Local Cohomology and the Cohen-Macaulay Property

Local Cohomology and the Cohen-Macaulay property

Abstract These lectures represent an extended version of the contents of a one hour introductory talk prepared by Florian Enescu and Sara Faridi for the Minnowbrook workshop to assist the lectures of one of the main speakers, Paul Roberts. Part of this talk was given earlier at the 2004 Utah mini-course on Classical problems in commutative algebra by the ﬁrst author. The notes for that talk have been been typed and prepared by Bahman Engheta and we used part of them quite extensively in preparing this version. We would like to thank Bahman for his work. The references listed at the end were used in depth in preparing these notes and the authors make no claim of originality. Moreover, the reader is encouraged to consult these references for more details and many more results that were omitted due to time constraints.

1 Injective modules and essential extensions

Throughout, let R be a commutative Noetherian ring. Deﬁnition 1 (injective module). An R-module E is called injective if one of the following equivalent conditions hold. 1. Given any R-module monomorphism f : N → M, every homomorphism u : N → E can be extended to a homomorphism g : M → E, i.e. g f = u.

f 0 w NMw u g u E

2. For each ideal J of R, any homomorphism u : J → E can be extended to a homomorphism R → E. 3. The functor Hom( ,E) is exact.

Example 2. From part 2 of Deﬁnition 1 it follows that Q and Q/Z are injective Z-modules. Deﬁnition 3 (divisible module). An R-module M is called divisible if for every m ∈ M and M-regular element r ∈ R, there is an m0 ∈ M such that m = rm0. Exercise 4. An injective R-module is divisible. The converse holds if R is a principal ideal domain; see the example above.

A note on existence: If R → S is a ring homomorphism and E is an injective R-module, then HomR(S, E) is an injective S-module. [The S-module structure is given by s·ϕ( ) := ϕ(s ).] In particular, HomZ(R, Q) is an injective R-module and any R-module can be embedded in an injective R-module. We now show how to embed a given module into a minimal injective module. Deﬁnition 5 (essential extension). Let M ⊆ N be R-modules. We say that N is an essential extension of M if one of the following equivalent conditions holds: a) M ∩ N 0 6= 0 ∀ 0 6= N 0 ⊆ N.

1 b) ∀ 0 6= n ∈ N ∃ r ∈ R such that 0 6= rn ∈ M. ϕ c) ∀ N → Q, if ϕ|M is injective, then ϕ is injective. If additionally M 6= N, then we say that N is a proper essential extension of M.

More generally, an injective map M →h N is called an essential extension if in the conditions above we replace M by h(M). Example 6. If R is a domain and Q(R) its fraction field, then R ⊆ Q(R) is an essential extension. Example 7. Let (R, m, k) be a local ring and N an R-module such that every element of N is annihilated by some power of the maximal ideal m. Let Soc(N) := AnnN (m) denote the socle of N (which is a k-vector space). Then Soc(N) ⊆ N is an essential extension. Exercise 8. An R-module N is Artinian if and only if Soc(N) is finite dimensional (as a k-vector space) and Soc(N) ⊆ N is essential. Exercise 9. Let M be a submodule of N. Use Zorn’s lemma to show that there is a maximal submodule N 0 ⊆ N containing M such that M ⊂ N 0 is an essential extension. Definition 10 (maximal essential extension). If M ⊆ N is an essential extension such that N has no proper essential extension, then M ⊆ N is called a maximal essential extension. The following proposition characterizes injective modules in terms of essential extensions. Proposition 11. a) An R-module E is injective if and only if it has no proper essential extension. b) Let M be an R-module and E an injective R-module containing M. Then any maximal essential extension N of M with N ⊆ E is a maximal essential extension. In particular, N is injective and thus a direct summand of E. c) If M ⊆ E and M ⊆ E0 are two maximal essential extensions, then there is an isomorphism E =∼ E0 that fixes M. Definition 12 (injective hull). A maximal essential extension of an R-module M is called an injective hull of M, denoted ER(M). Proposition 11 states that every module M has a unique injective hull up to isomorphism. Moreover, if M ⊆ E where E is an injective module, then M has a maximal essential extension ER(M) that is contained in E, and ER(M) is a direct summand of E.

