COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS

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1 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS W. FRANK MOORE Abstract. Let S and T be local rings with common residue field k, let R be the fiber product S k T, and let M be an S-module. The Poincaré series PM R of M has been expressed in terms of P M S, P k S and P k T by Kostrikin and Shafarevich, and by Dress and Krämer. Here, theorems on the structure of Ext R (k, k) and Ext R (M, k) that illuminate these equalities, as well as structure theorems for the cohomology modules of fiber products of modules are given. We also study the depth of cohomology modules over a fiber product. 1. Introduction In homological investigations one often has information on properties of a module over a certain ring, and wants to extract information on its properties over a different ring. In this paper we consider the following situation: S k T are surjective homomorphisms of rings, k is a field, R is the fiber product S k T, and M an S-module. In the introduction, we make the further hypothesis that S and T are commutative local noetherian rings with common residue field k, and M is finitely generated. First, we study the k-algebra Ext R (k, k) in terms of Ext S (k, k) and Ext T (k, k), as well the left Ext R (k, k)-module Ext R (M, k) in terms of Ext S (M, k) for an S- module M. Explicit computation of the left action of Ext R (k, k) on Ext R (M, k) leads to the following Theorem A. The homomorphism of graded k-algebras Ext S (k, k) Ext T (k, k) Ext S k T (k, k) defined by the universal property of coproducts of k-algebras is bijective. If M is an S-module, then the homomorphism of graded left Ext R (k, k)-modules Ext R (k, k) ExtS (k,k) Ext S (M, k) Ext R (M, k) defined by the multiplication map, is bijective. The numerical information carried by these isomorphisms which relates the Poincaré series PM R (t) of M over R to P M S (t), P k S(t) and P k T (t) was first proved in the case M = k by Kostrikin and Shafarevich in [6], while Dress and Krämer in [3] proved it in the setting presented here. In [8], Polishchuk and Positselski proved the preceding theorem, as well as Theorem D, when S and T are connected k-algebras, using cobar constructions. Our presentation handles both the local and graded setting, using an explicit resolution of M over R built from data over S and T. The construction works in slightly greater generality, see section 2. On the face of it, Theorem A seems to only cover a small class of R-modules. However, in [3], Dress and Krämer show that for each R-module L, its second Date: January 15, 2007, Preliminary Version. 1

2 2 W. FRANK MOORE syzygy is a direct sum of an S-module M and a T -module N. This leads to the following corollary. Corollary B. There exists an exact sequence of left graded Ext R (k, k)-modules 0 ( Σ 2 R S Ext S (M, k) ) ( Σ 2 R T Ext T (N, k) ) L L/L 2 0 where R = Ext R (k, k), S = Ext S (k, k),t = Ext T (k, k) and L = Ext R (L, k). In section 5 we use these structure theorems to compute the depth of cohomology modules. Introduced in [4] and developed extensively in [5], depth has been used in rational homotopy theory and local algebra to give information on homotopy Lie algebras of simply connected CW complexes, and of local rings. For the cohomology modules of a module over a fiber product, large depth is impossible. In fact, we prove Theorem C. Let M be an R-module. Then depth ExtR (k,k) Ext R (M, k) 1. If M is an S-module, then depth ExtR (k,k) Ext R (M, k) = 1. In particular, one has depth Ext R (k, k) = 1. Using Theorem A, we show that the functor Ext R (, k) enjoys a property similar to that of Ext (k, k) in Theorem A. Namely, Ext R (k, k) converts certain pullback diagrams of R-modules to pushout diagrams of Ext R (k, k)-modules. Theorem D. Let M, N, and H be S, T and k-modules respectively such that there exist surjective π S, respectively π T -equivariant homomorphisms M µ H ν N with ker µ = pm and ker ν = qn. The exact sequence of R-modules 0 M H N ι M N µ ν H 0 induces an exact sequence of graded left Ext R (k, k)-modules µ ν 0 Ext R (H, k) Ext R (M, k) Ext R (N, k) ι Ext R (M H N, k) 0. In particular, there is an equality of Poincaré series P R M H N(t) + (rank k H)P R k (t) = P R M (t) + P R N (t). 2. Resolutions over a fiber product We consider a diagram of homomorphisms of rings (2.a) S k T τ T where π S and π T are surjective, and S k T is the fiber product: σ S π S S k T = {(s, t) S T : π S (s) = π T (t)}. We set p = Ker π S, q = Ker π T, and m = Ker π S σ = Ker π T τ. One then has m = p q and we identify p and q with subsets of R. Every S-module is considered an R-module via σ, and similarly for T -modules. A matrix ϕ with entries in p defines a homomorphism S ϕ of free S-modules, as well as a homomorphism R ϕ of free R-modules. k π T

