical maps are quasi-isomorphisms, Eχ ⊕ · · · ⊕ Eχ c is the free DG E -module with basis χ , . . . , χ c and such that thedifferential of the elements χ , . . . , χ c is zero and the differential on the indeter-minates of E [ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2] is trivial. The isomorphism(Diff E e E e h Y i ) ⊗ E e h Y i E ∼ = EY , where EY is the free E -module with basis Y andtrivial differential on the elements of Y , follows directly from the construction ofthe module of differentials. The isomorphism E e h Y i ⊗ E e E ∼ = E h Y | ∂y i = 0 ∀ i i follows from the construction of the resolution E e h Y i . The bottom left verticalmap is the map θ P ci =1 θ ( y i ) χ i and similarly for the bottom right verticalmap. It is straightforward to check that this is a commutative diagram and that UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 11 χ i ∈ Eχ ⊕ · · · ⊕ Eχ c corresponds to the derivation θ ∈ Der E e ( E e h Y i , E e h Y i ) suchthat θ ( y j ) = δ i,j .For the remainder of the proof L will denote the homology of the complexDer E e ( E e h Y i , E e h Y i ); from 3.1 it follows that L is a graded color Lie H( E e )-algebra. The map induced in homology by the bottom map is the inclusion ofH( E e )-modules H( E ) χ ⊕· · ·⊕ H( E ) χ c → H( E )[ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | =2]. Therefore the top map induces, in homology, an injection of graded color LieH( E e )-algebras ι : L → Lie(HH( E | Q )). Let ι ′ : U L → HH( E | Q ) be the universalextension of ι , where U L is the universal enveloping algebra of L (see [22, Def-inition 7.6 and Remark 7.7]), we claim that ι ′ is an isomorphism of associativeH( E e )-algebras. Indeed, since L = H( E ) χ ⊕ · · · ⊕ H( E ) χ c and G ( χ i ) = G ( f i ) − for all i , it follows that U L ∼ = H( E )[ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2]. Thecomputations performed at the beginning of this proof show that HH( E | Q ) is iso-morphic to H( E )[ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2] as a H( E )-module, andthey also show that the map ι ′ is surjective. In each homological degree ι ′ is a sur-jective map of free H( E )-modules of the same rank, and since H( E ) is noetherianit follows that ι ′ must be injective in each homological degree, and therefore it isitself injective. (cid:3) Corollary 3.4.
When f is a regular sequence, there is an isomorphism of gradedcolor R -algebras HH ( R | Q ) ∼ = R [ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2 , i = 1 , . . . , c ] . Proof.
This follows from the previous theorem sinceHH( E | Q ) = HH ( R | Q ) and H( E ) = R ;therefore the isomorphism claimed in the Corollary is one of H( E e ) = Tor Q ( R, R )-algebras, and therefore one of R -algebras. (cid:3) Remark . As a consequence of Theorem 3.3, κ ( χ i , − ) = κ ( y i , − ) − = κ ( f i , − ) − . In the sequel we will make use of this fact in Section 5 to define a color DG S -modulewhose homology computes Ext over a skew complete intersection where S = Q [ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2 , i = 1 , . . . , c ] . Products on the derived braided Hochschild cohomology
In this section we clarify that for a skew complete intersection R = Q/ ( f ) thecomposition action of HH ( R | Q ) on Ext R ( M, N ) can be computed at the chain levelwith the so-called “cup product,” see Definition 4.2. The first action establishesproperties expected from a well-posed support theory, see Proposition 7.8; the latterallows us to prove the intersection formula in Theorem 7.12.We fix the following notation. Let A be a color DG Q -algebra. Let ε : P ≃ −→ A be a semiprojective resolution of A over A e .Recall that for any color DG A e -module X , Hom A e ( P, X ) is naturally a right DGHom A e ( P, P )-module via right composition of maps. This action induces a gradedright HH( A | Q )-module structure on Ext A e ( A, X ). First, we define a different rightDG Hom A e ( P, P )-module structure on Hom A e ( P, X ), provided that P admits a diagonal approximation; furthermore, we show that these two structures are thesame in homology. This will be a slight generalization of [15, Proposition 4.5].Namely, we adapt [15, Proposition 4.5] to the color setting, while also discussingactions on certain Ext-modules. Definition 4.1. A co-unital diagonal approximation for ε : P ≃ −→ A is a morphismΦ : P → P ⊗ A P of left color DG A e -modules so that PP P ⊗ A P P Φ ε ⊗ ⊗ ε commutes, where we have identified P ⊗ A A ∼ = P ∼ = A ⊗ A P with the appropriatemultiplication maps. Definition 4.2.
Given a co-unital diagonal approximation Φ : P → P ⊗ A P for ǫ we define a right cup action ∪ Φ (or simply ∪ ) of Hom A e ( P, P ) on Hom A e ( P, X ) as σ ∪ τ = µ ( σ ⊗ ( ετ ))Φ : P Φ −→ P ⊗ A P σ ⊗ ( ετ ) −−−−→ X ⊗ A A µ −→ X, where σ ∈ Hom A e ( P, X ) , τ ∈ Hom A e ( P, P ), and µ is the right multiplication in-duced from the A e -module structure of X .4.3 . Since P is semiprojective, the surjective quasi-isomorphism ε : P ≃ −→ A inducesa surjective quasi-isomorphism ε ∗ : Hom A e ( P, P ) ≃ −→ Hom A ( P, A ) , and so ε ∗ prescribes a right action of Hom A e ( P, A ) on Hom A e ( P, X ). Indeed, for σ ∈ Hom A e ( P, X ) and τ ∈ Hom A e ( P, A ) we first lift τ to ˜ τ in Hom A e ( P, P ) satisfying ε ˜ τ = τ. Now we define σ · τ := σ ∪ ˜ τ . From the definition of ∪ , it follows immediately that this action is independent ofthe choice of lifting ˜ τ . Proposition 4.4.
Let A be a color DG Q -algebra, let ε : P → A be a semiprojectiveresolution of A over A e , let X be a color DG A e -module. If P admits a co-unitaldiagonal approximation then, the cup and composition actions of Ext A e ( A, A ) on Ext A e ( A, X ) coincide.Proof. First, we claim that 1 ⊗ ε and ε ⊗ A e ( P ⊗ A P, P ) . Indeed, as P ⊗ A P is semiprojective over A e , ε induces a quasi-isomorphismHom A e ( P ⊗ A P, P ) ≃ −→ Hom A e ( P ⊗ A P, A )given by ψ εψ. Note that ε ◦ (1 ⊗ ε ) = ε ◦ ( ε ⊗
1) and so the quasi-isomorphismabove implies that 1 ⊗ ε − ε ⊗ A e ( P ⊗ A P, P ) , as needed.Now by the claim above, the square below commutes up to homotopy and hence,we have the following homotopy commutative diagram of DG A e -modules: UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 13
P P ⊗ A P P ⊗ A P X ⊗ A P X ⊗ A AP P X
Φ 1 ⊗ τε ⊗ σ ⊗ ⊗ ε ⊗ ε µτ στ σ where σ ∈ Hom A e ( P, X ) and τ ∈ Hom A e ( P, P ). Hence σ ∪ τ , which is given by thecomposition of the four horizontal maps at the top, is homotopic to στ . (cid:3) Now we specialize and return to the setting of Section 2. In the proof of thefollowing proposition we will make use of the divided powers DG E e -structure on E e h Y i ; see [22, Section 6] or [24, Chapter 1] for more details regarding DG algebraswith divided powers. Proposition 4.5.
