ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS YOUSUF A. ALKHEZI & DAVID A. JORGENSEN Abstract. We construct tensor products of complete resolutions of finitely generated modules over Noetherian rings. As applications we prove that there exists complete resolutions over artinian Gorenstein rings whose sequence of negative Betti numbers grow exponentially, while the sequence of positive Betti numbers grow polynomially of any degree 0. We also describe complete resolutions of the simple module over groups algebras of elementary abelian groups and quantum complete intersections. 1. Introduction Complete resolutions are doubly infinite exact sequences of projective modules (see, for example, [CE], [B] and [R].) They arose historically in group representation theory as a means of defining the Tate homology and cohomology groups. They are in fact easily seen to exist over any ring where the projectives and injectives coincide. In more general settings, one typically imposes additional conditions in the definition of complete resolution to allow for a viable theory of Tate (co)homology. In this paper we adopt the definition of complete resolution in [AM]. in particular, we assume that the projective modules in a complete resolution are finitely generated. The point of this paper is to construct complete resolutions as the tensor product of simpler complete resolutions. The main tool here is the pinched tensor product of complexes, introduced in [CJ]. As applications we construct asymmetric minimal complete resolutions over commutative artinian Gorenstein rings in which growth of the positive (negative) Betti numbers is polynomial, of any degree, while the growth of the negative (positive) Betti numbers is exponential. We also describe in terms of the pinched tensor product minimal complete resolutions of the simple module over the group algebra of a finite elementary abelian group, and over quantum complete intersections. 2. Complete resolutions and the pinched tensor product Let R be an associative ring. By a complex of (left) R-modules C we mean a chain complex C n+1 C : C n+1 C C n n C n 1 with (left) R-module homomorphisms C n.. Date: September 20, 2018. 2010 Mathematics Subject Classification. 16E05, 16E10, 13D02. Key words and phrases. Totally acyclic complex, Complete resolution, Pinched tensor product 1

2 YOUSUF A. ALKHEZI & DAVID A. JORGENSEN Let C and D be two complexes of R-modules. Then by a chain map f : C D we take to mean a family of homomorphisms f = {f n : C n D n } such that f n 1 n C = n D f n for all n. For a complex C of R-modules, we denote by C n the hard truncation below of C at n, that is, C n+1 C n : C n+1 C n 0 0 so that (C n ) i = 0 for i < n and i C = 0 for i n. For a complex C we define the shift (or suspension) ΣC of C as the complex with (ΣC) n = C n 1. There is a natural degree 1 isomorphism of complexes σ C : C ΣC which reduces by one the degree of every element of C. The differential of ΣC is given by ΣC n = σ C C n 1(σ C ) 1 Assume now that R is Noetherian on both sides. The following definitions are due to Avramov and Martsinkovsky [AM]. Definition 2.1. An acyclic complex C of finitely generated projective R-modules is called totally acyclic if the complex Hom R (C, R) is also acyclic. Definition 2.2. A complete projective (free) resolution of a finitely generated R- module M is a diagram C τ F π M, where F π M is a projective (free) resolution of M, C is a totally acyclic complex of finitely generated projective (free) R-modules, and τ is a chain map such that τ i is an isomorphism for all i 0. We often abuse terminology and say that C is a complete projective (free) resolution of M. Pinched tensor products. We now recall the main tool of this paper, a variant of the tensor product of complexes, introduced in [CJ], and called the pinched tensor product. We first recall the definition of the ordinary tensor product of complexes. Throughout this subsection k is a commutative ring. Definition 2.3. Let C and D be complexes of k-modules. C k D of C and D is specified by setting and the differential is defined by (C k D) n = i Z C i k D n i C kd n (c d) = i C (c) d + ( 1) i c n i(d) D for c C i and d D n i. The tensor product Definition 2.4. Let C and D a complexes of k-modules. We define the pinched tensor product C k D of C and D by setting { (C k D) (C 0 k D 0 ) n for n 0 n = (C 1 k (ΣD) 0 ) n for n 1 with differential defined by C k D n = C 0 k D 0 n for n 1 0 C σ 1 D 0 D for n = 0 C 1 k (ΣD) 0 for n 1 n