3 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS 3 Definition 2.1. Let A be a ring, and X a graded set, X = n 0 X n. We let A X be the graded free left A-module with basis X n in degree n, and set A X n = 0 when X n =. We call A X a graded based A-module on A with basis X. Homomorphisms of based modules are identified with their matrices in the chosen bases. For a based module A X, we identify A A A X and A X by means of the canonical isomorphism. We use A [XY] to denote the graded based A-module A X A A Y with graded basis XY = n (XY) n, where (XY) n = {xy x X i, y Y j, i + j = n}. Construction 2.2. Let M be an S-module. Let P M and E k be resolutions of M, respectively k, over S, and let T k be a resolution of k over T. Choose bases P, E, and F of the graded modules P, E, and F over S, S and T, respectively, so that E 0 = {1 S } and F 0 = {1 T }. Consider elements of P, E 1 and F 1 as letters of an alphabet. The degree of a word in this alphabet is defined to be the sum of the degrees of its letters. Let G be the set of words of the form {f 1 e 2 f 3 e 2l 2 f 2l 1 p 2l } and {e 1 f 2 e 3 e 2l 1 f 2l p 2l+1 } where e i, f i and p i range over E 1, F 1 and P respectively, and l 0. Form the free R-module R G. Every word w G has the form xw for some letter x and a (possibly empty) word w. Assume that one has (E) pe, (P ) pp, (F ) qf, and set P (x) for x P (w) = E (x)w for x E F (x)w for x F, and extend to a homomorphism of R G by R-linearity. Set i = R G i. Theorem 2.3. Assume that: M is an S-module with a free resolution P such that (P ) pp, S/p has a free resolution E such that (E) pe, T/q has a free resolution F such that (F ) qf. The maps of free modules i defined in Construction 2.2 R G: R G i i R G i 1 R G 1 1 R G 0 0, give a free resolution of the R-module M and satisfies ( R G) m( R G).

4 4 W. FRANK MOORE Remark 2.4. The first few degrees of the complex R G may be depicted as follows: R P 3 R P 2 R [F 1 P 2 ] R [E 1 F 1 P 1 ] R [F 1 P 1 ] R [F 2 P 1 ] R [E 2 F 1 P 0 ] R [E 1 F 1 P 0 ] R [F 1 E 1 F 1 P 0 ] R [E 1 F 2 P 0 ] R [F 2 P 0 ] R [F 3 P 0 ] R P 1 R [F 1 P 0 ] R P 0 Note that each map in the diagram acts on the leftmost letter of a word. Remark 2.5. Assume that k is a field, S and T are graded k-algebras with S 0 = k = T 0, the maps π S and π T are the canonical homomorphisms, and M is a graded left S-module. If P, E and F are complexes of graded S, S and T -modules respectively, then R G is a complex of graded R-modules; the internal degree of a word is the sum of the internal degrees of the letters in the word. Proof of Theorem 2.3. To show that R G is a complex, let w be a word of degree i, with i 2. Suppose w = xyw where w is a word, x is a letter of degree 1 and y is an arbitrary letter. For x E and y F, one has 2 (w) = ( E (x)yw ) (pyw ) = p (yw ) = p F (y)w pqw = 0. The cases with x F 1 and y P, and with x F 1 and y E are similar. If w = xw where x is a letter of degree greater than or equal to 2, then 2 (w) = 0, since R P, R E, and R F are complexes of R-modules, and hence one has 2 (x) = 0. Let R [E 2 G] denote the R-linear span of words whose first letter is in R E i for some i 2, see Definition 2.1. Let R [F 1 E 1 G] denote the span of words starting with a letter from R F 1, followed by a letter from E. Symbols R [F 2 G], R [E 1 F 1 G], etc. are defined similarly. Let a be an element of R G i, with i 1. It has a unique expression a = (x + x ) + (y + y ) + (z + z ) with x R [E 2 G], x R [F 1 E 1 G], y R [F 2 G], y R [E 1 F 1 G], z R P, z R [F 1 P]. Notice that one has (x+x ) R [E 1 G], (y+y ) R [F 1 G], and (z+z ) R P. Thus, (a) = 0 implies that each one of x + x, y + y and z + z is a cycle. Next, we show each is a boundary, by giving details for x + x ; the other cases are similar. Since one has p q = 0 in R, and R G i 1 is a free R-module, it follows that p( R G i 1 ) q( R G i 1 ) = 0. Therefore, (x + x ) = 0 implies that x and x are cycles