The resolution E e h Y i ≃ −→ E admits a co-unital diagonal approx-imation Φ : E e h Y i → E e h Y i ⊗ E E e h Y i , determined by y y ⊗ ⊗ y for all y ∈ Y .Proof. Let ϕ : E e → E e ⊗ E E e be the color DG Q -algebra map given by a ⊗ b ( a ⊗ ⊗ (1 ⊗ b ) . Notice that ϕ is compatible with the system of divided powers on E e and E e ⊗ E E e .Let y ∈ Y and let e ∈ E be such that ∂ ( y ) = e ⊗ − ⊗ e in E e h Y i . Observe thatin E e h Y i ⊗ E E e h Y i the following holds: ϕ ( ∂ ( y )) = ϕ ( e ⊗ − ⊗ e )= ( e ⊗ ⊗ (1 ⊗ − (1 ⊗ ⊗ (1 ⊗ e )= ( e ⊗ ⊗ (1 ⊗ − (1 ⊗ e ) ⊗ (1 ⊗ ⊗ ⊗ ( e ⊗ − (1 ⊗ ⊗ (1 ⊗ e )= ( e ⊗ − ⊗ e ) ⊗ (1 ⊗
1) + (1 ⊗ ⊗ ( e ⊗ − ⊗ e )= ∂ ( y ) ⊗ (1 ⊗
1) + (1 ⊗ ⊗ ∂ ( y ) . By [22, Lemma 6.4], ϕ uniquely extends to a morphism of color DG E e -algebraswith divided powers Φ : E e h Y i → E e h Y i ⊗ E E e h Y i given by y y ⊗ ⊗ y. Finally, the co-unital approximation condition isimmediate. (cid:3) . It follows from Proposition 4.4 and Proposition 4.5 that the composition actionof HH ( R | Q ) on Ext E e ( E, X ) is the same as the cup action for any color DG E e -module X .4.7 . Since Φ is a map of divided powers algebras, for any y ∈ Y we have thatΦ( y ( H ) ) = Φ( y ) ( h ) . . . Φ( y c ) ( h c ) where H = ( h , . . . , h c ) ∈ N c . Now a computation shows that(1) Φ( y ( H ) ) = X H ′ + H ′′ = H Y i Adopting the usual notation we also fix S = Q [ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2 , i = 1 , . . . , c ]for the remainder of the article. We will regard S as a color DG Q -algebra withtrivial differential.5.1 . We notice that S can be realized as Hom Q ( Q h Y i , Q ) by Proposition A.3 where χ i is the Q -linear dual of y i . We recall that the algebra structure on S is inducedby the coalgebra structure on Q h Y i . That is, the product χ i χ j is identified withthe composition Q h Y i ∆ −→ Q h Y i ⊗ Q Q h Y i χ i ⊗ χ j −−−−→ Q ⊗ Q Q µ −→ Q where the isomorphism is the multiplication map and ∆ is defined as Φ in (1),namely ∆( y ( H ) ) = X H ′ + H ′′ = H Y i Let X be a color DG E e -module. Define E X to be the DG S -module with underlying graded Q -module S ⊗ Q X and differential ∂ E X = 1 ⊗ ∂ X + c X i =1 χ i ⊗ ( λ i − λ ′ i )where λ i and λ ′ i are left multiplication by 1 ⊗ e i and e i ⊗ 1, respectively. Explicitly,as a graded S -module E X is E ♮X ∼ = M j ∈ Z Σ j ( S ⊗ Q X j )and on elements its differential is prescribed by ∂ E X ( s ⊗ x ) = s ⊗ ∂ X ( x ) + c X i =1 κ (1 ⊗ e i − e i ⊗ , s ) χ i s ⊗ (1 ⊗ e i − e i ⊗ x = s ⊗ ∂ X ( x ) + c X i =1 κ ( f i , s ) χ i s ⊗ (1 ⊗ e i − e i ⊗ x. Proposition 5.3. E X is a color DG S -module. UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 15 Proof. Note that ∂ := ∂ E X is (color) S -linear. Indeed, continuing from the compu-tation in Construction 5.2 ∂ ( s ⊗ x ) = s ⊗ ∂ X ( x ) + c X i =1 κ ( f i , s ) χ i s ⊗ (1 ⊗ e i − e i ⊗ x = s ⊗ ∂ X ( x ) + c X i =1 κ ( f i , s ) κ ( χ i , s ) sχ i ⊗ (1 ⊗ e i − e i ⊗ x = s ⊗ ∂ X ( x ) + c X i =1 sχ i ⊗ (1 ⊗ e i − e i ⊗ x = s∂ (1 ⊗ x )where the third equality holds because κ ( χ i , − ) := κ ( f i , − ) − .It remains to show ∂ = 0, but since ∂ is (color) S -linear it suffices to check ∂ (1 ⊗ x ) = 0 for each x in X . Consider ∂ (1 ⊗ x ) = ∂ ⊗ ∂ X ( x ) + c X i =1 χ i ⊗ (1 ⊗ e i − e i ⊗ · x ! = 1 ⊗ ∂ X ∂ X ( x ) + c X i =1 χ i ⊗ (1 ⊗ e i − e i ⊗ · ∂ X ( x )+ c X i =1 χ i ⊗ ∂ X ((1 ⊗ e i − e i ⊗ · x )+ c X i =1 c X j =1 κ ( f i , χ j ) χ i χ j ⊗ (1 ⊗ e i − e i ⊗ ⊗ e j − e j ⊗ x. The first summand in the last expression above is evidently zero. Also, as X is aDG E e -module ∂ X ((1 ⊗ e i − e i ⊗ x ) = ∂ E e (1 ⊗ e i − e i ⊗ x − (1 ⊗ e i − e i ⊗ ∂ X ( x )= − (1 ⊗ e i − e i ⊗ ∂ X ( x ) , and so(2) ∂ (1 ⊗ x ) = c X i =1 c X j =1 κ ( f i , χ j ) χ i χ j ⊗ (1 ⊗ e i − e i ⊗ ⊗ e j − e j ⊗ x. For i = j , κ ( f i , χ j ) χ i χ j ⊗ (1 ⊗ e i − e i ⊗ ⊗ e j − e j ⊗ x = 0 . For i = j , we show that the two terms on the right-hand side of (2) involving i and j cancel. Indeed, set f := f i , f ′ := f j , e := 1 ⊗ e i − e i ⊗ e ′ := 1 ⊗ e j − e j ⊗ χ = χ i and χ ′ := χ j for ease of notation. Since κ ( f, χ ′ ) χχ ′ = κ ( f, f ′ ) − χχ ′ = κ ( f, f ′ ) − κ ( f, f ′ ) χ ′ χ = χ ′ χ the first equality below follows κ ( f, χ ′ ) χχ ′ ⊗ ee ′ + κ ( f ′ , χ ) χ ′ χ ⊗ e ′ e = χ ′ χ ⊗ ee ′ + κ ( f ′ , χ ) χ ′ χ ⊗ e ′ e = χ ′ χ ⊗ ( ee ′ + κ ( f ′ , f ) − e ′ e )= χ ′ χ ⊗ ( ee ′ − κ ( f ′ , f ) − κ ( f ′ , f ) ee ′ )= 0 . Combining these calculations with (2) it follows that ∂ = 0, finishing the proofthat E X is a color DG S -module. (cid:3) Theorem 5.4. For any bounded above DG E e -module X , the map η X : E X Hom E e ( E e h Y i , X ) χ i ⊗ x ( y ( H ) κ ( x, y ( H ) ) χ i ( y ( H ) ) x ) is an isomorphism color DG Q -modules. Moreover, η X satisfies (3) η X ( χ i ⊗ x · χ j ) = η X ( χ i ⊗ x ) · η E ( χ j ⊗ , where x ∈ X (cf. 4.3). That is, the isomorphism η X is equivariant with respect tothe DG algebra map S → S ⊗ Q E η E −−→ Hom E e ( E e h Y i , E ) . Proof. Consider the following isomorphisms of graded S -modulesHom E e ( E e h Y i , X ) ♮ ∼ = Hom Q ( Q h Y i , X ) ♮ (4) ∼ = Hom Q ( Q h Y i , Q ) ⊗ Q X ♮ (5) ∼ = S ⊗ Q X ♮ (6)where (4) is adjunction, (5) follows as Q h Y i consists of degreewise finite rank free Q -modules and X is bounded above, and (6) is discussed in the appendix (cf.Proposition A.3).Now we check that the isomorphisms above send ∂ Hom E e ( E e h Y i ,X ) to ∂ E X . Indeed, ∂ Hom E e ( E e h Y i ,X ) = Hom( E e h Y i , ∂ X ) − Hom( ∂ E e ⊗ c X i =1 ( λ ′ i − λ i ) ⊗ χ i , X )and so (4) maps this differential toHom( Q h Y i , ∂ X ) − c X i =1 κ ( f i , λ ′ i − λ i ) − Hom( χ i , λ ′ i − λ i )which, using that κ ( − , λ ′ i − λ i ) = κ ( − , f i ), is simplyHom( Q h Y i , ∂ X ) − c X i =1 Hom( χ i , λ ′ i − λ i ) = Hom( Q h Y i , ∂ X ) + c X i =1 Hom( χ i , λ i − λ ′ i ) . Next, (5) maps the differential toHom Q ( Q h Y i , Q ) ⊗ ∂ X + c X i =1 Hom( χ i , Q ) ⊗ ( λ i − λ ′ i )and so (6) gives us exactly ∂ E X , as claimed. Finally, the composition of the isomor-phisms (4), (5), and (6) is exactly η X . UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 17 Now we prove (3) holds by showing both sides agree after evaluating at y ( H ) andusing the E e -linearity. Let x ∈ X and H ∈ N c . First, observe that η X ( χ i ⊗ x · χ j )( y ( H ) ) = κ ( x, χ j y ( H ) ) χ i χ j ( y ( H ) ) x where the product χ i χ j is interpreted as in 5.1.Assuming that i < j (the other cases are similar), by the definition of χ i χ j itfollows that the previous display is zero unless H has a 1 in position i and j , andzero everywhere else. Hence, the display above is equal to κ ( x, y i ) x. Similarly, using (1), we see that if H has 1 in position i and j and zero everywhereelse, then ( η X ( χ i ⊗ x ) ∪ η E ( χ j ⊗ y ( H ) ) = κ ( x, y i ) x, otherwise ( η X ( χ i ⊗ x ) ∪ η E ( χ j ⊗ y ( H ) ) = 0. (cid:3) Remark . Let X be a bounded above color DG E e -module. Combining 4.6 andTheorem 5.4, in homology η X induces an equivariant isomorphismH( E X ) H( η X ) −−−−→ Ext E e ( E, X )that respects the canonical projection π : S → S ⊗ Q R ∼ = HH ( R | Q ); the left-hand side is regarded with the obvious S -action from Construction 5.2 while theright-hand side has the composition action; see the beginning of Section 4.Viewing Ext E e ( E, X ) as an S -module via restriction of scalars along π , theprevious observation states that H( η X ) is an isomorphism of graded S -modules.Because of this we identify these S -actions on Ext E e ( E, X ) . . Let F and G be color DG E -modules. Then Hom Q ( F, G ) naturally inheritsthe structure of a DG E e -module. Namely, given α ∈ Hom Q ( F, G ) (homogeneouswith respect to all gradings) define1 ⊗ e i · α := ( − | α | κ ( e i , α ) α ( e i · − ) and e i ⊗ · α := e i · α ( − ) . Hence, E Hom Q ( F,G ) can be regarded as a DG S -module via Construction 5.2. Wewill write E F,G in lieu of E Hom Q ( F,G ) . Moreover, when F ♮ and G ♮ are free as graded Q -modules, it follows that E ♮F,G is a free graded S -module.6. Cohomology operators on Ext modules Definition 6.1. Let M be a color DG E -module. A surjective quasi-isomorphismof color DG E -modules F ≃ −→ M is called a Koszul resolution of M provided that F is semifree over Q via restriction of scalars along the structure map Q → E . When F is a finite DG E -module we say the Koszul resolution is finite . In this case, F isstrongly perfect over Q . Proposition 6.2. For each finite color DG E -module M , there exists a finiteKoszul resolution P ≃ −→ M .Proof. By 1.8, there exists a semifree resolution ǫ : F ≃ −→ M of M over E where F ♮ ∼ = ∞ M j = i Σ j ( E β j ) ♮ for some fixed i ∈ Z and nonnegative integers β j . In particular, when F is regardedas a complex of Q -modules via the structure map Q → E , F is a bounded belowcomplex of finite rank free Q -modules. As Q has finite global dimension and M isfinite, coker ∂ Fn +1 is free over Q for each n ≫ F ′ defined as . . . → F n +2 → F n +1 → im ∂ Fn +1 → F ′ is a DG E -submoduleof F . Furthermore, by possibly increasing n one can assume that F ′ is acyclic andconcentrated in degrees strictly larger than max { i : M i = 0 } .Next, we take P to be the quotient DG E -module F/F ′ → coker ∂ Fn +1 → F n − ∂ Fn − −−−→ . . . ∂ Fi +1 −−−→ F i → . As P is the quotient of F by an acyclic DG E -submodule F ′ such that ǫ | F ′ = 0, thereis a canonically induced quasi-isomorphism P ≃ −→ M of DG E -modules. Finally, weremark that by construction P is strongly perfect over Q . (cid:3) . Let M and N be finite color DG R -modules and fix F a Koszul resolution of M . The quasi-isomorphism E ≃ −→ R induces the first isomorphism of graded color R -modules below (see [20, Proposition 6.7] for a proof in the case that E has trivialcolor), while the second isomorphism follows from Proposition B.3(1)(7) Ext R ( M, N ) ∼ = −→ Ext E ( M, N ) ∼ = −→ Ext E e ( E, Hom Q ( F, N )) . The isomorphism in (7) provides Ext R ( M, N ) the structure of a HH ( R | Q )-module(and hence, via restriction of scalars, an S -module structure). Theorem 6.4. Let M and N be finite color DG R -modules. There exist the fol-lowing isomorphisms of graded S -modules Ext R ( M, N ) ∼ = H( E F,N ) ∼ = H( E F,G ) where F ≃ −→ M and G ≃ −→ N are any bounded below Koszul resolutions. Moreover, Ext R ( M, N ) is a finitely generated graded S -module.Proof. The isomorphisms follow from Theorem 5.4 and the discussion in 6.3.For the moreover statement, let F be a finite Koszul resolution of M , whichexists by Proposition 6.2. Since F is strongly perfect over Q , E ♮F,N = S ⊗ Q Hom Q ( F, N ) ♮ is a noetherian graded S -module. Any subquotient of a noetherian module is againnoetherian and so the already established isomorphism of graded S -modules, above,implies that Ext R ( M, N ) is a noetherian over S . (cid:3) Proposition 6.5. The following hold for color DG R -modules L, M, N : (1) The natural isomorphisms of graded color Q -modules Ext R ( L, M ⊕ N ) ∼ = Ext R ( L, M ) ⊕ Ext R ( L, N )Ext R ( L ⊕ M, N ) ∼ = Ext R ( L, N ) ⊕ Ext R ( M, N )Ext R ( M, Σ n N ) ∼ = Σ n Ext R ( M, N ) ∼ = Ext R ( Σ − n M, N ) are isomorphisms of graded S -modules; UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 19 (2) For any exact sequence of color DG R -modules → M → M → M → ,the exact sequences of graded Q -modules Ext R ( M , N ) → Ext R ( M , N ) → Ext R ( M , N ) → Σ Ext R ( M , N )Ext R ( L, M ) → Ext R ( L, M ) → Ext R ( L, M ) → Σ Ext R ( L, M ) are exact sequences of graded S -modules.Proof. All of these follow directly from Remark 5.5; we will prove the first isomor-phism in (1) while the rest are left to the reader. Fix a Koszul resolution F ≃ −→ L ,the claim follows from the commutativity of the following diagram where all arrowsare isomorphisms of graded color Q -modulesExt R ( L, M ⊕ N ) Ext R ( L, M ) ⊕ Ext R ( L, N )Ext E ( L, M ⊕ N ) Ext E ( L, M ) ⊕ Ext E ( L, N )Ext E e ( E, Hom Q ( F, M ⊕ N )) Ext E e ( E, Hom Q ( F, N )) ⊕ Ext E e ( E, Hom Q ( F, N )) . Both vertical maps at the top of the diagram are induced by the quasi-isomorphism E → R , see 6.3. The second set of vertical maps are those from Proposition B.3(1).The bottom horizontal map is clearly HH ( R | Q )-linear, and so the desired resultfollows from Remark 5.5. (cid:3) Support varieties Let A be a commutative noetherian graded k -algebra that is concentrated ineven nonnegative cohomological degrees. We let D f ( A ) denote the bounded derivedcategory of finite DG A -modules which is obtained in the standard way of formallyinverting quasi-isomorphisms between DG A -modules. Explicitly, the objects of D f ( A ) are DG A -modules