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS 3 It is clear from the definition that C k D n 1 C k D n = 0 for n 2 and n 1. The reader may check that the same holds for n = 0, 1. We record a key fact of the pinched tensor product, one that does not hold for the ordinary tensor product of complexes. Fact 2.5. Assume that C and D are complexes of finitely generated k-modules. Then C k D is a complex of finitely generated k-modules. Moreover, if C and D are complexes of projective (free) k-modules, then so is C k D. Definition 2.6. Let M be a k-module with a complete free resolution C F M. For a k-module N, the Tate homology of M with coefficients in N is defined as Tor k n(m, N) = H n (C k N). As is the case for ordinary Tor, Tate homology is independent (up to isomorphism) of the choice of complete free resolution. One also has Tor k n(m, N) = Tor k n(m, N) for n 0. The following theorem from [CJ, Theorem 3.5] is used in the proofs below. Theorem 2.7. Let C be a complete resolution of a k-module M, and D be an acyclic complex of k-modules with N = Im D 0. Then for all n Z we have isomorphisms H n (C k D) = Tor k n(m, N) 3. Tensor products of complete resolutions The goal of this section is to prove that, under suitable hypotheses, the pinched tensor product of complete resolutions is a complete resolution. We start with a discussion on tensor products and several key preliminary results. We assume throughout that k is a fixed commutative ring with unity. Tensor products of k-algebras. Let R and S be k-algebras. Then, R k S is a k-algebra with multiplication given by (r s)(r s ) = rr ss. We state some basic facts involving modules and maps over R k S. If M is a left R-module and N is a left S-module. Then M k N is a left R k S-module, with multiplication (r s)(m n) = rm sn. Let f : M M be a homomorphism of left R-modules, and g : N N be a homomorphism of left S-modules. Then there exist a homomorphism f g : M k N M k N of left R k S-modules defined by (f g)(m n) = f(m) g(n) for n N and m M. Note that if P is a projective R-module and Q is a projective S-module, then P k Q is a projective R k S-module; if both P and Q are finitely generated, then so is P k Q. In particular, the tensor product of a (finitely generated) free R-module and a (finitely generated) free S-module yields a (finitely generated) free R k S-module. Specifically, if F is a finitely generated free R-module with basis α 1,..., α m, and G is a finitely generated free S-module with basis β 1,..., β n. Then F k G is a finitely generated free R k S-module with basis {α i β j } 1 i m,1 j n. In particular, we have (1) rank R k S(F k G) = rank R (F ) rank S (G)

4 YOUSUF A. ALKHEZI & DAVID A. JORGENSEN 3.1. The previous paragraphs apply to show that if C is a complex of left R-modules and D is a complex of left S-modules, then C k D and C k D are complexes of left R k S-modules. Moreover, if C and D are complexes of projective (free) left R-modules, then C k D and C k D are complexes of projective (free) R k S- modules; if C and D are complexes of finitely generated projective (free) S-modules, then C k D is a complex of finitely generated projective (free) R k S-modules. Proposition 3.2. Let C be a complex of finitely generated projective left R-modules, and D be a complex of finitely generated projective left S-modules. Assume either R or S is flat as a k-module, and one of the following conditions holds. (1) C n = D n = 0 for n < 0 and H n (C) = H n (D) = 0 for n 0 (2) C n = D n = 0 for n > 0 and H n (C) = H n (D) = 0 for n 0. Then C k D is a complex of finitely generated