−1 0 Deﬁnition 13 (injective resolution). Let M be an R-module. Set E := M and E := ER(M). Induc- n En−1 tively deﬁne E := ER( /im(En−2)). Then the acyclic complex

0 1 n I : 0 → E → E → · · · → E → · · · is called an injective resolution of M, where the maps are given by the composition

n−1 n−1 n−2 n−1 n−2 E → E /im(E ) ,→ ER(E /im(E )).

Conversely, an acyclic complex I of injective R-modules is a minimal injective resolution of M if • M = ker(E0 → E1),

0 • E = ER(M),

n n−1 n • E = ER(im(E → E )). For a Noetherian ring R one can give a speciﬁc description of each Ei appearing in the injective resolution of M as a direct sum of ER(R/p) for p ∈ SpecR. We refer interested reader to the sources mentioned in the references for more on this.

2 2 Local cohomology

S n Deﬁnition 14 (I-torsion). Given an ideal I ⊆ R and an R-module M, set ΓI (M) := n(0 :M I ). The covariant functor ΓI ( ) over the category of R-modules is called the I-torsion functor, and for a homomor-

phism f : M → N, ΓI (f) is given by the restriction f . ΓI (M)

Proposition 15. ΓI ( ) is a left exact functor. Proof. Let f g 0 → L → M → N → 0 be a short exact sequence. We want to show that

ΓI (f) ΓI (g) 0 → ΓI (L) −→ ΓI (M) −→ ΓI (N) is an exact sequence. Exactness at ΓI (L): ΓI (f) is injective as it is the restriction of the injective map f. Exactness at ΓI (M): It is clear that im(ΓI (f)) ⊆ ker(ΓI (g)). Conversely, let m ∈ ker(ΓI (g)). Then m ∈ ker(g) and therefore k m = f(l) for some l ∈ L. It remains to show that l ∈ ΓI (L). As m ∈ ΓI (M), we have I m = 0 for some k k k k integer k. Then f(I l) = I f(l) = I m = 0. As f is injective, I l = 0 and l ∈ ΓI (L). √ √ Exercise 16. ΓI = ΓJ if and only if I = J. i Deﬁnition 17 (local cohomology). The i-th local cohomology functor HI ( ) is deﬁned as the i-th right derived functor of ΓI ( ). More precisely, given an R-module M, let I be an injective resolution of M: 0 1 n 0 d 1 d n d I : 0 → E → E → · · · → E → · · ·

Apply ΓI ( ) to I and obtain the complex: 0 1 n ΓI (I) : 0 → ΓI (E ) → ΓI (E ) → · · · → ΓI (E ) → · · · 0 i i i−1 i Then set HI (M) := ΓI (M) and HI (M) := ker(ΓI (d ))/ im(ΓI (d )) for i > 0. Note that HI ( ) is a covariant functor. i Clearly, if E is an injective R-module, then HI (E) = 0 for i > 0. Proposition 18. 1) If 0 → L → M → N → 0 is a short exact sequence, then we have an induced long exact sequence 0 0 0 0 → HI (L) → HI (M) → HI (N) → 1 1 1 HI (L) → HI (M) → HI (N) → · · · 2) Given a commutative diagram with exact rows: 0 w LMNw w w 0

u u u 0 w L0 w M 0 w N 0 w 0 then we have the following commutative diagram with exact rows:

w i w i w i+1 w ··· HI (M) HI (N) HI (L) ···

u u u w i 0 w i 0 w i+1 0 w ··· HI (M ) HI (N ) HI (L ) ··· √ √ i i 3) I = J if and only if HI ( ) = HJ ( ). −1 −1 4) Localization: Let S ⊆ R be a multiplicatively closed set. Then S ΓI (M) = ΓS−1I (S M) and the same holds for the higher local cohomology modules.

3 2.1 Alternate construction of local cohomology n ∼ An alternate way of constructing the local cohomology modules: Consider the module HomR(R/I ,M) = n n m (0 :M I ). Now, if n > m, then one has a natural map R/I → R/I , forming an inverse system. Applying HomR( ,M), we get a direct system of maps:

n ∼ [ n −→limn HomR(R/I ,M) = (0 :M I ) = ΓI (M). n As one might guess (or hope), it is also the case that

i n ∼ i −→limn ExtR(R/I ,M) = HI (M). This follows from the theory of negative strongly connected functors – see [R]. Deﬁnition 19 (strongly connected functors). Let R,R0 be commutative rings. A sequence of covariant {T i} R R0 functors i>0 from the category of -modules to the category of -modules is said to be negative (strongly) connected if (i) Any short exact sequence 0 → L → M → N → 0 induces a long exact sequence