5 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS 5 as well. Let l(w) denote the leftmost letter in the word w. We may express x according to the decomposition R [E 2 G] i = R [E j w]. 2 j i w G i j,l(w) F If w is basis element of degree i j with l(w) F and 2 j i, then one has ( R [E j w]) R [E j 1 w]. Hence, in the decomposition above, each component of x is a cycle. A similar argument shows that the components of x in R [F 1 E 1 G] are cycles. Therefore, it is enough to consider a cycle of one of the forms x = r e ew R [E j w], e E j x = r fe few R [F 1 E j 1 w], e E j 1 f F 1 are boundaries, where w is a fixed word, and r e, r ef are in R. We give details for x, the other case is similar. We first show that r e m for each e E j. Indeed, there is a commutative diagram of R-modules (2.b) R [E 3 w] R [E 2 w] R [E 1 w] Rw 0 eσ 3 eσ 2 E 3 E 2 E 1 E 0 0 eσ 1 eσ 0 where the vertical maps send e E j r e ew to e E j σ(r e )e. The image of x in E is a cycle, and hence a boundary, of E. As E is minimal, the claim follows. Suppose t = (f) for f F 1. Then tew = (few) is a boundary of R G. As F is a resolution of k over T, the images of F 1 form a minimal generating set for q. Hence qew consists entirely of boundaries. The claims above show it suffices to prove the theorem when the coefficients are in p. In diagram (2.b), we may also define a morphism of complexes γ : pe p R [Ew] by sending e E j s e e to e E j s e ew, viewing s e p R. Note that γ and σ pr [Ew] are inverses of one another. Suppose that x p R [Ew] is a cycle. Then σ(x) is a cycle in E. Hence there exists u so that E (u) = σ(x). Then one has ( γ(u)) = γ( E (u)) = γ( σ(x)) = x. Two special cases of the theorem are used in section 4. Example 2.6. When M = k, we can take P = E. Let R D be the resolution given by Theorem 2.3. Since P 0 = {1}, we can also replace all basis elements of the form w1 with a basis element w of the same degree, and set RD (1 R ) = 0. Therefore, in

6 6 W. FRANK MOORE low degrees, R D has the form R E 3 R E 2 R [F 1 E 2 ] R [E 1 F 1 E 1 ] R [F 1 E 1 ] R [F 2 E 1 ] R [E 2 F 1 ] R [E 1 F 1 ] R [F 1 E 1 F 1 ] R [E 1 F 2 ] R F 2 R F 3 R E 1 Example 2.7. When M = T, applying the theorem with the roles of S and T reversed, one has R P i = 0 for i 0, and P 0 = {1}. Let R C be the resolution given by Theorem 2.3. Letting w denote the basis element w1 as above, we see that R C has the following form in low degrees: R E 3 R E 2 R [F 1 E 2 ] R [E 1 F 1 E 1 ] R [F 1 E 1 ] R [F 2 E 1 ] R F 1 R E 1 Note that this is the top half of the diagram in Example Graded algebras In this section, we set notation needed for the remainder of this article. Let k be a ring A k-algebra A is (Z-)graded if there is a decomposition of A = i Z A i as abelian groups, and for all i, j Z, one has A i A j A i+j. Since we will use both upper and lower indexed graded objects, we adopt the notation A i = A i. A is said to be connected if A 0 = k and A i = 0 for i < 0 (or equivalently, A i = 0 for i > 0) A left module M over a graded k-algebra A is (Z-)graded if there is a decomposition M = i Z M i as abelian groups, and for all i, j Z, one has A i M j M i+j. We also assume that M i = 0 for i 0, and that M i is a finitely generated free k-module for each i Z. Note that this condition is satisfied for finitely generated graded A-modules. R R