projective R k S-modules with H n (C k D) = 0 for n 0 (and also possessing the respective boundedness condition). Proof. The hypotheses imply that C k D is a either a first or third quadrant bicomplex with either exact rows or exact columns (except on the x- or y-axis). The result on homology is thus well-known (see, for example, [R, Exercise 10.13].) Lemma 3.3. Let F be a finitely generated free R-module and G a finitely generated free S-module. Then there exists a natural isomorphism φ : Hom R k S(F k G, R k S) Hom R (F, R) k Hom S (G, S) of finitely generated free R k S-modules. Proof. Let {α i } 1 i f be a basis for F and {α i } 1 i f the dual basis of Hom R (F, R). Similarly, let {β j } 1 j g be a basis for G, and {β j } 1 j g its dual basis. Then a basis for F k G is {α i β j } 1 i f,1 j g, and we denote the dual basis of Hom R k S(F k G, R k S) by {(α i β j ) } 1 i f,1 j g. Now define a map φ : Hom R k S(F k G, R k S) Hom R (F, R) k Hom S (G, S) by φ((α i β j ) ) = αi β j, and extend by linearity. One easily sees that φ defines an R k S-linear homomorphism. Moreover, φ has the obvious inverse given by αi β j (α i β j ). Lemma 3.4. Let f : F F be a homomorphism of finitely generated free R- modules, and g : G G be a homomorphism of finitely generated free S-modules. Then for the isomorphisms φ defined in Lemma 3.3, the following diagram commutes Hom R k S(F k G, R k S) Hom R k S(f g,r k S) Hom R k S(F k G, R k S) φ Hom R (F, R) k Hom S (G, S) Hom R (f,r) k Hom S (g,s) Hom R (F, R) k Hom S (G, S) φ Proof. Let {α i }, {α i }, {β i}, {β i } be (finite) bases for F, F, G, and G, respectively. Write f(α i ) = l a liα l, and g(β i) = l b liβ l with a li, b li k. We will show that the square commutes for each basis element (α i β j ) of Hom R k S(F k G, R k S). Note that (α i β j ) (f g) = m,n (a im b jn )(α m β n ). Indeed, for α p β q

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS 5 F k G we have (α i β j ) (f g)(α p β q ) = (α i β j ) ( l a lpα l l b lqβ l ) = a ip b jq. Whereas m,n (a im b jn )(α m β n ) (α p β q ) = a ip b jq. Therefore we have φ ( Hom R k S(f g, R k S)((α i β j) ) ) = φ ( (α i β j) (f g) ) ( ) = φ (a im b jn )(α m β n ) m,n = m,n(a im b jn )φ((α m β m ) ) = m,n(a im α m b jn β n) On the other hand (Hom R (f, R) k Hom S (g, S)) ( φ ((α i β j) ) ) = (Hom R (f, R) k Hom S (g, S))(α i β j ) = α i f β j g = im αm m,n(a b jn βn) Thus the square commutes. Theorem 3.5. Let C be a complex of finitely generated free R-modules, and D be a complex of finitely generated free S-modules. Then there is an isomorphism of complexes Hom R k S(C k D, R k S) = Σ ( Hom R (ΣC, R) k Hom S(ΣD, S) ) of finitely generated free R k S-modules. Proof. By Lemma 3.4 we deduce that if C and D are both bounded below complexes or both bounded above complexes, then Hom R k S(C k D, R k S) = Hom R (F, R) k Hom S (G, S). We have ( Σ 1 Hom R k S(C k D, R k S) ) 0 = Hom R k S(C k D, R k S) 1 = Hom R k S((C k D) 1, R k S) = Hom R k S(C 1 k (ΣD) 0, R k S) = Hom R k S((ΣC) 0 k (ΣD) 0, R k S) = Hom R ((ΣC) 0, R) k Hom S ((ΣD) 0, S) = Hom R (ΣC, R) 0 k Hom S (ΣD, S) 0 = ( Hom R (ΣC, R) k Hom S(ΣD, S) ) 0

6 YOUSUF A. ALKHEZI & DAVID A. JORGENSEN and (Σ 1 Hom R k S(C k D, R k S) ) 1 = Hom R k S(C k D, R k S) 0 = Hom R k S((C k D) 0, R k S) = Hom R k S(C 0 k D 0, R k S) = Hom R k S((ΣC) 1 k (ΣD) 1, R k S) = Hom R ((ΣC) 1, R) k Hom S ((ΣD) 1, S) = Hom R (ΣC, R) 1 k Hom S (ΣD, S) 1 = Hom R (ΣC, R) 1 k (Σ Hom S (ΣD, S)) 0 = ( Hom R (ΣC, R) k Hom S(ΣD, S) ) 1 It remains to show the square in degrees 0 and 1 commutes: ( Σ 1 Hom R k S(C k D, R k S) ) ( 0 Σ 1 Hom R k S(C k D, R k S) ) 1 ( HomR (ΣC, R) k Hom S(ΣD, S) ) 0 ( HomR (ΣC, R) k Hom S(ΣD, S) ) 1 Unraveling the modules via the definition of pinched tensor products, it suffices to show the square Hom R k S(C 1 k (ΣD) 0, R k S) θ Hom R k S(C 0 k D 0, R k S) Hom R (C 1, R) k Hom S ((ΣD) 0, S) φ τ Hom R (C 0, R) k Hom S (D 0, S) φ commutes, where θ = Hom R k S( 0 C σ 1 D 0 D, R k S) and τ = Hom R ( 0 C, R) k Hom S (σ 1 D 0 D, S), but this is just Lemma 3.4. Corollary 