0 → T 0(L) → T 0(M) → T 0(N) → T 1(L) → T 1(M) → T 1(N) → · · ·

(ii) For any commutative diagram with exact rows

0 w LMNw w w 0

u u u 0 w L0 w M 0 w N 0 w 0

there is a chain map between the long exact sequences given in (i). Deﬁnition 20 (natural equivalence of functors). Let T and U be two covariant functors from a category C to a category D.A natural transformation ψ from T to U associates to every object X in C a morphism ψX : T (X) → U(X) in D, such that for every morphism f : X → Y in C we have ψY oT (f) = U(f)oψX . If, for every object X in C, the morphism ψX is an isomorphism in D, then ψ is said to be a natural equivalence. ψ0 : T 0 → U 0 {T i} , {U i} Theorem 21. Let be a natural equivalence, where i>0 i>0 are strongly connected. If T i(E) = U i(E) = 0 for all i > 0 and injective modules E, then there is a natural equivalence of functors ψ = {ψi} : {T i} → {U i} i>0 i i. lim Exti (R/In,M) ∼ Hi (M) The above theorem implies that −→n R = I . n Remark 22. i) One can replace the sequence of {I } by any decreasing sequence of ideals {Jt} which n n n are coﬁnal with {I }, i.e. ∀ t ∃ n such that I ⊆ Jt and ∀ n ∃ t such that Jt ⊆ I .

i i ii) Every element of HI (M) is killed by some power of I, as every m ∈ HI (M) is the image of some i n n ExtR(R/I ,M) which is killed by I .

·x i ·x i iii) For any x ∈ R, the homomorphism M −→ M induces a homomorphism HI (M) −→ HI (M). Proposition 23. Let M be a ﬁnitely generated R-module and I ⊆ R an ideal. Then IM = M ⇐⇒ i i HI (M) = 0 ∀ i. If IM 6= M, then min{i | HI (M) 6= 0} = depthI (M).

4 Proof. We have IM = M ⇐⇒ ItM = M ∀ t. So It +Ann(M) = R, as otherwise It +Ann(M) ⊆ m for t i t i t some m ∈ max-Spec(R). Since I + Ann(M) annihilates ExtR(R/I ,M), we have ExtR(R/I ,M) = 0 i and therefore HI (M) = 0. It sufﬁces to assume now that IM 6= M. Set d := depthI (M) and let x1, . . . , xd be a maximal M- i d regular sequence in I. We show by induction on d that HI (M) = 0 for i < d and HI (M) 6= 0. 0 If d = 0, that is, if depthI (M) = 0, then there is an 0 6= m ∈ M killed by I. So m ∈ HI (M) 6= 0. Now let d > 1 and set x := x1. Consider the short exact sequence 0 → M →·x M → M/xM → 0

and the induced long exact sequence

i−1 i ·x i · · · → HI (M/xM) → HI (M) → HI (M) → · · ·

i−1 i If i < d, then HI (M/xM) = 0 by induction hypothesis and x is a nonzerodivisor on HI (M). As all i i elements of HI (M) are killed by some power of I, we conclude that HI (M) = 0. d It remains to show that HI (M) 6= 0. This follows from the induction hypothesis and the long exact sequence ·x d−1 d−1 d ·x ··· → HI (M) → HI (M/xM) → HI (M) → · · · d−1 d which yield 0 6= HI (M/xM) ,→ HI (M).

2.2 The Koszul interpretation

Let K, L be two complexes of R-modules with differentials d0, d00, respectively. One can deﬁne the complex L 0 i 00 M := K ⊗ L via Mk := i+j=k Ki ⊗ Lj with differential d(ai ⊗ bj) := d (ai) ⊗ bj + (−1) ai ⊗ d (bj) (1) (n) (1) (n) where ai ∈ Ki, bj ∈ Lj. Similarly, if K ,..., K are n complexes, then K ⊗ · · · ⊗ K is deﬁned inductively as (K(1) ⊗ · · · ⊗ K(n−1)) ⊗ K(n). To any x ∈ R one can associate complexes

