Universities of Leeds, Sheffield and York

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1 promotng access to Whte Rose research papers Unverstes of Leeds, Sheffeld and York Ths s an author produced verson of a paper publshed n Journal of Algebra. Whte Rose Research Onlne URL for ths paper: Publshed paper Huneke, C., Katzman, M., Sharp, R.Y., Yao, Y. (2006 Frobenus test exponents for parameter deals n generalzed Cohen-Macaulay local rngs, Journal of Algebra, 305 (1, pp Whte Rose Research Onlne eprnts@whterose.ac.uk

2 FROBENIUS TEST EXPONENTS FOR PARAMETER IDEALS IN GENERALIZED COHEN MACAULAY LOCAL RINGS CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO arxv:math/ v1 [math.ac] 6 Jul 2006 Abstract. Ths paper studes Frobenus powers of parameter deals n a commutatve Noetheran local rng R of prme characterstc p. For a gven deal a of R, there s a power Q of p, dependng on a, such that the Q-th Frobenus power of the Frobenus closure of a s equal to the Q-th Frobenus power of a. The paper addresses the queston as to whether there exsts a unform Q 0 whch works n ths context for all parameter deals of R smultaneously. In a recent paper, Katzman and Sharp proved that there does exsts such a unform Q 0 when R s Cohen Macaulay. The purpose of ths paper s to show that such a unform Q 0 exsts when R s a generalzed Cohen Macaulay local rng. A varety of concepts and technques from commutatve algebra are used, ncludng uncondtoned strong d-sequences, cohomologcal annhlators, modules of generalzed fractons, and the Hartshorne Speser Lyubeznk Theorem employed by Katzman and Sharp n the Cohen Macaulay case. 0. Introducton Ths paper studes a certan type of unform behavour of parameter deals n a commutatve Noetheran rng R of prme characterstc p. One motvaton for our work comes from the theory of test exponents for tght closure ntroduced by M. Hochster and C. Huneke n [8, Defnton 2.2]. For an deal a of R and a non-negatve nteger n, the p n -th Frobenus power a [pn] of a s the deal of R generated by all p n -th powers of elements of a. Suppose, temporarly, that R s reduced. Recall that a test element for R s an element c of R outsde all the mnmal prme deals of R such that, for each deal a of R, and for r R, t s the case that r a, the tght closure of a, f and only f cr pn a [pn] for all n 0. It s a result of Hochster and Huneke [7, Theorem (6.1(b] that such a test element exsts f R s a (reduced algebra of fnte type over an excellent local rng of characterstc p. Let c be a test element for R, and let a be an deal of R. A test exponent for c, a s a power q p e 0 (where e 0 s a non-negatve nteger such that f, for an r R, we have cr pe a [pe ] for one sngle e e 0, then r a (so that cr pn a [pn ] for all n 0. In [8], t s shown that ths concept has strong connectons wth the major open problem about whether tght closure commutes wth localzaton; ndeed, to Date: February 2, Mathematcs Subject Classfcaton. Prmary 13A35, 13A15, 13D45, 13E05, 13H10, 16S36; Secondary 13C15, 13E10, 13F40. Key words and phrases. Commutatve Noetheran rng, prme characterstc, Frobenus homomorphsm, Frobenus closure, generalzed Cohen Macaulay local rng, uncondtoned strong d-sequence, flter-regular sequence, Artnan module, Frobenus skew polynomal rng, local cohomology module. Huneke was partally supported by the US Natonal Scence Foundaton (grant number DMS ; Sharp was partally supported by the Engneerng and Physcal Scences Research Councl of the Unted Kngdom (grant number EP/C538803/1. 1