whose homology is a finitely generated graded A -module.7.1 . Let Proj A denote the topological space consisting of homogeneous prime idealsnot containing A > equipped with the Zariski topology. The closed subsets ofProj A are of the form V ( g , . . . , g t ) = { p ∈ Proj A : g i ∈ p for all i } for some homogeneous elements g , . . . , g t ∈ A . For a graded A -module Y and p ∈ Proj A , we let Y p denote the homogeneous localization of Y at p . Also, for p ∈ Proj A define k ( p ) to be A p / p A p . . For a DG A -module X , its (small) support is supp A X = { p ∈ Proj A : X ⊗ L A k ( p ) } . By [18, Theorem 2.4], if X ∈ D f ( A ) then supp A X = { p ∈ Proj A : H( X ) p = 0 } . Furthermore, as H( X ) is a finitely generated graded A -module, the support of X over A is exactly V ( g , . . . , g t ) where g , . . . , g t is some list of homogeneousgenerators for ann A H( X ). Thus, supp A X is a closed subset of Proj A whenever X ∈ D f ( A ).7.3 . We recall the following well known property of cohomological support (see, forexample, [30, 2.1.5]). For DG A -modules X and X ′ , supp A ( X ⊗ L A X ′ ) = supp A X ∩ supp A X ′ . In particular, when X is semiprojective over A , X ⊗ A X ′ ≃ X ⊗ L A X ′ and so supp A X ∩ supp A X ′ = supp A ( X ⊗ A X ′ ) . . For the rest of the section we add to our fixed notation from Section 2 theassumption that the group of colors of Q is finite. We point out that this hypothesisis equivalent to saying that the skew commuting parameters of Q are roots of unity.In particular, there exists t > A := Q ′ [ χ t , . . . , χ tc ]of S is commutative where Q ′ is the subalgebra on the generators for Q raised tothe t th power. Moreover, it is clear that A ⊆ S is a module finite extension and A has finite global dimension.With 7.4 in place, we have a way to study graded Ext-modules over R as modulesover a commutative polynomial ring in variables of cohomological degree 2 t withcoefficients in Q ′ . This allows us to introduce a theory of support varieties analo-gous to the ones over commutative complete intersections as well as more general(commutative) settings (cf. [1, 4, 5, 16, 25, 30]). Definition 7.5. If M and N are color DG R -modules, we define the support varietyof ( M, N ) to be V R ( M, N ) = supp A (Ext R ( M, N )) . . Assume that M and N are finite color DG R -modules. In this case, Ext R ( M, N )is a finitely generated graded A -module. Indeed, by Theorem 6.4, Ext R ( M, N ) isa finitely generated graded color S -module. Since A → S is module finite, theclaim holds. That is, E F,N is an object of D f ( A ) where F ≃ −→ M is a finite Koszulresolution of M . Hence,V R ( M, N ) = { p ∈ Proj A : Ext R ( M, N ) p = 0 } = { p ∈ Proj A : E F,N ⊗ L A k ( p ) } = { p ∈ Proj A : E F,G ⊗ L A k ( p ) } where G ≃ −→ N is a finite Koszul resolution of N ; the first and second equalitieshold by 7.2, and the third equality is justified by Theorem 6.4.From 7.2 and 7.6, the support variety of a pair of finite color DG R -modules isin fact a closed subset of Proj A. This is recorded in the first proposition below. Proposition 7.7. If M and N are finite color DG R -modules, then V R ( M, N ) isa closed subset of Proj A. Proposition 7.8. The following hold for DG R -modules L, M, N : (1) V R ( L ⊕ M, N ) = V R ( L, N ) ∪ V R ( M, N ) . UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 21 (2) V R ( L, M ⊕ N ) = V R ( L, M ) ∪ V R ( L, N ) . (3) For any n ∈ Z , V R ( M, Σ n N ) = V R ( M, N ) = V R ( Σ n M, N ) . (4) For → M → M → M → an exact sequence of color DG R -modules V R ( M h , N ) ⊆ V R ( M i , N ) ∪ V R ( M j , N )V R ( N, M h ) ⊆ V R ( N, M i ) ∪ V R ( N, M j ) whenever { h, i, j } = { , , } . (5) If Ext nR ( M, N ) = 0 for all n ≫ , then V R ( M, N ) = ∅ ; the converse holdswhen both M and N are finite DG R -modules.Proof. The first four statements are clear from 6.5 and standard facts for homo-geneous supports of graded modules over commutative noetherian graded rings (as A is); for these facts see, for example, [5, 2.2]. For (5), the forward implication iselementary. For the converse, the assumption on M and N imply that Ext R ( M, N )is a finitely generated graded A -module, see 7.6, and so we can apply [5, 2.2(5)]directly. (cid:3) Example 7.9. We show that V R ( k , k ) = Proj( A ⊗ Q ′ k ). We first notice that by[22, Theorem 10.7] it follows that Ext R ( k , k ) is isomorphic as a S -module, andhence as a A -module, to S ⊗ Q V q ( k e ⊕ · · · ⊕ k e n ), see [22, Definition 10.3] for thedefinition of skew exterior algebra. This tensor product is isomorphic to a directsum of copies of A ⊗ Q ′ k , therefore V R ( k , k ) = Proj( A ⊗ Q ′ k ). Example 7.10. Let Q = C i [ x, y ] , R = Q ( x ,y ) and let M = R ( x ) , in this example weare going to calculate V R ( M, C ). A Q -resolution of M is given by the skew Koszulcomplex of the sequence x, y F : 0 −→ Q y x −−−−→ Q (cid:16) x y (cid:17) −−−−−−→ Q −→ . Let E be the skew Koszul complex over the ring Q of the sequence x , y , and let e , e be the basis elements that differentiate to x , y respectively. The complex F admits a structure of color DG E -module by defining the action in the followingway: e : F x −−−→ F , e : F (cid:16) x (cid:17) −−−−−→ F ,e : F −−−→ F , e : F (cid:16) (cid:17) −−−−−→ F . We calculate Ext R ( M, C ) by calculating the homology of the complex E F, C . As F is minimal, the differential given in 5.2 reduces to left multiplication by χ ⊗ λ ,using the actions given in 5.6.For the rest of the computation let ( − ) ∗ := Hom Q ( − , C ). It is directly observedthe DG S -module E F, C can be regarded as the skew Koszul complex over S ⊗ Q C of the sequence ( χ , → Σ − S ⊗ Q F ∗ χ −−−−→ Σ − S ⊗ Q F ∗ (cid:16) χ (cid:17) −−−−−−→ S ⊗ Q F ∗ → . Therefore, Ext R ( M, C ) is free over C [ χ ,χ ]( χ ) . Furthermore, A = C [ χ , χ ] and hence,V R ( M, C ) = supp A (cid:18) C [ χ , χ ]( χ ) (cid:19) = V R ( χ ) . Example 7.11. A finite color DG R -module M is perfect over R if and only ifV( M, k ) = ∅ (cf. Proposition 7.8(5)). In fact, for a perfect color DG R -module M ,V R ( M, N ) = ∅ for all finite color DG R -modules N . In particular, it follows fromthis remark and Proposition 7.8(4) that if M is a finitely generated color R -modulethen V R ( M, N ) = V R (Ω iR ( M ) , N )for any