·x • ·x K•(x; R) : 0 → R → R → 0 or K (x; R) : 0 → R → R → 0

• where the degrees of the components R, from left to right, are 1, 0 for K•(x; R) and 0,1 for K (x; R). Given a sequence x = x1, . . . , xn of elements in R, we deﬁne the (homological) Koszul complex Nn • Nn • K•(x; R) := i=1 K•(xi; R), and cohomological Koszul complex K (x; R) := i=1 K (xi; R). • • For an R-module M we deﬁne K•(x; M) := K•(x; R) ⊗ M and K (x; M) := K (x; R) ⊗ M. The • ∼ following isomorphism holds: K (x; M) = HomR(K•(x; R),M). Let x ∈ R and M an R-module. Consider the complex

M →·x M →·x M → · · ·

0 0 and set N := ker(M → Mx) and M := M/N. Note that N = HxR(M). Then ·x ·x ∼ −→lim(M → M → M → · · · ) = 0 ·x 0 ·x 0 −→lim(M → M → M → · · · ) = 0 −1 0 −t 0 ∼ 0 ∼ −→lim(M ⊆ x M ⊆ · · · ⊆ x M ⊆ · · · ) = Mx = Mx. t Notation: Whenever x = x1, . . . , xn denotes a sequence of elements in R, we will denote by x the t t sequence of the individual powers x1, . . . , xn For x ∈ R we have a chain map of complexes K•(xt; R) → K•(xt+1; R) via the commutative diagram

·xt K•(xt; R): 0 w RRw w 0

id ·x (1) u u ·xt+1 K•(xt+1; R): 0 w RRw w 0

5 • t • t+1 By tensoring these maps for all the xi, and then tensoring with M, we get K (x ; M) → K (x ; M). Take the direct limit and denote the resulting complex by K•(x∞; M). We can look at Hi(K•(x∞; M)) = i • t i ∞ −→limt H (K (x ; M)) for which we simply write H (x ; M). i ∞ ∼ Theorem 24. If I = (x1, . . . , xn) and M is an R-module, then there is a canonical isomorphism H (x ; M) = i HI (M). t Consider the case where x = x1, . . . , xn is an R-regular sequence. It is known that K•(x ; R) is a projec- t • t • t tive resolution of R/x R. Apply HomR( ,M) and note that on the one hand K (x ; M) gives H (x ; M) • t i ∞ ∼ i while on the other hand we get ExtR(R/x ; M). Now take the direct limit: H (x ; M) = HI (M). Discussion 25 (The Cechˇ complex and a detailed look at K•(x∞; M)). Let x ∈ R. Then K0(x∞; R) = R 1 ∞ • ∞ Nn and K (x ; R) = Rx. (Recall diagram (1).) So K (x ; R) = i=1(0 → R → Rxi → 0), that is,

j ∞ M K (x ; R) = Rx(S) |S|=j

Q j ∞ L where S ⊆ {1, . . . , n} and x(S) = xi. Similarly, K (x ; M) = |S|=j Mx(S). The map Rx(S) → i∈S Rx(T ), where |T | = |S| + 1, is the zero map unless S ⊂ T , in which case it is the localization map times (−1)a where a is the number of elements in S preceding the element in T \ S. Exercise 26. Let x, y, z ∈ R. Write down the maps in

0 → R → Rx ⊕ Ry ⊕ Rz → Rxy ⊕ Ryz ⊕ Rzx → Rxyz → 0.

Discussion 25 and Theorem 24 imply the following.

i Corollary 27. If I ⊆ R is an ideal which can be generated by n elements up to radical, then HI (R) = 0 for i > n. Remark 28. The modules occurring in K•(x∞; R) are ﬂat.

3 Properties of local cohomology

Proposition 29. Let R and S be Noetherian rings. i ∼ i 1. Let R → S be a ring homomorphism, I an ideal of R, and M an S-module. Then HI (M) = HIS(M) as S-modules. Λ {M } R lim Hi (M ) ∼ Hi (lim M ) 2. Let be a directed set and λ λ∈Λ a direct system of -modules. Then −→λ I λ = I −→λ λ . i i 3. If S is flat over R, then HI (M) ⊗R S = HIS(M ⊗R S). 4. If m ⊆ R is a maximal ideal, then Hi (M) ∼ Hi (M ). m = mRm m 5. If (R, m) is local, then Hi (M) =∼ Hi (Rˆ ⊗ M) which is isomorphic to Hi (Mˆ ) if M is finitely m mRˆ R mRˆ generated. Proposition 30. Let I ⊆ R be an ideal which is the radical of an ideal generated by a regular sequence of i ∼ n length n. Then HI (M) = Torn−i(M,HI (R)) for i 6 n. i • ∞ n Proof. If i < depthI (R) = n, then HI (R) = 0. Therefore K (x ; R) gives a flat resolution of HI (R), numbered backwards:

n−1 n n n n−1 · · · → K → K → HI (R) = K / im(d ) → 0.

