F-RATIONALITY MANOJ KUMMINI. 1. Double-complex spectral sequences

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1 F-RATIONALITY MANOJ KUMMINI Introduction These are notes from my lectures at the worshop Commutative Algebra and Algebraic Geometry in Positive Characteristics held at IIT Bombay in December 8. The goal is to give a proof of a theorem of K. Smith which asserts that F-rational rings have pseudorational singularities [Smi97]. Notation. By a ring we mean a commutative ring with multiplicative identity. Ring homomorphisms are assumed to tae the multiplicative identity to the multiplicative identity. : field R, S: rings.. Double-complex spectral sequences 3 In this lecture, we list some results, mostly without proofs, about double-complex spec- 4 tral sequences. References are [CE99, Chapter XV], [Eis95, Appendix A3], and [Wei94, 5 Chapter 5]. 6 Let A be an abelian category and C, a first-quadrant double complex in A, i.e., a 7 double complex with C i,j = if i < or j <. Write F = Tot(C, ). We wish to 8 understand H (F ). To this end, we tae a filtration F F F. Fix n. Write 9 M p = Im(H n ( F p ) H n (F )). Since H n is a functor from the category of complexes over A to A, we get an induced filtration H n (F ) M M on H n (F ). Using a spectral sequence, we start from ( ( ) ) H F p /F p+ p 3 and obtain the associated graded object M p /M p+ of the filtration of H n (F ). p 4 Filtration by columns. For p, define C i,j p = { C i,j, if i p;, otherwise, for every j. Write F p = Tot( C, ). This gives a filtration F = F F F with 5 p F p / F p+ = Cp,. Version of December 5, 8.

2 MANOJ KUMMINI Set E i,j = C i,j for every i, j. Thin of E, as the collection of complexes C i, i, with the horizontal arrows as maps of these complexes. Now E, is the homology of E, ; more precisely, the maps in E, are of the form C i,j+ C i,j C i,j so E i,j = Hj (C i, ). The horizontal maps of C,, which are thought of as maps of complexes C i, C i+,, give maps E i,j E i,j E i+,j. We define E, as the homology of E,. One can show that there are maps E i,j E i,j E i+,j and that these form a complex. Define E, 3 as the homology of E, ; there are maps E i 3,j+ 3 E i,j 3 E i+3,j 3. Inductively define E r, as the homology of E, r ; the maps are E i r,j+r r E i,j r E i+r,j r+ r. Note for each s r, and and each i, j, Es i,j is a subquotient of Er i,j and that E i,j is a subquotient of C i,j. Hence, for each i, j, there exists r such that for every s r, the map coming into E i,j s is from the second quadrant and the map leaving from E i,j s the fourth quadrant; therefore these maps are zero, which gives that E i,j s for this r. E i,j = Er i,j = E i,j r is to ; define.. Theorem. For the filtration on H n (F ) induced by the filtration of { F p } p of F, the associated graded object of H n (F ) has E i,n i as its ith component. Filtration by rows. For q, define C i,j q = { C i,j, if j q;, otherwise, for every i. Write F q = Tot( C, q ). This gives a filtration F = F F F 4 with 43 F q / F q+ = C,q.

3 F-RATIONALITY 3 Set E i,j = Ci,j for every i, j. Thin of E, 44 as the collection of complexes C,j j, with the vertical arrows as maps of these complexes. Now E, is the homology of E, 45 ; more precisely, the maps in E, 46 are of the form C i,j C i,j C i+,j so E i,j = Hi (C,j ). The vertical maps of C,, which are thought of as maps of complexes C,j C,j+, give maps E i,j+ E i,j 49 E i,j We define E, as the homology of E,. One can show that there are maps E i,j+ E i,j E i+,j and that these form a complex. Inductively define E r, as the homology of E, 5 r ; the 5 maps are Er i+r,j r+ Er i,j Er i r,j+r As with the filtration by columns, for each i, j, there exists r such that for every s r, Es i,j = Er i,j ; define E i,j = Er i,j for this r... Theorem. For the filtration on H (F ) induced by the filtration of { F q } q of F, the associated graded object of H n (F ) has E n i,i as its ith component. Terminology. We often refer to E r, and E r, as the rth page of the spectral sequence. We also say that the spectral sequences E r, and E r, converge to H (F ). We denote this by Er i,j H i+j (F ) and Er i,j H i+j (F )