finite color DG R -module N and any i ≥ iR ( M ) denotes the i th syzygy module of M over R .The following proof is adapated from [30, Theorem 4.3.1] by working over thesmaller commutative ring A , rather than S . We sketch the argument for the con-venience of the reader. Theorem 7.12. For finite color DG R -modules M, M ′ , N, N ′ , V R ( M, N ) ∩ V R ( M ′ , N ′ ) = V R ( M, N ′ ) ∩ V R ( M ′ , N ) . Proof. By 7.6, we can replace M, M ′ , N, N ′ with their finite Koszul resolutions andso in the sequel we assume these are all strongly perfect over Q . In particular, E ♮X,Y is a finite rank free graded S -module for X = M, M ′ and Y = N, N ′ . As A → S is a finite free extension of graded A -modules, E ♮X,Y is a finite rank freegraded A -module for X = M, M ′ and Y = N, N ′ . By applying [30, Proposition1.2.8], E X,Y is a semiprojective DG A -module; here the fact that A has finite globaldimension is essential (cf. 7.4). Thus,(8) E X,Y ⊗ L A E X ′ ,Y ′ = E X,Y ⊗ A E X ′ ,Y ′ whenever { X, X ′ } = { M, M ′ } and { Y, Y ′ } = { N, N ′ } . Also, we have the followingisomorphisms of graded A -modules E X,Y ⊗ A E X ′ ,Y ′ = ( S ⊗ Q Hom Q ( X, Y )) ⊗ A ( S ⊗ Q Hom Q ( X ′ , Y ′ )) ∼ = ( S ⊗ A S ) ⊗ Q Hom Q ( X, Y ) ⊗ Q Hom Q ( X ′ , Y ′ ) ∼ = ( S ⊗ A S ) ⊗ Q Hom Q ( X, Y ′ ) ⊗ Q Hom Q ( X ′ , Y ) ∼ = ( S ⊗ Q Hom Q ( X, Y ′ )) ⊗ A ( S ⊗ Q Hom Q ( X ′ , Y )) ∼ = E X,Y ′ ⊗ A E X ′ ,Y where the third isomorphism is induced from the natural evaluation mapHom Q ( P, Q ) ⊗ Q V ∼ = −→ Hom Q ( P, V )being an isomorphism for a strongly perfect Q -complex P and for any complex V .Tracing the differentials through the isomorphisms above verifies that this in factestablishes an isomorphism of DG A -modules E X,Y ⊗ A E X ′ ,Y ′ ∼ = E X,Y ′ ⊗ A E X ′ ,Y . Combining this with (8) establishes the following isomorphim of DG A -modules E M,N ⊗ L A E M ′ ,N ′ ∼ = E M,N ′ ⊗ L A E M ′ ,N . UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 23 Thus, by 7.3 supp A E M,N ∩ supp A E M ′ ,N ′ = supp A E M,N ′ ∩ supp A E M ′ ,N and so 7.6 yields the desired result. (cid:3) We obtain the following corollary from the symmetry of supports satisfied inTheorem 7.12. Namely, Corollary 7.13 is a consequence of Theorem 7.12; since theargument is the same as in [30, 4.3.1] we omit its proof here. Corollary 7.13. For any pair of finite color DG R -modules M, N , V R ( M, N ) =V R ( N, M ) . Moreover, the following closed subsets of Proj A coincide (1) supp A ⊗ Q ′ k (Ext R ( M, N ) ⊗ Q ′ k ) ; (2) supp A ⊗ Q ′ k (Ext R ( N, M ) ⊗ Q ′ k ) ; (3) V R ( M, N ) ∩ V R ( k , k )(4) V R ( M, k ) ∩ V R ( k , N ) ; (5) V R ( M, k ) ∩ V R ( N, k ) ; (6) V R ( k , M ) ∩ V R ( k , N ) .In particular, V R ( M, M ) ∩ V R ( k , k ) = V R ( M, k ) = V R ( k , M ) . Vanishing of Ext modules We continue with the usual hypothesis that R is a skew complete intersectionas in Section 2 and the hypothesis used in the last section that the group of colorsof Q is finite. We apply the facts from the previous section to obtain the followingresults over such skew complete intersections. Proposition 8.1. Let R be a skew complete intersection with a finite group ofcolors. For a finite color DG R -module M , the following are equivalent: (1) M is perfect over R ; (2) Ext ≫ R ( M, M ) = 0 .In particular, if M is a finitely generated color R -module, then M has finite pro-jective dimension over R if and only if Ext ≫ R ( M, M ) = 0 .Proof. The implication “(1) implies (2)” is trivial, so we assume Ext ≫ R ( M, M ) = 0 . By Proposition 7.8(5), it follows that V R ( M, M ) = ∅ . It follows from Corollary 7.13that ∅ = V R ( M, M ) = V R ( M, M ) ∩ V R ( k , k ) = V R ( M, k ) . Now the desired result is obtained from Example 7.11. (cid:3) Proposition 8.2. Let R be a skew hypersurface with finite group of colors. If M and N are finite color DG R -modules such that Ext ≫ R ( M, N ) = 0 , then M or N is a perfect DG R -module. In particular, if M and N are finitely generated color R -modules such that Ext ≫ R ( M, N ) = 0 , then pd R M < ∞ or pd R N < ∞ . Proof. By Corollary 7.13 it follows that(9) ∅ = V R ( M, N ) = V R ( M, N ) ∩ V R ( k , k ) = V R ( M, k ) ∩ V R ( N, k ) . The assumption that R is a hypersurface implies Proj k [ χ t ] = { (0) } and so V R ( M, k )and V R ( N, k ) are naturally identified with subsets of { (0) } . Therefore, from (9) itfollows that one of V R ( M, k ) or V R ( N, k ) must be empty. Hence, M or N is perfectover R (cf. Example 7.11). (cid:3) Theorem 8.3 (Generalized Auslander-Reiten Conjecture) . Let R be a skew com-plete intersection with finite group of colors and let M be a finitely generated color R -module. If Ext iR ( M, M ⊕ R ) = 0 for all i > r , then pd R M ≤ r .Proof. It follows from Proposition 8.1 that M has finite projective dimension. Thetheorem now follows from the following(10) pd R M = sup { i | Ext iR ( M, R ) = 0 } , as the right hand side is clearly at most r by assumption. Hence, we prove that(10) holds.First, by the graded version of Nakayama’s Lemmapd R M = sup { i | Ext iR ( M, k ) = 0 } . Since M has finite projective dimension (10) holds by the fact above and by usingthe exact sequence 0 → R + → R → k → 0, where R + is the ideal generated by theelements of R of positive internal degree. (cid:3) Remark . It is proved in [33] that the ring k q [ x , x ] / ( x , x ) does not satisfythe Generalized Auslander-Reiten Conjecture whenever q , is not a root of unity.We point out that the module considered in [33] is ( x + x ), which is not a colormodule. It is unknown whether a skew complete intersection with infinite group ofcolors satisfies the Generalized Auslander-Reiten Conjecture on color modules.9. Symmetry in complexity We continue with the usual assumption that R = Q/ ( f , . . . , f c ) is a skew com-plete intersection.9.1 . Let { b i } be a sequence of nonnegative integers. Recall that the complexity of { b i } , denoted by cx { b i } , is the least integer d such that there exists a > b i ≤ ai d − for all i ≫ . Let M and N be finite color DG R -modules. The complexity of the pair ( M, N )is defined as cx R ( M, N ) = cx { dim k (Ext iR ( M, N ) ⊗ R k ) } . In [22, Corollary 10.10] the first two authors show cx R ( k , k ) = c . In fact, astronger statement about the Poincar´e series of