R n i • ∞ On the one hand Torn−i(M,HI (R)) = H (K (x ; R) ⊗R M) by deﬁnition of Tor. On the other hand, by i • ∞ ∼ i the preceding theorem, H (K (x ; R) ⊗R M) = HI (M).

6 i ∼ R n Corollary 31. Let (R, m, k) be a Cohen-Macaulay local ring of dimension n. Then Hm(M) = Torn−i(M,Hm(R)). Proof. The maximal ideal m is the radical of an ideal generated by a(ny) regular sequence of length n. Theorem 32 (Grothendieck’s Theorems).

i (Vanishing Theorem) Let I ⊆ R be an ideal and M an R-module. Then HI (M) = 0 for i > dim(M). (Non-Vanishing Theorem) Let (R, m, k) be a local ring and M a finitely generated R-module. Then i Hm(M) 6= 0 for i = dim(M). Proof of 1. We may assume that R is local with maximal ideal m. Further, as M is the direct limit of its finitely generated submodules, we may also assume that M is finitely generated. Set S := R/ Ann(M) so that n := dim(M) = dim(S). The maximal ideal of S is generated by n elements up to radical, so i HmS(M) = 0 for i > n. i We want to show that HI (M) = 0 for i > dim(M). By induction, we assume the theorem is true for all finitely generated modules of dimension less than n. We leave the case n = 0 as an exercise and assume n > 0. Note that if a module is I-torsion, then all its higher local cohomology modules vanish. So, as ΓI (M) is I-torsion, without loss of generality ΓI (M) 6= M. Also, the long exact sequence induced by

M 0 → ΓI (M) → M → /ΓI (M) → 0

i ∼ i yields that HI (M) = HI (M/ΓI (M)) for all i > 0. Hence, by passing to M/ΓI (M), we may assume that M 6= 0 is I-torsionfree. It follows that I contains an M-regular element r. (Otherwise I is contained in the union of the associated primes of M, and by prime avoidance I is contained in one of those primes which is of the form (0 :R m) for some 0 6= m ∈ M. That is Im = 0 — a contradiction.) Let i > n and let t be an integer. Consider the short exact sequence

t ·r M 0 → M → M → /rtM → 0 and the induced long exact sequence

t i−1 t i ·r i · · · → HI (M/r M) → HI (M) → HI (M) → · · ·

t i−1 t t Since dim(M/r M) < dim(M), HI (M/r M) = 0 by induction hypothesis and r is a nonzerodivisor i i i on HI (M). But any element of HI (M) is killed by a power of I. As r ∈ I, that implies HI (M) = 0. n ∼ n Remark 33. If dim(R) = n and M is an R-module, then HI (M) = M ⊗R HI (R). This follows from the n fact that Hm(−) is a right exact functor, a consequence of Grothedieck’s Vanishing Theorem.

4 Kunneth formula and applications

R = ⊕ R N We would like to discuss the local cohomology modules of a graded ring. Let n>0 n be a -graded ring over R0 = K, where K is an algebraically closed field. Assume that R is finitely generated over K. Let m be the maximal ideal ⊕ R . Do the modules Hi (R) inherit a natural grading from the ring R? The R n>1 n mR answer is yes and we plan to explain how this goes. Let I be a graded ideal of R. We can consider the category of graded R-modules and homogeneous R-homomorphisms which is an Abelian category which has enough projective objects and injective objects so the construction used to define the local cohomology modules can be imitated in this setting. Let M,N be two Z-graded R-modules. Let i be an integer and denote HomR(M,N)i the Abelian group of R-linear maps f : M → N such that f(Mn) = Nn+i for all n. The R-module of homogeneous M N Hom (M,N) := ⊕ Hom (M,N) homomorphisms between and is by definition R i>0 R i. One can check that, for a fixed graded R-module N, HomR(−,N) defines a functor on the category of graded R-modules. i For every graded R-module N, the ith right derived functor of HomR(−,N) will be denoted ExtR(−,N). i i n Then, HI (N) = lim−→ExtR(R/I ,N).