4 4 MANOJ KUMMINI Edge maps. Fix n and consider the filtration on H n (F ) induced by the filtration of { F p } p of F. Since this is a decreasing filtration, we see that E n, is a submodule of H n (F ). For r, there is a surjective morphism Er n, E n,. The composite map Er n, E n, H n (F ) is called an edge homomorphism. Similarly, we get an edge homomorphism Er,n E,n H n (F ) Grothendiec spectral sequence. We give an application of the double complex spectral sequence to obtain a relation between the derived functors of a composite of two functors. Let A, B, C be abelian categories such that A and B have enough injectives. Let F : A B and G : B C be left-exact covariant additive functors such that F taes injectives in A to G-acyclic objects in B, i.e., objects Y of B such that R i GY = for every i >..3. Theorem. With notation as above, there is a spectral sequence for every object X of A. E i,j = Rj G(R i F(X)) R i+j (GF)(X) Proof. Let X be an object of A. Let I be an injective resolution of X. Let J, be a Cartan- Eilenberg injective resolution (double complex) of F(I ). (See [CE99, Chapter XVII] and [Wei94, Section 5.7] for the construction of Cartan-Eilenberg resolutions.) Let C, = G(J, ). Then { E i,j = Hj (G(J i, )) = R j G(F(I i (GF)(I i ), if j = ; )) =, otherwise, by the hypothesis on F. Hence the E page is the complex (GF)(I ), from which we conclude that { E i,j R i (GF)(X), if j =, =, otherwise, In particular, for every n, the associated graded object of H n (Tot(C, )) has only one potentially non-zero term R n (GF)(X); it follows that H n (Tot(C, )) = R n (GF)(X). In the spectral sequence associated to filtration by rows of C,, we have E i,j = Hi G(J,j ) One can chec, using the definition and properties of Cartan-Eilenberg resolutions that Hence Set E = E. H i G(J,j ) = G(an injective resolution of H i (F(I ))). E i,j = Rj G(R i F(X)) The edge homomorphisms of the above spectral sequence are R n G(F(X)) R n (GF)(X).. Pseudo-rational rings In this lecture, we loo at pseudo-rational rings [LT8]. We begin with some remars on local cohomology.

5 F-RATIONALITY 5 Cohomology with supports. Let X be a topological space, Z a (locally) closed subset of X and F a sheaf of abelian groups on X. We denote the category of abelian groups by Ab and, for a topological space Y, the category of sheaves of abelian groups on Y by Ab Y. Write U = X Z. Define Γ Z (X, F) := er (Γ(X, F) Γ(U, F)). This is a functor from Ab X to Ab. It is left exact (Exercise 5.5). Define cohomology groups 96 with support in Z, denoted H Z (X), to be its right-derived functors... Proposition. Suppose that X = Spec R, that Z is defined by a finitely generated R-ideal I and that F is the sheaf defined by an R-module M. Then for every i. H i Z (X, F ) = Hi I (M) For a proof, see [Har67, Proposition.] or [ILL + 7, Theorem.47]... Proposition. Let f : X X be a continuous map, Z a closed subset of X, Z := f (Z) and F a sheaf of abelian groups on X. Then we have a spectral sequence E i,j = Hj Z (X, Ri f F ) converging to H i+j Z (X, F ). The edge homomorphisms of this page are the maps H n Z (X, f F ) H n Z (X, F ). Proof. Use Theorem.3 with A = Ab X, B = Ab X, C = Ab, F = f and G = Γ Z (X, ). Note that f taes injectives in Ab X to injectives in Ab X, which are acyclic for Γ Z (X, ). See [Har67, Proposition 5.5] for details. The assertion about edge homomorphisms follows from the definition. Pseudo-rational rings..3. Definition. Let (R, m) be a d-dimensional Cohen-Macaulay, normal, analytically unramified local ring. Then R is said to be pseudo-rational if the edge homomorphism H d m(r) δ f H d f ({m}) (Z, O Z) is injective, for every proper birational map f : Z Spec R with Z normal..4. Example. Regular local rings are pseudo-rational [LT8, Section 4]..5. Example. Let (R, m) be a d-dimensional Cohen-Macaulay, normal local ring that is essentially of finite type over a field of characteristic zero. Suppose that R has rational singularities, i.e., there exists a proper birational morphism h : Z Spec R such that Z is nonsingular (such a morphism is called a desingularization) and R i h O Z = for every i >. In fact, if this holds for one desingularization, it holds for every desingularization. Let f : W Spec R be a proper birational morphism with W normal. Let g : Z W be a desingularization. Then h = f g is a desingularization of Spec R. Then the edge homomorphism δ f is injective. (Exercise 5.8)..6. Example. Let be a field of characteristic different from 3, S = [x, y, z] and R = S/(x 3 + y 3 + z 3 ). Write m for the homogeneous maximal ideal of R. After replacing by an algebraic closure and using the jacobian criterion [Eis95, 6.9] we see that the