7 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS Let M be a graded A-module. The Hilbert series of M is the formal Laurent series M (t) = rank k M i t i. i 3.4. Let k be a field. If A and B are connected graded k-algebras, a coproduct A B in this category exists, and can be dipicted as follows: The vector space (A B) n has a basis {a 1 b 2 a l } {a 1 b 2 b m } {b 1 a 2 b p } {b 1 a 2 a q } where a i and b j are elements of a basis of A and B respectively, and the total degree of an elementary tensor(given by summing the degrees of its terms) is n. Multiplication of elementary tensors in A B is given by if both x and y are v xy w in A or in B (v x)(y w) = v x y w otherwise. Using this basis, one can show there is an equality of Hilbert series 1 A B (t) = 1 A (t) + 1 B (t) 1. One also sees that A B is a free right A-module with A-basis the elementary tensors listed above ending in an element from B. Therefore, there is a graded k-vector space V such that A B = V k A, and hence one has V (t) = A B (t). A (t) 3.5. If M is a graded left A-module, then (A B) A M is a graded left A B- module. The vector space ((A B) A M) n has a basis {a 1 b 2 b l p l+1 } {b 1 a 2 b m p m+1 } where a i, b j, and p k are elements of a basis of A, B and M respectively, and the total degree of an elementary tensor is n. The action of A B on (A B) A M in terms of the bases given is (v x)(y w) = v xy w v xw v x y w 4. The Yoneda algebra if both x and y are in A or in B if x in A and y in P otherwise. In this section, we assume k is a field, and one of the following two sets of hypotheses: 4.1. (S, p) and (T, q) are noetherian local rings with unique two-sided maximal ideals p and q respectively, S/p = k = T/q, M is a finitely generated left S-module, and S k T the canonical projections.

8 8 W. FRANK MOORE 4.2. S and T are connected graded k-algebras, M an S-module as in 3.2, p = S + and q = T + the ideals of S and T of positive degree elements, and S k T are the canonical projections For a ring A as in 4.1 or 4.2, we say a complex of free A-modules C is minimal if (C) mc, where m is the (homogeneous) maximal ideal of A. Every S-module has a degree-wise finite minimal free resolution over S, meaning that each free module used in the resolution is finitely generated. Furthermore, if the complexes used as input for Construction 2.2 are minimal and degree-wise finite, so will be the resulting complex R G. If L is an A-module, the Poincaré series of L over A is the formal power series P A L (t) := i dim k Ext i A(L, k)t i. Thus the coefficient of t i is the rank of the i th free module in a minimal resolution of L over A For the remainder of the paper, we let R, S and T denote the Ext algebras of R, S, and T respectively. For an S-module M, we let M S be the graded left S-module Ext S (M, k), and let M R be the graded left R-module Ext R (M, k). The diagram of homomorphisms of rings R τ T σ S πs k π T induces a diagram of graded algebras R σ S τ π T T k π S by applying the functor Ext (k, k). Hence, there is a homomorphism of graded k-algebras φ : S T R, given by the universal property of S T. Theorem 4.5. The homomorphism of connected k-algebras φ : S T R is an isomorphism. In particular, there is an equality of formal power series 1 Pk R(t) = 1 Pk S(t) (t) 4.6. One also has an induced homomorphism of graded S-modules P T k σ M : M S M R. Since σ M is σ -equivariant, the below pairing is S-bilinear, and so there is a homomorphism of graded left R-modules (4.c) θ : R S M S M R ξ µ ξ σ M (µ). Theorem 4.7. The homomorphism of graded left R-modules θ is an isomorphism. In particular, there is an equality of formal power series P R M (t) = P T k (t)p S M (t) P S k (t) + P T k (t) P S k (t)p T k (t).