3.6. Let C be a totally acyclic complex of finitely generated free R- modules, and D be a totally acyclic complex of finitely generated free S-modules. Set N = 0 D. Assume that R is finitely generated projective, and N, Hom S (N, S) are flat as a k-modules. Then C k D is a totally acyclic complex of finitely generated free R k S-modules. Proof. We already know by 3.1 that C k D is a complex of free R k S-modules. We just need to know that it is totally acyclic. Since C is an acyclic complex of finitely generated free R-modules, and R is a finitely generated projective k-module, we see that C is also an acyclic complex of finitely generated projective k-modules. Since N is flat as a k-module, we obtain H n (C k N) = 0 for all n Z. Therefore Theorem 2.7 shows that C k D is acyclic. To show that the dual Hom R k S(C k D, R ks) is acyclic, by Theorem 3.5 it suffices to prove that Hom R (ΣC, R) k Hom S(ΣD, S) is acyclic. Since ΣC is a totally acyclic complex of R-modules, Hom R (ΣC, R) is acyclic, and Hom S (ΣD, S) is acyclic since D is a totally acyclic complex of S-modules. Since Hom S (N, S) is k-flat, we obtain acyclicity of Hom R (ΣC, R) k Hom S (N, S). Noting that Im Hom S(ΣD,S) 0 = Hom S (N, S), we see by Theorem 2.7 that the complex Hom R (ΣC, R) k Hom S(ΣD, S) is acyclic, and this is what we needed to show.

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS 7 Definition 3.7. Suppose that A is a commutative local ring with maximal ideal m. A complex C of finitely generated free A-modules is called minimal if one has (C) m C. A complete free resolution C F M of the A-module M is called minimal if C and F are minimal complexes of finitely generated free A-modules. If M has a minimal complete free resolution C F M, then it is unique up to isomorphism [AM]. Lemma 3.8. Suppose that (R, m) and (S, n) are commutative local rings, and R k S is local with maximal ideal m k S + R k n. Suppose that C is a minimal complex of finitely generated free R-modules, and D is a minimal complex of finitely generated free S-modules, then C k D is a minimal complex of finitely generated free R k S-modules. Proof. We check that C k D is minimal by looking at three cases: n 1, n = 0 and n 1. For n 1, let c d C i k D n i, n i 0. Then C k D n (c d) = i C (c) d + ( 1) i c n i(d) D m C i 1 D n i + C i n D n i 1 For n = 0, let c d C 0 k D 0. Then (m k S + R k n)(c k D) n 1 C k D 0 (c d) = C 0 (c) σ D 1 D 0 (d) m C 1 n(σd) 0 (m k S + R k n)(c k D) 1 For n 1, let c d C i k (ΣD) n i, 1 i n. Then C k D n (c d) = i C (c) d + ( 1) i c n i(d) ΣD m C i 1 (ΣD) n i + C i n(σd) n i 1 (m k S + R k n)(c k D) n 1 Thus we have shown that C k D n (C k D) (m ks + R k n)(c k C k D is minimal. D), and so Theorem 3.9. Let C F M be a complete free resolution of a finitely generated R-module M over R, and D G N be a complete free resolution of a finitely generated S-module N over S. Assume that R is a finitely generated projective k-module, and that Im 0 D and Hom S (Im 0 D, S) are flat as k-modules. Then C k D F k G M k N is a complete resolution of M k N over R k S. Moreover, if (R, m) and (S, n) are local, and R k S is local with maximal ideal m k S +R k n, and C and D are minimal, then so is C k D. Proof. Given the minimality of C and D, Lemma 3.8 shows that C k D is a minimal complex of finitely generated free R k S-modules. The fact that C k D is a totally acyclic complex is exactly Corollary 3.6. Now it is clear that C k D F k G M k N