3 2 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO quote Hochster and Huneke, roughly speakng, test exponents exst f and only f tght closure commutes wth localzaton. In a recent paper [19], R. Y. Sharp has shown that, for a test element c n a reduced equdmensonal excellent local rng R (of characterstc p, there exsts a non-negatve nteger e 0 such that p e 0 s a test exponent for c, a for every parameter deal a of R. (In such an R, a parameter deal s smply an deal that can be generated by part of a system of parameters. We can thnk of p e 0 as a unform parameter test exponent for R. It s natural to ask whether there s an analogous result for Frobenus closures. Return to the general stuaton where we assume only that R s a commutatve Noetheran rng of prme characterstc p. The Frobenus closure a F of an deal a of R s defned by a F : { r R : there exsts 0 n Z such that r pn a [pn ] }. Ths s an deal of R, and so s fntely generated; therefore there exsts a power Q 0 of p such that (a F [Q0] a [Q0], and we defne Q(a to be the smallest power of p wth ths property. Note that, for r R, t s the case that r a F f and only f r Q(a a [Q(a]. In [10, 0], M. Katzman and Sharp rased the followng queston: s the set {Q(b : b s a proper deal of R} of powers of p bounded? In the case where R s Artnan, that s, has dmenson 0, t s easy to see that ths queston has an affrmatve answer, because n that case R/ 0 s a drect product of felds, and one can deduce easly from that that, f Q 1 s a power of p such that ( 0 [Q 1 ] 0, then Q(b Q 1 for every deal b of R. For ths reason, we shall assume that dm R > 0 for the remander of the paper. H. Brenner [1] has recently shown that the answer to the queston of Katzman and Sharp, as stated above, s negatve. Nevertheless, t mght not be too unreasonable to hope that, n the case where R s local, the set {Q(b : b s a parameter deal of R} s bounded. In [10, Theorem 2.5], Katzman and Sharp showed that ths s the case when R s a Cohen Macaulay local rng: they showed that, then, there exsts an nvarant η(r of R such that (b F [pη(r ] b [pη(r ] for all parameter deals b of R. The purpose of ths paper s to prove the correspondng result when R s a generalzed Cohen Macaulay local rng, that s, when all the local cohomology modules Hm (R ( 0,...,t 1 (where m denotes the maxmal deal of R and t : dm R have fnte length. Specfcally, n Theorem 5.2 we prove the followng. Theorem. Let R be a generalzed Cohen Macaulay local rng of prme characterstc p. Then there exsts a power Q of p such that ((b F [Q] b [Q] for every deal b of R that can be generated by part of a system of parameters of R. In the Cohen Macaulay case, the nvarant η(r was defned by means of the Hartshorne Speser Lyubeznk Theorem (see [10, Theorem 1.4] about a certan type of unform behavour of a left module over the Frobenus skew polynomal rng (assocated to R that s Artnan as an R-module: Katzman and Sharp appled ths Hartshorne Speser Lyubeznk Theorem to the top local cohomology module of a Cohen Macaulay local rng.

4 FROBENIUS TEST EXPONENTS 3 In ths paper, we make smlar use of the Hartshorne Speser Lyubeznk Theorem, although we apply t to all the local cohomology modules of the generalzed Cohen Macaulay local rng R. We also use a varety of other concepts and technques from commutatve algebra, ncludng uncondtoned strong d-sequences and work of S. Goto and K. Yamagsh [4] about them, flter-regular sequences, cohomologcal annhlators, and modules of generalzed fractons. Other motvaton for ths work s provded n [10, 0]. There s no doubt n our mnds that unform behavour of Frobenus closures of the type establshed n ths paper s both desrable for ts own sake and also relevant to the vgorous and ongong development of tght closure theory. 1. Notaton and termnology Throughout ths paper, R wll denote a Noetheran commutatve rng wth dm R t > 0, and a wll denote an deal of R. We shall use Var(a to denote the varety of a; thus Var(a : {p Spec(R : p a}. We shall use mn(r to denote the set of mnmal prme deals of R and R to denote R \ p mn(r p. The annhlator of an R-module M wll be denoted by Ann R (M. We shall sometmes use the notaton (R, m to ndcate that R s local wth maxmal deal m; then, ( R, m wll denote the m-adc completon of R. Also n the local case, we say that x x 1,...,x l s a system of parameters of (R, m f l 1 x R s m-prmary and l t; we say x s a