6 6 MANOJ KUMMINI singular locus of Spec R is {m}, which has codimension two. Since it is Cohen-Macaulay, it satisfies the Serre condition (S ). Hence R is a normal domain. Let A be the Rees algebra R[mt] and X = Proj A. Write f for the natural map X Spec R. We now mae several observations and conclude that R is not pseudo-rational. () X is nonsingular: X has an affine open covering (( [ mt ]) ) (( [ ]) ) mt Spec R Spec R. xt yt (Observe that zt (xt, yt).) Note that ( [ mt ]) [ y R xt R x, z ]. x Write u = y x and y = z x to see that [ y R x, z ] [x, u, v]/( + u 3 + v 3 ) x which is non-singular; similarly for the other open set. () H (X, F ) = for every coherent sheaf F on X, since X has an affine cover with two open sets. (3) The map f is birational: for, let a m. Then A a R a [t], so f (Spec R a ) Proj(R a R A) Proj(R a [t]) Spec R a. Write U = Spec R {m} and V = f (U). Then f V : V U ( is an isomorphism, ) since U has an affine covering by Spec R a, a m, a. (4) Supp H (X, F ) {m} for every coherent sheaf F on X. This follows from applying the flat-base change theorem for cohomology [Har77, III.9.3] for the flat (in fact open) morphism U Spec R, and noting that all higher direct images vanish for the isomorphism V U. (5) Let E = Proj (R/m R A), the scheme-theoretic pre-image of Spec(R/m) Spec R. Note that R/m R A [x, y, z]/(x 3 + y 3 + z 3 ), so E Proj R. Note that we have an exact sequence mo X O X O E. (6) H (E, O E ) : Since E Proj R, it suffices [ILL + 7, 3.] to show that Note that we have an exact sequence H m(r). H m(r) H 3 m(s)( 3) H 3 m(s) A description of H 3 m(s) as a graded S-module is given in [ILL + 7, Example 7.6], whence we conclude that H m(r) H 3 m(s) 3. (7) H m(h (X, O X )) = H (X, O X ), since H (X, O X ) is a finite-length non-zero module. (8) The exact sequence of low-degree terms (Exercise 5.) for the spectral sequence of Proposition. Hm(R j i f O X ) H i+j E (O X) is H m(r) edge H E (O X) H (X, O X ) H m(r) edge H E (O X) (9) H E (O X) = [Lip78, Theorem.4, p. 77].

7 F-RATIONALITY Hence the edge map H m(r) H E (O X) is non-zero, and Spec R is not pseudo-rational. What we essentially used is the fact that H m(r) j for some j. See Exercise 5. in this context. 3. F-rationality Tight closure. For this lecture and the next, p is a prime number and R is a noetherian ring of characteristic p. Let I be an R-ideal. By q, we mean a power of p. By I [q], we mean the ideal generated by {x q x I}. By R o, we mean the set R p Min(R) p. 3.. Definition. The tight closure of I, denoted I, is the set {x R there exists c R o such that cx q I [q] for all q }. We say that I is tightly closed if I = I. Some facts: () I is an ideal containing I; (I ) = I. () x I if and only if x (IR/p) for every p Min(R). (3) Every ideal in a regular local ring is tightly closed. F-rational rings. Let x,..., x n R. We say that (x,..., x n ) is a parameter ideal if the images of x,..., x n in R p form part of a system of parameters for R p for every prime ideal p of R containing x,..., x n. We say that R is F-rational if every parameter ideal is tightly closed. Some facts: () Every F-rational ring is normal. () Every ideal in a Gorenstein F-rational ring is tightly closed. (3) If R is a quotient of a Cohen-Macaulay ring and is F-rational, then R is Cohen- Macaulay, and localizations of R are F-rational. (4) Let R be a local ring that is a quotient of a Cohen-Macaulay ring. Then R is F- rational if and only if R is equi-dimensional and there exists a system of parameters that generates a tightly closed ideal. (5) Let R be a local ring and R its completion. If R is F-rational, then R is F-rational. The converse is true if R is excellent (e.g., essentially of finite type over a field). 8 8 Frobenius action on local cohomology. The Frobenius map F : R R, r r p commutes with localization. Let I = (x,..., x n ); then F commutes with the maps in Č(x,..., x n ; R), so it induces a map on H i I (R) for every i. On Hn I (R), this map is ] [ ] z z p x t xt xt n x tp xtp. xtp n 83 [ Č (x,..., x n ; R) is also the limit of the Koszul complexes K (x t,..., xt n; R) [ILL + 7, Chapter 7]. We have ( ) R x x x n R lim t (x t,..., xt n) (x t+ = H i I,..., xt+ n ) (R).