8 YOUSUF A. ALKHEZI & DAVID A. JORGENSEN is a complete resolution of M k N over R k S: F k G is a free resolution of M k N over R k S (see Proposition 3.2) with C k D F k G eventually an isomorphism since both C F and D G are eventually isomorphisms. Definition 3.10. If C is a complex of free R-modules, then the ranks of the free modules can be encoded into the power series P C +(t) = n 0 rank R C n t n and P C (t) = n 1 rank R C n t n which call the the positive and negative Poincaré series of C, respectively. Fact 3.11. Let C be a complex of free R-modules, and D a complex of free S- modules. Then (using (1) above) we have the following formulae of Poincaré series P R k S + (t) = rank R C n t n rank S D n t n n 0 n 0 and P R k S (t) = rank R C n t n rank S D n t n+1 n 1 n 1 4. Asymmetric minimal complete resolutions Every totally acyclic complex C of finitely generated free R-modules contains a minimal such complex C as a direct summand, and this minimal complex C is unique up to isomorphism of complexes. In fact, C may be obtained from C, up to isomorphism of complex, by adding summands of the form R 1 R. Thus the Poincaré series of a nonminimal totally acyclic complex of finitely generated free modules can be quite arbitrary. It is therefore only of interest to understand the possible shapes of Poincaré series of minimal totally acyclic complexes of finitely generated free R-modules. It is proved in [AB] that the coefficients of both the positive and negative Poincaré series of a minimal totally acyclic complex C, that is, the ranks of the free modules of C, grow polynomially of the same degree. In contrast to this fact, it is proved in [JS] that there exists an artinian Gorenstein ring R, which is not a complete intersection and defined below, which admits a minimal totally acyclic complex C in which {rank R C n } n 1 grows exponentially, while rank R C n = 2 for all n 0. In this section, using the pinched tensor product, we will extend this result to show that for all d 0 there exist artinian Gorenstein rings A admitting minimal totally acyclic complexes E such that {rank R E n } n 1 grows exponentially whereas {rank R D n } n 0 grows polynomially of degree d. Let k be a field which is not algebraic over a finite field and let α k be an element of infinite multiplicative order. The ring R = k[x 1, X 2, X 3, X 4, X 5, X 6 ]/I

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS 9 is defined as the quotient of a polynomial ring in six variables (each of degree one) by the ideal I minimally generated by the following fifteen quadratic polynomials: X 6, X 2 X 6 X 1 X 4 αx 2 X 3, X 2 2, X 5 X 6 + X 3 X 5, X 2 X 3, X 2 5 X 1 X 4 (α 1)X 2 X 3, X 4 X 6 + αx 3 X 4, X 2 X 4, X 4 X 5, X 2 4 X 1 X 4 X 1 X 3, X 1 X 6 + X 1 X 5 + αx 3 X 4, X 1 X 2 X 1 X 5 X 3 X 4 + X 1 X 3, X 2 1 + (α + 1)X 2 X 3 X 3 X 5, X 2 3 Let x 1, x 2, x 3, x 4, x 5, x 6 denote the residue classes of the variables modulo I, and let m denote the ideal they generate. Proposition 4.1. ([JS, 3.1]) The ring R is local, with maximal ideal m and satisfies the following properties: (1) R is graded and has Hilbert series H R (t) = 1 + 6t + 6t 2 + t 3 ; a basis of R over k is given by the following fourteen elements: 1, x 1, x 2, x 3, x 4, x 5, x 6, x 1 x 3, x 2 x 3, x 3 x 4, x 3 x 5, x 3 x 6, x 1 x 4, x 3 x 1 x 4 ; (2) R is Gorenstein, with socle(r) = (x 1 x 3 x 4 ); (3) R is a Koszul algebra. Proved in [JS] is the following foundation for our construction in this section. Theorem 4.2. ([JS, 3.2]) Over the ring R defined above, there exists a minimal totally acyclic complex C of finitely generated free modules such that {rank R C n } n 1 grows exponentially, while rank R C n = 2 for all n 0 Now consider the k-algebra S = k[y]/(y 2 ), and S d k its d-fold tensor product, d 0, that is, S d k = S k S k k S (d times) The next proposition discusses relevant properties of A. Proposition 4.3. The ring A = R k (S d k ) is local, with maximal ideal m k S d k + d i=1 and satisfies the following properties: (1) A is graded and has Hilbert series R k S k k ith (y) k k S H A (t) = (1 + 6t + 6t 2 + t 3 )(1 + t) d (2) A is Gorenstein, with socle generated by x 1 x 2 x 3 y y. (3) A is a Koszul algebra. Proof. Property (1) is easily proved, for example, via induction. Property (2) and (3) are well-known. See, for example, [WITO] and [BF]. It is easily seen that the ring S admits the minimal totally acyclic complex D : y S y S y S y Theorem 4.4. Let A be the ring defined in Proposition 4.3. Then there exists a minimal totally acyclic complex E over A such that {rank R E n } n 0 grows polynomially of degree d, while {rank R E n } 1 grows exponentially.