9 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS 9 Remark 4.8. Theorem 4.7 shows that for any S-module M, there is an equality of Poincaré series PM R (t) = P S R(t)P M S (t). This says that the map R S is a large homomorphism, as defined in [7]. In order to prove Theorems 4.5 and 4.7, we set up some notation and prove some technical lemmas regarding the multiplication table in R and M R. Notation 4.9. In the notation of Construction 2.2, one sees that M R = HomR ( R G, k) = Hom R ( R G/m R G, k) = Hom k ( R G/m R G, k) is a k-vector space. Let {ξ R w w G} be the graded basis dual to the image of G in R G/m( R G). Also, let {ξ S e e E} be the graded basis dual to the basis given by the image of E in E/pE. We will abuse language and say that ξ R w starts (respectively ends) with a letter from E if the first (respectively last) letter of w is in E. Also, we will say that ξ R w has length n if w has length n. Our first lemma concerns the image of words of length one. Lemma Let e E, f F, and p P. One has σ (ξ S e ) = ξ R e, τ (ξ S f ) = ξ R f and σ M (ξ S p ) = ξ R p. Proof. Let ɛ R : R D k and ɛ S : E k be the augmentation maps. Set D = R (D\ E) + q(e) R D. The definition of shows that D is a subcomplex. Also, since R/q = S one has R D/D = E as complexes of R-modules. Let ψ be canonical surjection ψ : R D R D/D = E. Then one has ɛ R = ɛ S ψ, and hence σ (ξe S ) := ψξe S = ξe R, as desired. The other cases are similar. Our next lemma provides a partial multiplication table for the left action of R on M R. Lemma Let w D G, and let l(w) denote the starting letter of w. Then one has ξ R f ξr w = ξ R fw if l(w) E P, and ξr e ξ R w = ξ R ew if l(w) F. Proof. Suppose w = ew G i, with e E, and let f F. We define a chain map ψ w Hom R (G, D) i such that ɛ R ψ w = ξ R w as follows. Set G w = {x G x (Gw (GE j+1 )w )}, where Gw denotes elements of G that end in w (including w), and (GE j+1 )w denotes elements of G ending in a letter of E j+1, followed by w. Let R [G w ] be the free R-module generated by G w. The definition of shows R [G w ] is a subcomplex of G. Note that R G/ R [G w ] is a complex of free R-modules with basis Gw (GE j+1 )w and differential given by restricting to R [Gw] and R [(GE j+1 )w ]. If v = v w Gw, define α (v) = v, and extend α by R-linearity to all of R [Gw]. Then for v = v w Gw, one has α ( (v)) = α ( (v w)) = α ( (v )w) = (v ) = (α (v)). Let B be the subcomplex of R G/ R [G w ] spanned by (GE j+1 )w {w}, and let R C be the resolution of T as an R-module given in Example 2.7. As R C is acyclic and B is a free complex, we may define α : B R D by lifting the map that sends w B i to 1 C 0 and composing with the inclusion R C R D. Note that the words in the image of α end in letters from E. Since α (w) = 1 R = α (w), we may define { α α(v) := (v) v R [Gw] α (v) v R [(GE j+1 )w ]