8 8 MANOJ KUMMINI If x,..., x n is an R-regular then the maps in the above system are injective, so Under this map the element R (x t,..., xt n) Hi I (R). [ z x t xt xt n corresponds to z mod (x t,..., xt n). For a proof, see [LT8, p. 4 5]. ] Definition. A submodule M of H i I (R) is said to be F-stable if F(M) M Example. Let η H n I (R). Then the R-submodule of Hn I (R) generated by Fe (η), e is F-stable. In the proof of the theorem below, we will denote it by M η Theorem ([Smi97, Theorem.6]). Let (R, m) be a d-dimension excellent Cohen-Macaulay local ring of characteristic p. Then R is F-rational if and only if Hm(R) d has no proper non-zero F-stable submodules. A special case of this was proved by R. Fedder and K. i. Watanabe: assuming that R is an isolated singularity and that H i m(r) has finite length for every i < d; see [FW89, Theorem.8]. Proof. Only if : Since R is excellent and F-rational, R is Cohen-Macaulay and F-rational. Since Hm(R) d is both an R-module and an R-module (compatibly), we may assume that R is complete. By way of contradiction suppose that M Hm(R) d is an F-stable R-submodule of Hm(R). d Let C = Hm(R)/M. d Taing Matlis duals, we get ( C Hm(R)) d M where ω R is a canonical module of R. The isomorphism ω R ( H d m(r)) ωr is local dual- ity [ILL + 7, Theorem.44]. Note that M C, so C M. Since ω R is a torsion-free R-module of ran, K R M =. Since M is a finitely generated R-module, there exists c R such that cm =, so cm =. Let [ ] z η := x t xt M xt d be a non-zero element. Hence cf e (η) = [ cz q x tq xtq xtq d for every q. This means that cz q (x tq 8, xtq,..., xtq d ) for every q, i.e., z (xt, xt,..., xt d ) = 9 (x t, xt,..., xt d ), so η =, a contradiction. If : To mae the argument simple, we will assume that R is a domain. By way of contradiction, assume that R is not F-rational. Let x,..., x d be a system of parameters ] =.

9 and z (x,..., x d ) (x,..., x d ). Write [ ] z η = H x x x m(r). d d F-RATIONALITY 9 Note that η, so M η. Let c R be such that cz q (x q 3, xq,..., xq ) for every d 4 q. Then cη q = for every q, so cm η =. Note that M η Hm(R) d since the 5 annihilator of Hm(R) d is. This contradicts the hypothesis Example. Suppose that R is positively graded with R. Write m for the homogeneous maximal ideal. Then ( ) Hm(R) d is a proper F-stable submodule of H d m(r). Using this, we see that the ring is not F-rational for any field. j [x, y, z]/(x 3 + y 3 + z 3 ) 3.6. Example. Let R = F [x, y, z]/(x +y 3 +z 5 ). This is a Cohen-Macaulay normal domain. It is a graded ring if we set deg x = 5, deg y = and deg z = 6. Let S = F [x, y, z]. Then ] [H 3 (x,y,z) [H ] 3 (S) and 3 (x,y,z) (S) = for every j 3. Hence ] [H [H 3(x,y,z) ] (R) and 3(x,y,z) (R) However, R is not F-rational, as we see now. Since x (y, z), [ ] x H yz (x,y,z) (R). On the other hand, x (y, z ), so [ x ] y z = F j j ([ ]) x =. yz j = for every j. Hence F has a non-zero ernel. It is easy to chec that ernel of F is an F-stable submodule of H (x,y,z) (R) F-rationality implies pseudo-rationality 5. Exercises 5.. Derive the exact sequence of low-degree terms for the E page: E, 5.. Place an exact sequence edge H (F ) E, M M M 3 d, E, edge H (F ) on the horizontal axis and tae a Cartan-Eilenberg injective resolution. Let F be a leftexact covariant functor. Show that the E 3 page is zero and that the maps on the E and E pages give the familiar exact sequence in R i F.