10 YOUSUF A. ALKHEZI & DAVID A. JORGENSEN Proof. Consider the complex E = C d k D k where D d k = D k D k k D is the d-fold pinched tensor product of D. We note that the pinched tensor product of complexes is associative [CJ], so this product makes sense. By Theorem 3.9, E is a minimal totally acyclic complex over A. We have by Fact 3.11 the following equality. rank A E n t n = n 0 ( 2 ) ( 1 ) d 1 t 1 t so that {rank A E n } n 0 grows polynomially of degree d. We also have rank A E n t n = ( ) d rank R C n t n 1 1 t 1 n 1 n 1 so that {rank A E n } n 1 grows exponentially, since {rank R C n } n 1 does. Suppose temporarily that (A, m, k) is any commutative local ring. Recall from [A] that the complexity of an A-module M is the least integer d such that there exists a real number γ for which Tor A n (M, k) γn d 1 holds for all sufficiently large n. If no such integer d exists, we say that M has infinite complexity. If F is a minimal free resolution of M, then we have rank A F n = Tor A n (M, k) for all n 0. In terms of complexity, Theorem 4.4 can be reformulated as follows. Corollary 4.5. Let A be the ring defined in Proposition 4.3, and M = Im 0 E where E is the minimal totally acyclic complex of Theorem 4.4. Then M has complexity d + 1, whereas Hom A (M, A) has infinite complexity. As remarked earlier in this section, but now stated in terms of complexity, it is proved in [AB] (see also [BJ]) that for M = Im 0 C in a minimal totally acyclic complex C over a complete intersection R, the complexities of M and Hom R (M, R) must be the same nonnegative integer, not greater than the codimension of R. Thus the same statement fails when R is Gorenstein and not a complete intersection. 5. Applications to group algebras and quantum complete intersections In this section we give further applications of the pinched tensor product of complexes, specifically to group algebras of finite groups and quantum complete intersections. Group algebras. Let G be a finite group, and k a field. It is well-known that every finitely generated kg-module M has complete resolution (see, for example, [AM, Construction 3.6], and also [B, 5.15] or [R, 9.7]). In fact, since kg is selfinjective, complete resolutions may be obtained easily through the following construction: let P 1 P 0 be a projective resolution of M over kg, and Q 1 Q 0 be a projective resolution of (the right kg-module) Hom kg (M, kg) over kg. Then Hom kg (Q 0, kg) Hom kg (Q 1, kg) is an exact sequence of projective left kg-modules, and one may splice to obtain the complete resolution C : P 1 P 0 P 1 P 2 of M over kg, where P i 1 = Hom kg (Q i, kg), i 0. Note that we have M = Im C 0, so that C 0 coincides with a projective resolution of M. In this case we write the complete resolution simply as C M instead of C C 0 M. We note that

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS 11 a nonzero finitely generated kg-module M has a finite complete resolution if and only if it is projective. Thus in the case where the characteristic of k divides G, the order of G, (so that kg is not semisimple) there are non-projective modules, and so modules with no finite complete resolution. Now assume that G = G 1 G 2, for finite groups G 1 and G 2. We use the pinched tensor product to construct (minimal) complete resolutions of certain kg-modules from complete resolutions over kg 1 and kg 2. Indeed, since G = G 1 G 2 we have a natural isomorphism of group algebras kg = kg 