10 10 W. FRANK MOORE Now set ψ w = R G R G/ R [G w α ] R D. Clearly, one has ɛ R ψ w = ξw, R hence ξf R ξr w = ξf Rψ w. Let v G be a word. If v G w, then ψ w (v) = 0. If v Gw, then write v = v w. Then ξ R f ψ w (v) = ξ R f (v ) = { 1 if v = f 0 otherwise. If v (GE j+1 )w, then ψ w (v) is in the span of words with rightmost letters in E. Hence ξf Rψ w(v) = 0. Therefore { } ξf R ξw R = ξf R 1 if v = fw ψ w (v) = = ξ R 0 otherwise. fw(v). The other cases are similar, and often easier. Proof of Theorem 4.5. Under the hypothesis of the theorem, the algebras R, S, and T are degree-wise finite, so that their Hilbert series can be defined, and agree with the Poincaré series of R, S, and T, respectively. One may describe the basis D in Construction 2.2 as a basis of the based R-module R F R T (R E 1 R R F 1 ) R R E, where T ( ) denotes the tensor algebra. Therefore, one has R D (t) = R F (t) R E (t) 1 ( R E (t) 1)( R F (t) 1). Since R D (t) = Pk R(t), R E (t) = Pk S(t) and R F (t) = Pk T (t) we have simplifying to P R k (t) = P T k (t)p S k (t) P S k (t) + P T k (t) P S k (t)p T k (t), 1 Pk R(t) = 1 Pk S(t) (t) Hence, the Hilbert series of the algebras R and S T agree. As φ is homogeneous, it suffices to show that φ is surjective. Lemma 4.11 shows that {ξe R e E} {ξf R f F} generates R as an algebra, and Lemma 4.10 shows that φ maps onto these generators. Hence φ is surjective. Proof of Theorem 4.7. By remarks made in section 3.4, R is free as a right S- module, and there exists a k-vector space V such that R = V k S. Then one has R S M S = V k S S M S = V k M S, and hence R S M S (t) = V (t)p S M (t) = P R k (t) P S k (t) P S M (t) = = P T k P S k (t)p T k (t)p S M (t) ( P S k (t) + P T k (t) P S k (t)p T k (t)) P S k (t) P T k (t)p S M (t) P S k (t) + P T k (t) P S k (t)p T k (t) = P R M (t) = M R (t).

11 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS 11 where the third and fifth equalities are given by arguing as in the proof of Theorem 4.5. Hence, R S M S and M R have the same k-vector space dimension in each degree, and so it is enough to show that θ is surjective. By Lemma 4.11, M R is generated as a left R-module by {ξp R p P}. By Lemma 4.10, θ(1 ξp S ) = ξp R, and hence θ is surjective Recall that a graded module M over a connected algebra A is Koszul provided Ext i A(M, k) j = 0 if j i. A connected algebra A is Koszul if k is Koszul as an A-module. By Remark 2.5, the homomorphisms φ and θ preserve the internal gradings of S T and R, giving the next corollary, proved in [2]. Corollary The following conditions are equivalent. (1) The algebra R is Koszul. (2) The algebras S and T are Koszul. (3) There exists an S-module M that is Koszul as a R-module. The functor Ext R (, k) has a property similar to the one given in Theorem 4.5. Theorem Let M, N and H be S,T and k-modules respectively, such that there exist surjective π S and π T -equivariant homomorphisms M µ H ν N with ker µ = pm and ker ν = qn. The exact sequence of R-modules 0 M H N ι M N µ ν H 0 induces an exact sequence of graded left R-modules 0 R k H µ ν ι M R N R Ext R (M H N, k) 0 In particular, there is an equality of Poincaré series P R M H N(t) + (rank k H)P R k (t) = P R M (t) + P R N (t). Proof. Let n = rank k H. One has a sequence of maps S n M µ H which induces the sequence S k H M S k n. Tensoring with R over S on the left, one has R k H µ M R R S k n since M R = R S M S by Theorem 4.7. Under the isomorphism in Theorem 4.5, the kernel of this composition consists of elements of the dual basis ending in a letter from E, and hence the kernel of µ is contained in those elements of R k H ending in a letter from E. Similarly, one can show that the kernel of ν is contained in those elements of R k H ending in a letter from F. Hence Ker(µ ν ) = Ker µ Ker ν = Depth of cohomology modules In order to describe the cohomology module for an arbitrary R-module L, we use an elementary observation of Dress and Krämer in [3]. Recall that the syzygy Ω R L of an R-module L is the kernel of a free cover F L where F ; it is defined uniquely up to isomorphism, and one sets Ω R n L = Ω R Ω R n 1L for n 2. Proposition 5.1. Let L be a left R-module. Then Ω R 2 (L) = M N where M is an S-module, and N is a T -module.