10 MANOJ KUMMINI Show that Tor R (M, N) Tor R (N, M) by looing at the third quadrant double complex C i, j = F i G j where F and G are projective resolutions of M and N respectively The following is an example of a step in the construction of pure resolutions by Eisenbud and Schreyer. 38 Let X = P P, where is a field. Give homogeneous coordinates u, v and x, y respectively. Let S = [u, v, x, y], with deg u = deg v = (, ) and deg x = deg y = (, ) () The Koszul complex on S with respect to ux, uy + vx, vy gives an exact sequence K : O X ( 3, 3) O X (, ) 3 O X (, ) 3 O X (Hint: X can be thought of as the set of bigraded ideals not containing the irrelevant ideal (u, v) (x, y). The two projection maps from X are given by contraction to [u, v] and [x, y].) () Let π : X P be the projection to the first factor. Let I, be a Cartan-Eilenberg injective resolution of K. Let C, = π (I, ). (This is a first-quadrant double complex.) Use the projection formula to see that E i,j = O P ( 3), if i = 3 and j = ; O P ( ) 3, if i = and j = ; O P, if i = and j = ;, otherwise. (3) Use the E spectral sequence to conclude that E i,j = for every i, j. (4) Conclude that the non-zero terms of the E page give an exact sequence O P ( 3) O P ( ) 3 O P (Getting a pure resolution over [u, v] from the above exact sequence requires a little more wor, which we omit in this exercise.) (5) Using the same strategy, construct an exact sequence O P ( 3) a O P ( ) b O P () c Do a diagram-chasing in the commutative diagram below Γ Z (X, F ) Γ Z (X, F ) Γ Z (X, F 3 ) Γ(X, F ) Γ(X, F ) Γ(X, F 3 ) Γ(U, F ) Γ(U, F ) Γ(U, F 3 ) to conclude that Γ Z (X, ) is left-exact Let (R, m) be a two-dimensional analytically unramified normal domain and f : W Spec R a proper birational morphism with W normal.

11 F-RATIONALITY () Show that Supp(H (W, O W )) {m}. (Hint: Localize in the base and use flat basechange for cohomology and the fact that over a DVR, every proper birational map is an isomorphism.) () R is pseudo-rational if and only if H (Z, O Z ) = for every Z that has a proper birational map to Spec R. (Use: H f ({m})(o Z ) = [Lip78, Theorem.4, p. 77].) 5.7. Let (R, m) be a normal local ring. Let W g f Z Spec R be a proper birational morphisms with Z and W normal. Write h = f g. Then the edge map 63 factors as H d m(r) H d m(r) δ h H d h ({m}) (W, O W) δ f H d f ({m}) (Z, O Z) H d h ({m}) (W, O W) Show that rational singularities (in characteristic zero) are pseudo-rational. You need to use the fact that if R has rational singularities, then R i f O Z = for every desingularization f : Z Spec R and every i > Let R be a Cohen-Macaulay ring of characteristic zero. Show that R has rational singularities if and only if there exists a proper birational morphism f : Z Spec R such that Z has rational singularities and R i f O Z = for every i >. 5.. Let (R, m) be a noetherian ring. Let X = {p Spec R dim R/p = dim R}. Let a m p X p. If R/(a) is regular, then so is R. Show that the hypothesis on a is necessary. 5.. Let R be a two-dimensional standard graded normal domain, with R =, with homogeneous maximal ideal m. Assume that Spec R {m} has pseudo-rational singularities. Show that R has pseudo-rational singularities if and only if H m(r) j = for every j as follows: () Let X = Proj R[mt]. Then X has pseudo-rational singularities, and there is a proper birational map f : X Spec R. () Let h : W Spec R with W normal. Let W be the blow-up of W along the ideal sheaf mo W, and h the composite map W W Spec R. It suffices to show that the edge map δ h is injective. Hence replacing W by W, we may assume that h factors as g f W X Spec R. (3) R g O W =. (4) Let E be the divisor of X defined by mo X. Write Ẽ = h ({m}). The map is an isomorphism. (5) The map is injective if and only if the map H E (O X) H (O Ẽ W) H m(r) H (O Ẽ W) H m(r) H E (O X) is injective, which holds if and only if H (X, O X ) = which holds if and only if H (E, O E (j)) = for every j which holds if and only if H m(r) j = for every j. You will need to use two facts: E Proj R and that m j O X X O E O E (j).