k is determined. To elaborate onthis we introduce the following notation.9.2 . Let M and N be finite color R -modules. The Poincar´e series of ( M, N ) isP RM,N ( t ) := X i ≥ dim k (Ext iR ( M, N ) ⊗ R k ) t i . Also, we define the Poincar´e series of M to be P RM ( t ) := P RM, k ( t ) . Finally, for a(cohomologically) graded k -module M we let its Hilbert series beH M ( t ) = X i ∈ Z dim k M i t i . The stronger statement from [22, Corollary 10.8], mentioned above, saysP R k ( t ) = (1 + t ) n (1 − t ) c UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 25 and so there exists a polynomial p of degree c − p ( i ) = dim k Ext iR ( k , k )for all i ≫ . The following theorem generalizes the facts above to arbitrary pairsof modules over a skew complete intersection. We emphasize that Theorem 9.3,and its corollaries, do not require the assumption from the previous section thatthe parameters q i,j should be roots of unity. Theorem 9.3. Let R be a skew complete intersection of codimension c and M and N be finite color R -modules. If Ext R ( M, N ) ⊗ R k = 0 , then the formal powerseries (1 − t ) cx R ( M,N ) P RM,N ( t ) is a polynomial with integer coefficients that has no root at t = 1 . In particular, cx R ( M, N ) ≤ c .Proof. By Theorem 6.4, Ext R ( M, N ) is a finitely generated color module over S = Q [ χ , . . . , χ c | G ( χ i ) = G ( f i ) − , | χ i | = 2], a color commutative polynomial ring.Hence, M := Ext R ( M, N ) ⊗ Q k is a finitely generated color module over S := S ⊗ Q k . As each χ i has cohomological degree 2, M decomposes as a direct sum of finitelygenerated color S -modules M = M even ⊕ M odd . Also, we have the equalitiesP RM,N ( t ) = H M ( t ) = H M even ( t ) + H M odd ( t ) , and cx R ( M, N ) = max (cid:0) cx { dim k M i } , cx { dim k M i +1 } (cid:1) . By the previous two displays, it suffices to prove the desired result when M isconcentrated solely even degrees or solely in odd degrees; so we assume, withoutloss of generality, M is concentrated in even degrees. Therefore, for the rest of theproof we can regrade S by assigning each χ i cohomological degree 1, and M willbe a finitely generated S -module.Next, as S has finite global dimension it follows that M admits a boundedresolution by finite rank free color S -modules. Now using that the Hilbert series isadditive along exact sequences and H S ( t ) = (1 − t ) − c it follows thatH M ( t ) = q ( t )(1 − t ) c for some polynomial q ( t ) with integer coefficients. By canceling the common factorsof (1 − t ) we can write H M ( t ) = p ( t )(1 − t ) c ′ for some polynomial with integer coefficients p ( t ), where p (1) = 0 and c ′ ≤ c . Nowa direct calculation shows that c ′ is exactly cx { dim k M i } , as needed. (cid:3) Corollary 9.4. Every finitely generated color R -module over a skew complete in-tersection has rational Poincar´e series. Corollary 9.5. Let M be a finite color R -module with a minimal free resolution F ≃ −→ M over R . Then cx { rank R F i } ≤ c. Proof. This follows immediately from Theorem 9.3 with N = k . (cid:3) Remark . More can be said about specific resolutions over R . Namely, let M be a finite color DG R -module and fix a Koszul resolution F ≃ −→ M . The sameargument as in [3, Theorem 2.4] shows that the resolution of the diagonal in The-orem 2.3 determines an R -semifree resolution of M , depending on the choice of F . In particular, M will admit a semifree R -resolution whose underlying graded R -module is R ⊗ Q Hom Q ( S, Q ) ⊗ Q F. From this one obtains a second proof of Corollary 9.5 since F can be taken to be astrongly perfect Q -complex (cf. 6.3).For the remainder of the section we assume Q is a skew polynomial ring witheach q i,j a root of unity. Theorem 9.7. Let R be a skew complete insersection with finite group of colors.If M and N are finite color DG R -modules, then cx R ( M, N ) = cx R ( N, M ) .Proof. Adopting the notation set from 7.4, Ext R ( M, N ) is a finitely generated mod-ule over the S -subalgebra A . The upshot is that we may instead compute complexityusing A rather than S . Also, since Q ′ → Q is a module finite extension we have(11) cx R ( M, N ) = cx { dim k (Ext iR ( M, N ) ⊗ Q ′ k ) } . Therefore, cx R ( M, N ) = cx { dim k (Ext iR ( M, N ) ⊗ Q ′ k ) } = dim A Ext R ( M, N ) ⊗ Q ′ k = dim supp A Ext R ( M, N ) ⊗ Q ′ k = dim supp A Ext R ( N, M ) ⊗ Q ′ k = dim A Ext R ( N, M ) ⊗ Q ′ k = cx { dim k (Ext iR ( N, M ) ⊗ Q ′ k ) } = cx R ( N, M );where the first and last equalities are justified by (11), the fourth equality is fromCorollary 7.13, and the rest are standard since we are working over the gradedcommutative ring A (see, for example, [13, Section 4.1]). (cid:3) Corollary 9.8. Let R be a skew complete intersection with finite group of colors.Let M and N be finite color DG R -modules, then Ext ≫ R ( M, N ) = 0 ⇐⇒ Ext ≫ R ( N, M ) = 0 . Acknowledgements We thank Benjamin Briggs for references regarding braided Hochschild cohomol-ogy and Jason Gaddis for sharing his notes on the Generalized Auslander-ReitenConjecture. Appendix A. Skew divided powers algebra In this appendix, we show that the dual of a color polynomial ring under theconvolution product is isomorphic (as an algebra) to a color divided powers algebraand vice versa. For background regarding skew divided powers algebras, see [22,Section 6]. UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 27 Let Q be a color commutative k -algebra. That is, we assume that Q admits a G -grading where G is an abelian group, and that ab = κ ( σ, τ ) ba for all G -homogeneouselements a ∈ Q σ and b ∈ Q τ , where κ : G × G → k ∗ is an alternating bicharacter of G . This bicharacter κ is fixed throughout. When necessary, we denote the G -degreeof a homogeneous element by G ( x ), and we will abuse notation and write κ ( a, b )for κ ( G ( a ) , G ( b )).Next, we let A = Q [ x , . . . , x n | G ( x i ) = σ i , | x i | = d i , i = 1 , . . . , n ], with σ i ∈ G and d i positive integers. We denote a monomial in A by x α where α = ( α , . . . , α n ) ∈ N n is an exponent vector.Let ∆ : A → A ⊗ Q A be the coproduct defined by setting ∆( x i ) = x i ⊗ ⊗ x i and extending by products and linearity. For a general element of A , weuse Sweedler’s notation and let ∆( a ) = P a (1) ⊗ a (2) . Note that this