12 12 W. FRANK MOORE Proof. Recall that m = p q is the maximal ideal of R. minimal presentation of L over R. One has Ω R 2 (L) = Ker ϕ = Ker ϕ ma = Ker ϕ (pa qa) = (Ker ϕ pa) (Ker ϕ qa) Let ϕ : A B be a To see the last equality, suppose that (x 1, x 2 ) pa qa A has ϕ((x 1, x 2 )) = 0 = ϕ((x 1, 0)) + ϕ((0, x 2 )). Note that ϕ((x 1, 0)) ϕ(pa) pb and ϕ((0, x 2 )) ϕ(qa) qb. Also, pb qb = 0, hence ϕ((x 1, 0)) = 0 = ϕ((0, x 2 )). Taking M = Ker ϕ pa and N = Ker ϕ qb gives the desired result. On the face of it, Theorem 4.7 only describes the Ext modules of those R- modules that are also S-modules. However, the previous proposition gives us a way to describe the Ext modules of any R-module. Indeed, for an R-module L, there is an inclusion of graded R-modules Σ 2 Ext R (Ω R 2 (L), k) L that is an isomorphism in degrees greater than or equal to two. One has the following corollary, whose proof is evident by the previous proposition, together with Theorem 4.7. Corollary 5.2. Set L = Ext R (L, k), M S = Ext S (M, k), and N T = Ext T (N, k). There is then an exact sequence of graded left R-modules 0 (Σ 2 R S M S ) (Σ 2 R T N T ) L L/L 2 0 Corollary 5.2 allows us to study the depth of the cohomology module of an R- module, whose definition is given below. For basic properties of this invariant, see [4] or [1]. Definition 5.3. For a graded connected k-algebra E and a graded left E-module M, one defines the depth of M over E by means of the formula One sets depth E = depth E E. depth E M = inf{n N Ext n E(k, M) 0}. Theorem 5.4. Suppose that M is a nonzero S-module. If neither π S or π T are isomorphisms, then depth M R 1. If S and T are commutative or have global dimension at least two, then depth M R = 1. In this case, one has Ext 0 R(k, M R ) = 0 and either Ext 1 R(k, M R ) j 0 or Ext 1 R(k, M R ) j+1 0 for each j 2, unless M is free and S and T both are discrete valuation rings, or S and T both are polynomial rings in a single variable. In the latter case, one has Ext i R(k, M R ) j = { k rank M for i = j 1 0 otherwise Proof. We first show that for each µ M i R, there is an element ξ R so that ξµ 0. By the argument in Theorem 4.7, M R = R S M S = V k M S as left R-modules, where V is the basis of R as a right S-module given by elements of the dual basis corresponding to words ending in a letter from F. We say an elementary tensor in M R starts with a letter from E if it is of the form ξ e µ for some e E and µ M S, and similarly for F. We also say the length of an elementary tensor ς 1 ϑ 1 µ n is n.

13 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS 13 Arrange the terms in µ so that µ = α + β + γ, where the terms in α start with a letter from E, the terms in β start with F and the terms in γ start with 1. As S and T are not fields, there exists ς S 1 \ {0} and ϑ T 1 \ {0}. If γ 0, then the terms in ϑγ start with a letter of degree 1 from F, and the length of each of its terms is 2. But ϑα has terms of length greater than or equal to 3, and ϑβ has terms with the degree of their leading letters greater than 1, hence ϑµ 0. One can similarly argue that if α or β are nonzero, then ϑα or ςβ are nonzero, respectively. Choose free resolutions of k over S and T of the form V (2) k S V (1) k S S k 0 W (2) k T W (1) k T T k 0, where V (i) and W (i) are graded k-vector spaces. One then gets a free resolution of k over R (V (2) W (2)) k R (V (1) W (1)) k R R k 0 where the differentials are given by restricting those in the resolutions of k over S and T, see [5, Example 36.e.2]. Let α, β M i R, and define φ α,β : (V (1) W (1)) k R M R v ( v)α w ( w)β. Then φ α,β is a 1-cocycle and hence defines a class in Ext R (k, M R ). If there exists v V (1) or w W (1) so that ( v)β 0 or ( w)α) 0, then φ α,β represents a nonzero cohomology class, as long as φ α,β is not the zero map. As S and T are not fields, there exists ς S 1 \ {0} and ϑ T 1 \ {0}, and there exists µ M 0 S \ {0} as M is nonzero. If M is not free over S, then there also exists µ M 1 S \ {0}. If gldim S 2 (respectively gldim T 2), then there exists ς S 2 \ {0} (respectively ϑ T 2 \ {0}). In each of these cases, and for each i 1, define (ϑς) i 1 ϑµ if M is not free α i = (ϑς) i 1 ϑ µ if gldim T 2 (ϑς) i ϑµ if gldim S 2 and (ςϑ) i 1 µ if M is not free β i = (ςϑ) i µ if gldim T 2 (ςϑ) i 1 ς ϑµ if gldim S 2 Then in each case, and for each i 1, φ αi,β i defines a distinct nonzero class of Ext 1 R(k, M R ). Indeed, the internal degree of φ αi,β i is either 2i 1, 2i, or 2i + 1 depending on the situation. Hence one has rank k Ext 1 R(k, M R ) = in the three cases listed in the theorem. Therefore, it suffices to prove the theorem when M is free of rank one, and S and T are either discrete valuation rings or polynomial rings in a single variable. In this case, S = k[x]/x 2, T = k[y]/y 2, M R = R/Rx, and z = x y + y x is central, and both M R and R-regular. For each i 0, one has the standard Rees isomorphisms Ext i+1 R (k, M R) = Ext i Λ(k, M R /zm R )