7 n i n i n Since R/I are finitely generated for all n, then ExtR(R/I ,N) = ExtR(R/I ,N) as R-modules i (ignoring the grading on the first module). As a result, we see that HI (N) has a natural Z-grading, and, if we i ignore this grading, we recover HI (N). The graded local cohomology can also be defined via Cechˇ or Koszul complexes and all these approaches agree with each other. In the end, we obtain a naturally graded object such that if we drop its grading we recover the standard local cohomology module. Important information resides in this extra structure given by the grading on the local cohomology module and the a-invariant of a R a(R) a(R) = max{n :(Hd (R)) 6= 0} d = dim(R) ring , captures some of it. By definition, mR n , where . This number will become relevant in the discussion that follows. Let A = ⊕An and B = ⊕Bn be two N-graded rings such that A0 = B0 = K. The Segre product of these two rings is the ring A]B = ⊕nAn ⊗K Bn. This ring is naturally N-graded and a direct summand in the ring A ⊗K B. More generally, if M is a Z-graded A-module and N is a Z-graded B-module, Segre product of M and B is the graded A]B-module defined by M]N = ⊕nMn ⊗K Nn. Goto and Watanabe [GW] have proved a formula describing the local cohomology of the Segre product R with support at the maximal ideal mR = mA ] mB. In fact, their result is more general: Hj (M) = Hj (N) = 0 Theorem 34 (Goto-Watanabe). With notations as above, assume that mA mB for j = 0, 1. Then for all i > 0, we have the following isomorphism: Hi (M]N) ' (Hi (M) ]N) ⊕ (M]Hi (N)) ⊕ (⊕ Hr (M) ]Hs (N)). mR mA mB r+s=i+1 mA mB Corollary 35. Let A and B as above and assume that dim(A) = d, dim(B) = e. Then

1. If d, e > 1, then dim(A]B) = d + e − 1. d, e 2 A, B A]B Hd (A) ]B = 2. If > and if are Cohen-Macaulay, then is Cohen-Macaulay if and only if mA A]He (B) = 0 mB .

3. If d = 1 and e > 2, then if A, B are Cohen-Macaulay then A]B is Cohen-Macaulay if and only if H1 (A) ]B = 0 mA . 4. If d = e = 1, and if A, B are Cohen-Macaulay, then A]B is Cohen-Macaulay . This corollary is a source of examples of normal rings that are not Cohen-Macaulay: if A, B are normal and Cohen-Macaulay, then A]B is normal as a direct summand in a normal ring A ⊗K B. But often A ]B dim(A) = d 2, dim(B) = e 2 Hd (A) ]B = A]He (B) = 0 is not Cohen-Macaulay. If > > , then mA mB is a restrictive condition equivalent to a(A) < 0, a(B) < 0. For example, A = k[x, y, z]/(x3 + y3 + z3), then the a-invariant of A is 0, and hence, for any Cohen-Macaulay ring B, A]B is not Cohen-Macaulay. Two recent papers dealing with Segre products and local cohomology are [Si, SW], where in [SW] an example of a normal, but not Cohen-Macaulay ring, obtained as above as a Segre product is studied in detail.

References

[BS] M. P. Brodmann, R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, 60. Cambridge University Press, Cambridge, 1998. [BH] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambdrige University Press, 1994. [GW] S. Goto, K.-i. Watanabe, On graded rings I, J. Math. Soc. Japan, vol. 30, no. 2, 1978. [H] M. Hochster, Local cohomology, lecture notes, University of Michigan. [Hu2] C. Huneke, Tight closure, parameter ideals, and geometry, Six lectures on commutative algebra (Bel- laterra, 1996), 187–239, Progr. Math., 166, Birkhauser, Basel, 1998. [R] J.J. Rotman, An introduction to homological algebra. Pure and Applied Mathematics 85. Academic Press, 1979.

8 [Si] A. K. Singh, Cyclic covers of rings with rational singularities, Trans. Amer. Math. Soc. 355 (2003) 1009-1024. [SW] A. K. Singh, U. Walther, On the arithmetic rank of certain Segre products, preprint.