1 k kg 2 given by r(g 1, g 2 ) rg 1 k g 2 for r k and g i G i, i = 1, 2. The following is a direct consequence of Theorem 3.9. Corollary 5.1. Assume that G = G 1 G 2 is a finite group, and k is a field. If C M is a complete resolution of a finitely generated kg 1 -module M, and D N is a complete resolution of a finitely generated kg 2 -module N, then C k D is a complete resolution of the finitely generated kg-module M k N. If G 1 and G 2 are p-groups, and Char k = p > 0 (so that G 1, G 2 and G are local) then C k D is minimal provided both C and D are. A key case is when M = N = k. In this case the kg-module M k N is just k, and we obtain Corollary 5.2. Assume that G = G 1 G 2 is a finite group, and k is a field. If C k is a complete resolution of k as a kg 1 -module, and D k is a complete resolution of k as a kg 2 -module, then C k D is a complete resolution of k as a kg-module. If G 1 and G 2 are p-groups, and Char k = p > 0 (so that G 1, G 2 and G are local) then C k D is minimal provided both C and D are. Finite elementary abelian groups. Recall that a finite elementary abelian group G has the form G = (Z/pZ) n = Z/pZ Z/pZ. It is well-known in this case that if k is a field with Char k = p, then kg is isomorphic to the truncated polynomial ring k[x 1 ]/(x p 1 ) k k k[x n ]/(x p n) = k[x 1,..., x n ]/(x p 1,..., xp n). We now describe a minimal complete resolution of the simple module k over this ring, via the pinched tensor product. It is straightforward to see that a minimal complete resolution of k over R i = k[x i ]/(x p i ) is given by the complex where k = Coker(R i x p 1 i D i : R i x i Ri ). R i x i Ri x p 1 i R i Corollary 5.3. Let G = (Z/pZ) n be an elementary abelian group, and k a field with Char k = p. Then, in the notation above, a minimal complete resolution of the simple module k over kg = k[x 1 ]/(x p 1 ) k k k[x n ]/(x p n) is given by the complex D = D 1 k k D n. The following acts are easily read off of the corollary: (1) P D +(t) = ( 1 1 t )n ; (2) P D (t) = t 1 1 ( 1 t ) n ; 1 (3) in fact, D is self-dual, meaning that Hom kg (D, kg) = ΣD.

12 YOUSUF A. ALKHEZI & DAVID A. JORGENSEN Quantum complete intersections. In this subsection we generalize the results of Corollary 5.3 to algebras known as quantum complete intersections. These algebras are defined as Λ = k x 1,..., x n /I where k is a field, I is the two-sided ideal of the free algebra k x 1,..., x n generated by x a i i and x j x i q ij x i x j wth a 1,..., a n integers greater than or equal to 2, and q ij k for 1 i < j n. Then Λ is self-injective and admits complete resolutions in the same way that the exist for group algebras of finite groups. Note that Λ is a graded algebra. We denote the degree of a homogeneous element r Λ by r. In [BO] Bergh and Oppermann define the twisted tensor product k[x 1 ]/(x a 1 1 ) t k k[x 2 ]/(x a 2 2 ) of algebras k[x 1 ]/(x a 1 1 ) and k[x 2]/(x a 2 2 ) to have as its underlying set the ordinary tensor product k[x 1 ]/(x a1 1 ) kk[x 2 ]/(x a2 2 ), but with twisted multiplication defined by (r s) t (r s ) = q s r 12 rr ss for some q 12 k, and for r, r k[x 1 ]/(x a 1 1 ) and s, s k[x 2 ]/(x a 2 2 ). This construction can inductively be extended to the twisted tensor product of n algebras k[x 1 ]/(x a 1 1 ) t k t k k[x n ]/(x an n ) by (r 1 r n ) t (r 1 r n) = q rj r i ij r 1 r 1 r n r n j>i One sees easily that Λ = k[x 1 ]/(x a 1 1 ) t k t k k[x n]/(x a n n ) as k-algebras. Now let