subsystem of parameters of (R, m f t s a subsequence of a system of parameters of R. Notaton 1.1. Throughout the paper, x x 1, x 2,...,x l wll denote a sequence of l elements of R. ( We use N to denote the set of all non-negatve ntegers, and N + to denote the set of all postve ntegers. ( The man results concern the case where R has prme characterstc p, but ths hypothess wll only be n force when explctly stated; then, q, q, Q, Q and Q ( N wll always denote powers of p wth non-negatve nteger exponents. ( For ntegers j, we denote the subset {,...,j} of Z by [, j], and we agree that [, j] f > j. (v We adopt the normal conventon that a 0 1 for all a R. (v For each Λ [1, l], we set x Λ : Λ x R. In case Λ, we agree that x n 1 and xn R (0 for all n N. (v For any Λ [1, l] and n 1,...,n l N, the sequence ((( Λ xn +j R : x j Λ j0,1,2,... forms an ascendng chan of deals, and we denote ts ultmate constant value by ( Λ xn R lm. Thus ( Λ xn R lm j N (( Λ xn +j R : x j Λ. In partcular ( x R lm (0. (v For any Λ [1, l] and n 1,...,n l N, we set ( Λ xn R (x-unm : (( Λ xn R : [1,l]\Λ x R

5 4 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO and refer to ths as the unmxed part of Λ xn R relatve to the sequence x x 1, x 2,...,x l. Next, we recall some defntons of varous concepts n commutatve algebra. Defnton 1.2. Recall that a denotes an deal of R. ( We say that x s an a-flter regular sequence f there exsts an nteger n N ((( such that a n Ann j 1 R 1 x R : x j / j 1 1 x R for all j 1, 2,..., l. It s easy to see that x s an a-flter regular sequence f and only f, for all p Spec(R \ Var(a, the natural mages of x 1,...,x l form a possbly mproper regular sequence n R p. ( We say that x s a d-sequence f (( j 1 x R (( : x j+1 x k j 1 x R : x k for all j, k such that 0 j < k l. ( If x n 1 1,...,x n l l form a d-sequence n any order and for any postve ntegers n 1, n 2,...,n l, then we say that x s an uncondtoned strong d-sequence, or a d + -sequence. (v Generalzed Cohen Macaulay local rngs were studed by P. Schenzel n [16] (where they were called quas-cohen Macaulay local rngs ([16, Defnton 2] and by Schenzel, N. V. Trung and N. T. Cuòng n [18]. When (R, m s local, we say that R s a generalzed Cohen Macaulay local rng f t 1 0 Ann R(Hm (R (recall that t denotes dm R contans an m-prmary deal; snce all the local cohomology modules of R wth respect to m are Artnan, ths s the case f and only f Hm(R has fnte length for all 0,...,dm R 1. Note that R s a generalzed Cohen Macaulay local rng f and only f ts completon R s. (v Agan when (R, m s local, we say that R s s Cohen Macaulay on the punctured spectrum f R p s Cohen Macaulay for every p Spec(R \ {m}. Notaton 1.3. Suppose that R has prme characterstc p. In these crcumstances, we shall always denote by f : R R the Frobenus homomorphsm, for whch f(r r p for all r R. We shall use the skew polynomal rng R[T, f] assocated to R and f n the ndetermnate T over R. Recall that R[T, f] s, as a left R-module, freely generated by (T N, and so conssts of all polynomals n 0 r T, where n N and r 0,...,r n R; however, ts multplcaton s subject to the rule Tr f(rt r p T for all r R. We refer to R[T, f] as the Frobenus skew polynomal rng assocated to R. If G s a left R[T, f]-module, then the set Γ T (G : { g G : T j g 0 for some j N + } s an R[T, f]-submodule of G, called the T-torson submodule of G; we say that G s T-torson precsely when G Γ T (G. Note that R tself has a natural structure as a left R[T, f]-module under whch Tr f(r for all r R (that s, n whch the acton of the ndetermnate T on an element of R s just the same as the acton of the Frobenus homomorphsm. We