12 MANOJ KUMMINI Let R = [x, x 3 ] where is a field of characteristic p >. Show that x 3 (x ) (x ) Let R = [x, y, z]/(x 3 + y 3 + z 3 ) where is a field of characteristic p >, p 3. Show that z (x, y) (x, y) Let R be a noetherian ring, and I an R-ideal. Show that if I is tightly closed, then (I : J) is tightly closed for every ideal J Show that an intersection of tightly closed ideals is tightly closed Let (R, m) be a Gorenstein ring, I an unmixed R-ideal, and x,..., x c I a maximal regular sequence. Write J = (x,..., x c ). Show that (J : (J : I)) = I Show that every ideal in a Gorenstein F-rational ring is tightly closed Let R be a local ring and R its completion. If R is F-rational, then R is F-rational Let (R, m) be a two-dimensional pseudo-rational ring. Following [LT8, Section 5], prove the (special case of) Brian con-soda theorem: I n+ I n for every n, as follows: () We may assume that R/m is an infinite field [LT8, Example (c), p. 3]. () I has a reduction generated by two elements, i.e., there exists J = (x, y) I such that I n+ = JI n for every n. (Hint: tae a Noether normalization of R[It]/mR[It].) (3) The ideal generated by xt, yt in A := R[It] is primary to the irrelevant ideal. (4) Let X = Proj A. The Koszul complex K (xt, yt; A) gives an exact sequence O X (n) O X (n + ) O X (n + ) for every n Z. (Here, by O X () we mean the invertible sheaf IO X.) (5) For n, this gives an exact sequence I n [x y] In+ I n+. (Use: I n = H (X, O X (n)) for every n, I := R = H (X, O X ).) (6) I n+ = II n+ for every n. (7) I n+ I n for every n. References [CE99] H. Cartan and S. Eilenberg. Homological algebra. Princeton Landmars in Mathematics. Princeton University Press, Princeton, NJ, 999. With an appendix by David A. Buchsbaum, Reprint of the 956 original., 4 [Eis95] D. Eisenbud. Commutative algebra, with a View Toward Algebraic Geometry, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New Yor, 995., 5 [FW89] R. Fedder and K. Watanabe. A characterization of F-regularity in terms of F-purity. In Commutative algebra (Bereley, CA, 987), volume 5 of Math. Sci. Res. Inst. Publ., pages Springer, New Yor, [Har67] R. Hartshorne. Local cohomology, volume 4 of A seminar given by A. Grothendiec, Harvard University, Fall 96. Springer-Verlag, Berlin, [Har77] R. Hartshorne. Algebraic geometry. Springer-Verlag, New Yor, 977. Graduate Texts in Mathematics, No [ILL + 7] S. B. Iyengar, G. J. Leusche, A. Leyin, C. Miller, E. Miller, A. K. Singh, and U. Walther. Twentyfour hours of local cohomology, volume 87 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 7. 5, 6, 7, 8

13 F-RATIONALITY [Lip78] J. Lipman. Desingularization of two-dimensional schemes. Ann. Math. (), 7():5 7, , [LT8] J. Lipman and B. Teissier. Pseudorational local rings and a theorem of Briançon-Soda about integral closures of ideals. Michigan Math. J., 8():97 6, 98. 4, 5, 8, [Smi97] K. E. Smith. F-rational rings have rational singularities. Amer. J. Math., 9():59 8, 997., 8 [Wei94] C. A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 994., 4 Chennai Mathematical Institute, Siruseri, Tamilnadu 633. India address: mummini@cmi.ac.in