F i be a graded free k[x i ]/(x ai i )-module for 1 i n. Then we can also form the twisted tensor product F 1 t k t k F n where the underlying set is the ordinary tensor product and the Λ action is defined by (r 1 r n ) t (m 1 m n ) = j>i q r j m i ij r 1 m 1 r n m n where r i k[x i ]/(x ai i ) and m i F i are homogeneous. Suppose that f i : F i F i are k[x i ]/(x a i i )-linear homomorphisms between free k[x i]/(x a i i )-modules for 1 i n. Then f 1 k k f n : F 1 t k t k F n F 1 t k t k F n is a homomorphism of free Λ-modules. We want to now describe the complete resolution of k over Λ. For that we twist the pinched tensor product. Let C be a complex of free k[x 1 ]/(x a 1 1 )-modules and D be a complex of free k[x 2 ]/(x a2 2 )-modules. We define the twisted pinched tensor product C t k D of C and D by { (C t k D) (C 0 t k n = D 0) n for n 0 (C 1 t k (ΣD) 0) n for n 1 with differential defined by C t k D n = C 0 k D 0 n for n 1 0 C σ 1 D 0 D for n = 0 C 1 k (ΣD) 0 for n 1 n

ON TENSOR PRODUCTS OF COMPLETE RESOLUTIONS 13 One inductively defines the n-fold twisted pinched tensor product (using associativity of the pinched tensor product). A minimal complete resolution of k over R i = k[x i ]/(x a i i ) is given by the complex where k = Coker(R i x a i D i : R i 1 i x i Ri ). R i x i Ri x a i 1 i R i Corollary 5.4. Consider the quantum complete intersection Λ = k[x 1 ]/(x a 1 1 ) t k t k k[x n ]/(x a n n ) Then, in the notation above, a minimal complete resolution of the simple module k over Λ is given by the complex D = D 1 t k t k D n. We list some facts easily read from the complete resolution: (1) P D +(t) = ( 1 1 t )n ; (2) P D (t) = t 1 1 ( 1 t ) n ; 1 (3) in fact, D is self-dual, meaning that Hom Λ (D, Λ) = ΣD. References [A] L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), no. 1, 71 101. [AB] L. L. Avramov and R-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285 318. [AM] L. L. Avramov, A. Martinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc., (3) 85 (2002), no. 2, 393 440. [BF] J. Backelin, R. Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 2, 85 97. [B] D. J. Benson, Representations and cohomology II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge, 1991. [BJ] P. A. Bergh, D. A. Jorgensen, On growth in minimal totally acyclic complexes, J. Commut. Algebra 6 (2014), no. 1, 17 31. [BO] P. A. Bergh, S. Oppermann, Cohomology of twisted tensor products, J. Algebra 320 (2008), no. 8, 3327 3338. [CE] H. Cartan, S. Eilenberg Homological algebra, Princeton University Press. ISBN: 0-691- 07977-3. [CJ] L. W. Christensen, D. A. Jorgensen, Tate (co)homology via pinched complexes, Trans. Amer. Math. Soc., 366 (2014), 667 689. [JS] D. A. Jorgensen, L. Şega, Asymmetric complete resolutions and vanishing of Ext over Gorenstein rings, Int. Math. Res. Not. 2005, no. 56, 3459 3477. [R] J. J. Rotman, An introduction to homological algebra, Second edition. Universitext. Springer, New York, 2009. xiv+709 pp. ISBN: 978-0-387-24527-0. [WITO] K.-i. Watanabe, T. Ishikawa, S. Tachibana, K. Otsuka, On tensor products of Gorenstein rings, J. Math. Kyoto Univ. 9 (1969) 413 423. Yousuf A. Alkhezi, Department of mathematics, Public Authority for Applied Education and Training, college of basic education, Kuwait E-mail address: ya.alkhezi@paaet.edu.kw David A. Jorgensen, Department of mathematics, University of Texas at Arlington, Arlington, TX 76019, USA E-mail address: djorgens@uta.edu