6 FROBENIUS TEST EXPONENTS 5 shall use R[T,f] Mod to denote the category of all left R[T, f]-modules and R[T, f]- homomorphsms between them. The category of all R-modules wll be denoted by Mod R. 2. Modules of generalzed fractons The concept of module of generalzed fractons (due to Sharp and H. Zaker [21] wll be used n ths paper. The constructon and basc propertes of these modules can be found n [21], but, at the request of the referee, we nclude n ths secton explanaton of some of the man deas. Remnder 2.1 (R. Y. Sharp and H. Zaker [21, 2]. Let k N +. Let U be a trangular subset of R k [21, 2.1], that s, a non-empty subset of R k such that ( whenever (u 1,...,u k U and n 1,...,n k N +, then (u n 1 1,...,un k k U also; and ( whenever (u 1,..(., u k, (v 1,..., v k U, then there exsts (w 1,..., w k U ( such that w j1 u jr j1 v jr for all 1,...k, so that there exst k k lower trangular matrces H and K wth entres n R such that H[u 1,...,u k ] T [w 1,...,w k ] T K[v 1,...,v k ] T. (Here, T denotes matrx transpose, and [z 1,...,z k ] T (for z 1,..., z k R s to be nterpreted as a k 1 column matrx n the obvous way. It wll be convenent for us to use D k (R to denote the set of k k lower trangular matrces wth entres n R; we use det(h to denote the determnant of an H D k (R. Let M be an R-module. Defne a relaton on M U as follows: for m, n M and (u 1,...,u k, (v 1,...,v k U, wrte (m, (u 1,...,u k (n, (v 1,...,v k precsely when there exst (w 1,...,w k U and H,K D k (R such that H[u 1,..., u k ] T [w 1,...,w k ] T K[v 1,..., v k ] T and det(hm det(kn k 1 j1 w jm. Then s an equvalence relaton; for m M and (u 1,...,u k U, we denote the equvalence class of (m, (u 1,..., u k by the generalzed fracton m (u 1,..., u k. The set of all equvalence classes of s an R-module, called the module of generalzed fractons of M wth respect to U, under operatons for whch, for m, n M and (u 1,...,u k, (v 1,...,v k U, m (u 1,...,u k + n det(hm + det(kn (v 1,...,v k (w 1,...,w k for any choce of (w 1,...,w k U and H,K D k (R such that H[u 1,...,u k ] T [w 1,...,w k ] T K[v 1,...,v k ] T, and, for r R, m r (u 1,..., u k rm (u 1,...,u k. Ths module of generalzed fractons s denoted by U k M.

7 6 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO Remark 2.2. Use the notaton of 2.1. It s worth bearng n mnd, when one s calculatng wth generalzed fractons n U k M, that, whenever (u 1,..., u k, (y 1,...,y k U and L D k (R are such that L[u 1,...,u k ] T [y 1,...,y k ] T, then, for all m M, m (u 1,...,u k det(lm (y 1,...,y k n U k M. Ths sometmes permts us to change a denomnator to one that s, n some sense, more convenent. Example 2.3. For our gven sequence x x 1, x 2,..., x l and any N +, we set U(x : {(x n 1 1,...,x n : for some j [0, ], n 1,...,n j N +, n j+1 n 0}, where x k s nterpreted as 1 when k l + 1. Then U(x s a trangular subset of R for each N +. Dscusson 2.4. We use the notaton of 2.1. ( The constructon of a module of generalzed fractons can be vewed as a generalzaton of ordnary fracton formaton n commutatve algebra: see [21, 3.1]. ( For r 1,...,r k R, we shall denote the dagonal k k matrx n D k (R whose dagonal entres are r 1,...,r k by dag(r 1,...,r k. Notce that det(dag(r 1,...,r k r 1 r k, and that, for non-negatve ntegers n 1,...,n k, m 1,...,m k, we have dag(r m 1 1,...,rm k k [rn 1 1,...,rn k k ]T [r m 1+n 1 1,...,r m k+n k k ] T. ( The comments n ( can be useful. For example, f (u 1,...,u k, (w 1,...,w k U are such that there exst H,K D k (R wth then H[u 1,...,u k ] T [w 1,...,w k ] T K[u 1,...,u k ] T, DH[u 1,...,u k ] T [w 2 1,...,w2 k ]T DK[u 1,...,u k ] T, where D : dag(w 1,...,w k, and t turns out (see [21, Lemma 2.3] that det(dh det(dk k 1 1 w2 R. Ths can be helpful when one has rather lttle nformaton about H but full knowledge of K. To llustrate ths, suppose that t s known, for some r R, that det(hr k 1 1 w R; then det(dhr w 1 w k det(hr k 1 1 w2 R, and the above consderatons show that det(dkr det(dhr (det(dh det(dkr k 1 1 w2 R. (v The equvalence relaton of 2.1 s such that, whenever (u 1,...,u k U and the element m of M actually belongs to k 1 1 u M, then m (u 1,...,u k 0 n U k M. (v Note that, by (v, f k 2 and (v 1,...,v k 2, 1, 1 U, then n (v 1,...,v k 2, 1, 1 0 n U k M, for all n M.