14 14 W. FRANK MOORE where Λ = R/zR is the exterior algebra on {x, y}. The exact sequence 0 M R /zm R Λ(1) M R /zm R (1) 0 gives rise to an injective resolution of M R /zm R over Λ Therefore, one has 0 Λ(1) Λ(2). Ext i Λ(k, M R /zm R ) j = { k for j = i otherwise As a corollary, we obtain the fact that the depth of the cohomology algebra of a fiber product is one, a fact that was already available to us once Theorem 4.5 was in hand, using [5, Example 36.(e).2]. Corollary 5.5. If π S and π T are not isomorphisms and S and T are either commutative or have global dimension at least two, then depth Ext R (k, k) = 1. Using the short exact sequence given in Corollary 5.2, one can show that the depth of the cohomology module of any R-module is no bigger than 1, showing that the cohomology modules of S-modules have the biggest depth possible. Corollary 5.6. Let S and T be commutative or have global dimension at least two, set R = S k T, and let L be a nonzero R-module. Then depth R L 1. Proof. If pd R L <, L is finite dimensional over k, hence depth R L = 0. If pd R L =, then Ω 2 R (L) 0, and by Proposition 5.1, we have that Ω2 R (L) = M N for some S-module M and some T -module N. Suppose that either gldim S 2 or gldim T 2. Then rank k Ext R (k, Ext R (M N, k)) =. Set X := Ext R (M N, k). Then by Corollary 5.2, together with Theorem 5.4, there is an exact sequence Hom R (k, L/L 2 ) ϕ Ext 1 R(k, X ) ψ Ext 1 R(k, L). Since rank k Hom R (k, L/L 2 ) β0 R (L)+β1 R (L) < = rank k Ext 1 R(k, X ), ϕ cannot be surjective, and hence ψ is nonzero. If both S and T are discrete valuation rings or polynomial rings in a single variable, as in the proof of Theorem 5.4, L is finitely generated and graded over A, the central polynomial subring of R of dimension one generated by z, see [9]. Hence depth R L = depth A L dim A = 1. References 1. Luchezar L. Avramov and Oana Veliche, Stable cohomology over local rings, Preprint (2005). 2. Jörgen Backelin and Ralf Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30 (1985), Andreas Dress and Helmut Krämer, Bettireihen von faserprodukten lokaler ringe, Math. Ann. 215 (1975), Yves Felix, Stephen Halperin, C. Jacobsson, C. Löfwall, and Jean-Claude Thomas, The radical of the homotopy Lie algebra, Amer. J. Math. 110 (1988), Yves Felix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Springer- Verlag, Berlin-New York, Alexei I. Kostrikin and Igor R. Shafarevich, Groups of homologies of nilpotent algebras (Russian), Dokl. Akad. Nauk. SSSR 115 (1957), Gerson Levin, Large homomorphisms of local rings, Math. Scand. 46 (1980),

15 COHOMOLOGY OVER FIBER PRODUCTS OF LOCAL RINGS Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, p. 57, American Mathematical Society, Gunnar Sjödin, A set of generators for Ext R (k, k), Math. Scand. 38 (1976), no. 2,