coproduct isbihomogeneous for the canonical bigrading on A ⊗ Q A . Using the definition of ∆,a straightforward calculation shows that(12) ∆( x α ) = X β + γ = α C ( x β , x γ ) − (cid:18) αβ (cid:19) x β ⊗ x γ , where (cid:0) αβ (cid:1) = (cid:0) α β (cid:1) · · · (cid:0) α n β n (cid:1) , and C ( x β , x γ ) is the element of k ∗ that satisfies x β x γ = C ( x β , x γ ) x β + γ . Note that C ( − , − ) is a bicharacter defined on the monoid ofmonomials (but is not alternating), and satisfies(13) κ ( x β , x γ ) = C ( x β , x γ ) C ( x γ , x β ) − . See [21] for more details regarding the C ( − , − ) pairing.Let A ∗ = Hom Q ( A, Q ), note that A ∗ is a free left and right Q -module, and isspanned by the dual basis of monomials ( x α ) ∗ ; we denote the element ( x α ) ∗ by ξ α . Further, if the bidegree of x α is ( σ, d ), then the bidegree of ( x α ) ∗ is ( σ − , − d ).We may now define a product on A ∗ using the coproduct on A . Indeed, for any ϕ, ψ ∈ A ∗ and a ∈ A bihomogeneous:(14) ( ϕψ )( a ) = X κ ( ψ, a (1) ) ϕ ( a (1) ) ψ ( a (2) ) . The reason for the appearance of the κ factor is due to its presence in thecanonical isomorphism V ∗ ⊗ Q W ∗ ∼ = ( V ⊗ Q W ) ∗ in the category of color Q -modules.Before continuing, we prove a lemma regarding this product, as it demonstratessome of the ideas needed to justify claims which follow. Lemma A.1. Let β and γ be exponent vectors, and let ξ β , ξ γ be the duals of x β and x γ , respectively. Then one has ξ β ξ γ = C ( x γ , x β ) − (cid:18) β + γβ (cid:19) ξ β + γ . Proof. The binomial coefficient present on the right hand side comes from thebinomial coefficient present in the coproduct formula (12) above. By the coproductformula and the definition of the convolution product (14), the additional unitfactor is equal to κ ( ξ γ , x β ) C ( x β , x γ ) − . Since κ ( ξ γ , x β ) C ( x β , x γ ) − = κ ( x β , x γ ) C ( x β , x γ ) − = C ( x γ , x β ) − , the result follows. (cid:3) The algebra A ∗ also carries a system of skew divided powers. These are definedby using divided power binomial expansion and the following definition of dividedpower of a monomial: ( ξ α ) ( k ) = C ( x α , x α ) − ( k ) (cid:20) α k (cid:21) ξ k α where for h and k integers, one has (cid:20) hk (cid:21) = ( hk )! k !( h !) k , and for α an exponent vector,one has (cid:20) α k (cid:21) = (cid:20) α k (cid:21) · · · (cid:20) α n k (cid:21) .The proof that A ∗ satisfies the axioms of a skew divided powers algebra followsfrom careful use of the fact that C ( − , − ) and κ ( − , − ) are bicharacters, togetherwith identities involving binomial coefficients and the bracket notation introducedabove. We provide here a proof of the following equality as an example. Proposition A.2. Let x and y be elements of A ∗ that are homogeneous with respectto all gradings. Then one has ( xy ) ( k ) = κ ( y, x )( k ) x k y ( k ) . Proof. Using divided powers binomial expansion, it is enough to prove this formulafor x = ξ β and y = ξ γ . In this case, it is clear that both sides of the claimedequality evaluate to a scalar multiple of ξ k ( β + γ ) . The scalar on the left hand sideis made up of a constant involving C ( − , − ), and combinatorial constants: C ( x γ , x β ) − k C ( x β + γ , x β + γ ) − ( k ) (cid:18) β + γβ (cid:19) k (cid:20) β + γ k (cid:21) . By induction, it follows that( ξ β ) k = C ( x β , x β ) − ( k ) (cid:18) ββ (cid:19) · · · (cid:18) k ββ (cid:19) ξ β . Therefore, the scalar on the right hand side is therefore the product of κ ( x γ , x β )( k ) C ( x β , x β ) − ( k ) C ( x γ , x β ) − ( k ) C ( x k γ , x k β ) − and the combinatorial constant (cid:18) ββ (cid:19) · · · (cid:18) k ββ (cid:19)(cid:20) γ k (cid:21)(cid:18) k ( β + γ ) k β (cid:19) . It is a straightforward matter to check that the combinatorial constants on eitherside of the equality agree. To check that that the constants involving κ ( − , − ) and C ( − , − ) agree, one uses that C ( − , − ) is a bicharacter (13). (cid:3) We may use the product on A to define a coproduct on A ∗ as well, by declaring∆( ξ i ) = ξ i ⊗ ⊗ ξ i and extending ∆ to all of A ∗ as in (12). Using this coproduct,we may in turn consider the algebra A ∗∗ as before. However, one has the followingresult, whose proof follows along the same lines as in the case of a commutativering Q : Proposition A.3. The algebras A and A ∗∗ are isomorphic as color commutativegraded Q -algebras. In particular, the graded Q -dual of a skew divided powers algebraover Q is isomorphic to a color polynomial extension of Q . UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 29 Appendix B. An adjunction isomorphism The goal of this appendix is to prove Proposition B.3. In the commutative case,this follows immediately from [27, (8.7)].Let A be a graded color commutative DG Q -algebra where Q is a color commu-tative connected graded algebra over a fixed base field k ; set A e := A ⊗ Q A op to bethe enveloping DG algebra of A over Q and, as usual, A is regarded as a color DG A e -algebra via the multiplication map µ : A e → A given by a ⊗ b ab. Fix a color DG A-module M , then we define a pair of functors (depending on M ) h : Mod ( A ) → Mod ( A e ) and t : Mod ( A e ) → Mod ( A )given by h := Hom Q ( M, − ) and t := −⊗ A M, respectively. The A e -module structureon h ( N ) = Hom Q ( M, N ) is given by a ⊗ b · f := ( − | f || b | κ ( b, f ) af ( b − )while the A -action on t ( N ) is obvious one on the left of N in N ⊗ A M .B.1 . Let N be a color DG A -module. We let ev N : Hom Q ( M, N ) ⊗ A M → N bethe evaluation map. Namely, ev N ( g ⊗ m ) := g ( m ) . It is straightforward to check that ev N is a morphism of color DG A -modules.B.2 . Fix a color DG A e -module X and a color DG A -module N . Let f : X ⊗ A M → N be a morphism of color DG A -modules and consider ˜ f : X → Hom Q ( M, N ) givenby x f ( x ⊗ − ) . Observe that˜ f ( a ⊗ b · x ) = f (( − | b || x | κ ( b, x ) axb ⊗ − )= ( − | b || x | κ ( b, x ) f ( axb ⊗ − )= ( − | a || f | + | b || x | κ ( b, x ) κ ( f, a ) af ( xb ⊗ − )= ( − | a || f | + | b || x | κ ( b, x ) κ ( f, a ) af ( x ⊗ b − )= ( − | a || f | + | b || x | + | b || f | + | b || x | κ ( b, x ) κ ( f, a ) κ ( ˜ f ( x ) , b ) a ⊗ b · ˜ f ( x )= ( − ( | a | + | b | ) | ˜ f | κ ( ˜ f , a ⊗ b ) a ⊗ b · ˜ f ( x ) , that is, ˜ f is left color A e -linear. Proposition B.3. With the notation above, h is a right adjoint to t . In particular,for each DG A e -module X and DG A -module N the following maps are inverseisomorphisms that are natural in X , M and N : (1) Φ : Hom A ( t ( X ) , N ) → Hom A e ( X, h ( N )) given by f ˜ f (2) Ψ : Hom A e ( X, h ( N )) → Hom A ( t ( X ) , N ) given by g ev N ◦ ( g ⊗ id M ) . Proof. We directly check the maps defined in (1) and (2) are mutually inverse toone another. To see this considerΨΦ( f )( x ⊗ m ) = ev N ◦ ( ˜ f ⊗ id M )( x ⊗ m )= ev N ( f ( x ⊗ − ) ⊗ m )= f ( x ⊗ m ) , and ΦΨ( g )( x ) = Φ (cid:16) ev N ◦ ( g ⊗ id M ) (cid:17) ( x )= ev N ( g ( x ) ⊗ − )= g ( x ) . Therefore ΦΨ = id and ΨΦ = id, justifying the proposition. (cid:3) References 1. Luchezar L Avramov, Modules of finite virtual projective dimension , Inventiones mathemati-cae (1989), no. 1, 71–101.2. Luchezar L. Avramov, Infinite free resolutions , Six lectures on commutative algebra (Bel-laterra, 1996), Progr. Math., vol. 166, Birkh¨auser, Basel, 1998, pp. 1–118. MR 16486643. Luchezar L Avramov and Ragnar-Olaf Buchweitz, Homological algebra modulo a regular se-quence with special attention to codimension two , Journal of Algebra (2000), no. 1, 24–67.4. , Support varieties and cohomology over complete intersections , Invent. Math. (2000), no. 2, 285–318.5. Luchezar L. Avramov and Srikanth B. Iyengar, Constructing modules with prescribed coho-mological support , Illinois J. Math. (2007), no. 1, 1–20. MR 23461826. Luchezar L. Avramov, Srikanth B. Iyengar, Joseph Lipman, and Suresh Nayak, Reduction ofderived Hochschild functors over commutative algebras and schemes , Adv. Math. (2010),no. 2, 735–772. MR 25655487. Luchezar L. Avramov, Srikanth B. Iyengar, Saeed Nasseh, and Sean Sather-Wagstaff, Homol-ogy over trivial extensions of commutative DG algebras , Comm. Algebra (2019), no. 6,2341–2356. MR 39571018. John C. Baez, Hochschild homology in a braided tensor category , Trans. Amer. Math. Soc. (1994), no. 2, 885–906. MR 12409429. Hans-Joachim Baues and Teimuraz Pirashvili, Comparison of Mac Lane, Shukla andHochschild cohomologies , J. Reine Angew. Math. (2006), 25–69. MR 227056610. Petter Andreas Bergh, Ext-symmetry over quantum complete intersections , Arch. Math.(Basel) (2009), no. 6, 566–573. MR 251616211. Petter Andreas Bergh and Karin Erdmann, Homology and cohomology of quantum completeintersections , Algebra Number Theory (2008), no. 5, 501–522. MR 242945112. , The Avrunin-Scott theorem for quantum complete intersections , J. Algebra (2009), no. 2, 479–488. MR 252910013. Winfried Bruns and J¨urgen Herzog, Cohen-Macaulay rings , Cambridge Studies in AdvancedMathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 125195614. Ragnar-Olaf Buchweitz, Edward L. Green, Dag Madsen, and Ø yvind Solberg, FiniteHochschild cohomology without finite global dimension , Math. Res. Lett. (2005), no. 5-6, 805–816. MR 218924015. Ragnar-Olaf Buchweitz and Collin Roberts, The multiplicative structure on Hochschild co-homology of a complete intersection , J. Pure Appl. Algebra (2015), no. 3, 402–428.MR 327936316. Jesse Burke and Mark E. Walker, Matrix factorizations in higher codimension , Trans. Amer.Math. Soc. (2015), no. 5, 3323–3370. MR 331481017. Jon F. Carlson, The varieties and the cohomology ring of a module , J. Algebra (1983),no. 1, 104–143. MR 72307018. Jon F. Carlson and Srikanth B. Iyengar, Thick subcategories of the bounded derived categoryof a finite group , Trans. Amer. Math. Soc. (2015), no. 4, 2703–2717. MR 3301878 UPPORT VARIETIES OVER SKEW COMPLETE INTERSECTIONS 31 19. Karin Erdmann and Magnus Hellstrøm-Finnsen, Hochschild cohomology of some quantumcomplete intersections , J. Algebra Appl. (2018), no. 11, 1850215, 22. MR 387909120. Yves F´elix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory , GraduateTexts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 180284721. Luigi Ferraro, Desiree Martin, and Frank Moore, The Taylor resolution over a skew polyno-mial ring , in preparation.22. Luigi Ferraro and W. Frank Moore, Differential graded algebra over quotients of skew poly-nomial rings by normal elements , Trans. Amer. Math. Soc. (2020), 7755–7784.23. Eric M. Friedlander and Brian J. Parshall, Support varieties for restricted Lie algebras , Invent.Math. (1986), no. 3, 553–562. MR 86068224. Tor H. Gulliksen and Gerson Levin, Homology of local rings , Queen’s Paper in Pure andApplied Mathematics, No. 20, Queen’s University, Kingston, Ont., 1969. MR 026222725. David A Jorgensen, Support sets of pairs of modules , Pacific journal of mathematics (2002), no. 2, 393–409.26. Saunders Mac Lane, Homologie des anneaux et des modules , Colloque de topologie alg´ebrique,Louvain, 1956, Georges Thone, Li`ege; Masson & Cie, Paris, 1957, pp. 55–80. MR 009437427. Saunders MacLane, Homology , first ed., Springer-Verlag, Berlin-New York, 1967, DieGrundlehren der mathematischen Wissenschaften, Band 114. MR 034979228. D. K. Nakano and J. H. Palmieri, Support varieties for the Steenrod algebra , Math. Z. (1998), no. 4, 663–684. MR 162195529. Steffen Oppermann, Hochschild cohomology and homology of quantum complete intersections ,Algebra Number Theory (2010), no. 7, 821–838. MR 277687430. Josh Pollitz, Cohomological supports over derived complete intersections and local rings , arXivpreprint, arXiv:1912.12009.31. Daniel Quillen, On the (co-) homology of commutative rings , Applications of Categorical Alge-bra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), Amer. Math. Soc., Providence,R.I., 1970, pp. 65–87. MR 025706832. , The spectrum of an equivariant cohomology ring. I, II , Ann. of Math. (2) (1971),549–572; ibid. (2) 94 (1971), 573–602. MR 29869433. Rainer Schulz, A nonprojective module without self-extensions , Arch. Math. (Basel) (1994),no. 6, 497–500. MR 127410334. Umeshachandra Shukla, Cohomologie des alg`ebres associatives , Ann. Sci. ´Ecole Norm. Sup.(3) (1961), 163–209. MR 0132769 Department of Mathematics, Texas Tech University, Lubbock, TX 79409, U.S.A. Email address : [email protected] Department of Mathematics & Statistics, Wake Forest University, Winstom-Salem,NC 27109, U.S.A. Email address : [email protected] Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A. Email address ::