8 FROBENIUS TEST EXPONENTS 7 (v In the case where U conssts entrely of possbly mproper regular sequences on M, there s the followng converse of (v, proved by Sharp and Zaker n [23, Theorem 3.15]: f (u 1,...,u k U and m M are such that m (u 1,..., u k 0, then m k 1 1 u M. Dscusson 2.5. A chan of trangular subsets on R s a famly U : (U 1 such that ( U s a trangular subset of R for all N + ; ( whenever (u 1,...,u U (for any N +, then (u 1,..., u, 1 U +1 ; ( whenever (u 1,...,u U wth > 1, then (u 1,...,u 1 U 1 ; and (v (1 U 1. Gven such a famly U and an R-module M, we can construct a complex C(U, M of modules of generalzed fractons 0 e 1 M e0 U1 1 M e1 e 1 U M e U 1 +1 M, n whch e 0 (m m U 1 (1 1 M for all m M and, for N +, ( e m m (u 1,...,u (u 1,...,u, 1 U 1 +1 M for all m M and (u 1,...,u U. The comment n 2.4(v shows that C(U, M s ndeed a complex. The Exactness Theorem of Sharp Zaker ([23, Theorem 3.3], but see L. O Carroll [15, Theorem 3.1] for a subsequent shorter proof states that the complex C(U, M s exact f and only f, for all N +, each element of U s a possbly mproper regular sequence on M. Example 2.6. For our sequence x x 1, x 2,...,x l, the famly U(x : (U(x 1, where the U(x ( N + are as defned n 2.3, s a chan of trangular subsets on R. In partcular, we can construct the complex of modules of generalzed fractons C(U(x, R, whch we shall wrte as 0 e 1 R e0 U(x 1 1 R e1 e 1 U(x R e U(x 1 +1 R. Note that U(x R 0 whenever l+2. When workng wth a cohomology module H (C(U(x, R Ker e / Im e 1 (where N of the complex C(U(x, R, we shall use [ ] to denote natural mages, n ths cohomology module, of elements of Ker e. In the specal case n whch (R, m s local, l dm R t and x 1,...,x l s a system of parameters of R, t follows from the Exactness Theorem mentoned n 2.5 that R s Cohen Macaulay f and only f C(U(x, R s exact. In [24, Theorems 2.4 and 2.5], Sharp and Zaker proved that (R, m s a generalzed Cohen Macaulay local rng f and only f there s a power of m that annhlates all the cohomology modules of the complex C(U(x, R, and that when ths s the case, the th cohomology module H (C(U(x, R of the complex C(U(x, R s somorphc to Hm (R for all 0,...,dm R 1 l 1. The latter result s relevant to ths paper, although n 4 we provde a refnement for use n the case where R has prme characterstc p.

9 8 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO Some of the applcatons of modules of generalzed fractons can be found n [22], [23], [24], [15], [5] and [12]. Ths paper provdes some more. 3. Background results In ths secton, we shall collect together some observatons and results, wth approprate references, that concern topcs mentoned n 1 and whch we plan to use. Remark 3.1. Suppose that x x 1,...,x l s an uncondtoned strong d-sequence n R. ( It s mmedate from the defntons that, for any n 1,..., n l N +, Λ [1, l] and j [1, l] \ Λ, we have (( Λ xn R : x n (( j j Λ xn R ( : x j R (x-unm. l Λ xn ( Therefore, f a s an deal of R such that a 1 x R, then every permutaton of x 1,...,x l s an a-flter regular sequence, because l ( 1 x R annhlates Λ x R : [1,l]\Λ x R /( Λ x R for all Λ [1, l]. Theorem 3.2 (P. Schenzel [17, Satz 2.4.2]. Suppose that (R, m s local; recall that dm(r t. Then, for all systems of parameters a 1,...,a t of R, the deal t 1 0 Ann R(Hm (R of R annhlates all the R-modules (( j 1 a R ( : a j+1 / j 1 a R (j 0,...,t 1. In [3], S. Goto and T. Ogawa proved that, n a generalzed Cohen Macaulay local rng (R, m, there exsts a postve nteger h such that every system of parameters of R contaned n m h s a d-sequence. The followng corollary (of Schenzel s Theorem 3.2 s a varaton on that theme. Corollary 3.3. Suppose that (R, m s local wth dm(r t, and that x x 1,...,x l s a subsystem of parameters of R such that l 1 x R t 1 0 Ann R(Hm (R. Then x s an uncondtoned strong d-sequence. Proof. Let n 1,...,n l be postve ntegers, and let j N be such that 0 j < l. By Schenzel s Theorem 3.2, the R-module (( j 1 xn R : x n ( j+1 j+1 / j 1 xn R s annhlated by l 1 x R and so, n partcular, by x j+1. Hence (( j 1 xn R : x n (( j+1 j+1 j 1 xn R : x j+1. Moreover, the hypotheses on x 1,...,x l do not depend on the order n whch ths sequence s wrtten. Therefore, f j, k are such that 0 j < k l, then the fact that x k annhlates (( j ( 1 xn / j 1 xn ensures that (( j 1 xn R : x n j+1 j+1 xn k k R : x n j+1 j+1 R (( j 1 xn R : x n k+1 k paragraph shows that (( j 1 xn R (( : x n k+1 k j 1 xn R : x n k k., and the precedng Therefore x n 1 1,...,xn l l s a d-sequence. Snce the hypotheses on x 1,...,x l do not depend on the order n whch ths sequence s wrtten, we see that x s an uncondtoned strong d-sequence.

10 FROBENIUS TEST EXPONENTS 9 The next proposton s an extenson of a well-known result. For an explanaton of what t means to say that a local rng s formally catenary, and for a proof of L. J. Ratlff s Theorem that a unversally catenary local rng s formally catenary, the reader s referred to [14, p. 252]. Proposton 3.4. Assume that (R, m s a formally catenary local rng all of whose formal fbres are Cohen Macaulay. (These hypotheses would be satsfed f R was an excellent local rng. Suppose that R s equdmensonal and Cohen Macaulay on the punctured spectrum. Then R s a generalzed Cohen Macaulay local rng. Proof. Snce R s equdmensonal and formally catenary, R s equdmensonal; the hypothess concernng the formal fbres ensures that R s Cohen Macaulay on the punctured spectrum. Thus one can assume that R s complete, and n that case the clam follows from work of P. Schenzel, N. V. Trung and N. T. Cuòng [18, (2.5 and (3.8]. The next two results are entrely due to S. Goto and K. Yamagsh [4]. Unfortunately, as far as we are aware, [4] only exsts as a preprnt that has been crculatng nformally for more than 15 years, wthout formal publcaton. T. Kawasak ncluded a proof of [4, Lemma 2.2] n [11, Theorem A.1]; as we have not been able to fnd a formally publshed proof of [4, Theorem 2.3], we have ncluded one below. Theorem 3.5 (S. Goto and K. Yamagsh [4, Lemma 2.2]. (See also [11, Theorem A.1]. Suppose that x x 1,...,x l s an uncondtoned strong d-sequence n R. Then, for each [1, l] and each j [1, l] \, and for all postve ntegers n 1,...,n l, we have ( xn R (x-unm (( xn R : x j Λ ( Λ xn 1 ( Λ x R (x-unm. Proof. Ths result, orgnally due to Goto and Yamagsh, s proved n [11, Theorem A.1] n the case where n 1,...,n l 2, and one can check that that proof works for all choces of postve ntegers n 1,..., n l. Theorem 3.6 (S. Goto and K. Yamagsh [4, Theorem 2.3]. Suppose that x x 1,...,x l s an uncondtoned strong d-sequence n R and let n 1, n 2,...,n l N + be any postve ntegers. ( When l 1, (x n 1 1 Rlm x n 1 1 R + j N (0 : xj 1 xn 1 1 R + (0 : x 1. ( When l 2, ( l 1 xn R lm l 1 l 1 (( j [1,l]\{} xn j j R : R x ( j [1,l]\{} xn j j R (x-unm. Proof. (Goto Yamagsh. We use nducton on l; the result s easy when l 1, and so we suppose that l 2 and that the result has been proved for smaller values of l.

11 10 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO Let a ( l 1 xn R lm, so that, wthout loss of generalty, there exsts m N + such that x m [1,l] a l R. Thus there exst b l 1 R and c R such that x m [1,l] a b + xn l+m l 1 xn +m c, and so x m [1,l 1] a xn l l c (( l 1 Therefore, by 3.5, we have 1 xn +m R : x m l ( x m [1,l 1] a xn l l c Λ [1,l 1] 1 xn +m (( l 1 1 xn +m R : x l. Λ xn +m 1 h Λ, where h Λ ( Λ x R (x-unm for all Λ [1, l 1]. However, for each Λ [1, l 1] and j [1, l 1] \ Λ, we have (( Λ x R : x l ( Λ x R (x-unm (( Λ x R : x j by 3.1(, so that x [1,l 1] h Λ Λ x2 R. We now multply both sdes of equaton ( by x [1,l 1] to obtan that x m+1 [1,l 1] ( a l 1 1 xn 1 h [1,l 1] l 1 1 xn +m+1 R + x n l l R. Snce the natural mages of x 1,...,x l 1 n the rng R/x n l l R form an uncondtoned strong d-sequence n that rng, t follows from the nductve hypothess that a l 1 1 xn 1 h [1,l 1] l 1 1 l 1 1 (( j [1,l 1]\{} xn j 1 R j R + xn l l R : R x + x n l l R + x n 1 (( j [1,l 1]\{} xn j j R + xn l l R : R x + x n 1 1 R. (The presence of the deal x n 1 1 R on the rght hand sde ensures that the argument apples to the case where l 2. Snce ( l 1 1 x R (x-unm (( l 1 1 x R : x l, we see that l 1 1 xn 1 h [1,l 1] (( l 1 (even n the case where l 2 and so a l 1 j R : R x l (( j [1,l]\{} xn j j R : R x. Therefore j1 xn j ( l 1 xn R lm l 1 (( j [1,l]\{} xn j j R : R x l 1 ( j [1,l]\{} xn j j R (x-unm, and the reverse ncluson s easy. The followng corollary s mmedate from 3.5 and 3.6. Corollary 3.7. Suppose that x x 1,...,x l s an uncondtoned strong d-sequence n R (wth l 1 and let n 1, n 2,...,n l N + be any postve ntegers. Then ( x [1,l] ( l 1 xn R lm l ( ( l 1 xn R lm Λ [1,l] R; and 1 xn +1 ( Λ xn 1 ( Λ x R (x-unm when l 2. Theorem 3.8. Let x x 1, x 2,...,x l be a d-sequence n R. ( (C. Huneke [9, Proposton 2.1]. Then (( r 1 x R ( : x l r+1 1 x R r 1 x R for all r 0,...,l 1. ( If x x 1, x 2,...,x l s an uncondtoned strong d-sequence n R, then ( Λ xn R (x-unm ( l 1 xn R Λ xn R for all postve ntegers n 1, n 2,..., n l and each Λ [1, l].

12 FROBENIUS TEST EXPONENTS 11 Proof. ( Ths s mmedate from Huneke s result quoted n part (, because ( Λ xn R (x-unm (( Λ xn R (( : x j Λ xn R : x n j j for j [1, l] \ Λ: see 3.1(. As n [10], use wll be made of the followng extenson, due to G. Lyubeznk, of a result of R. Hartshorne and R. Speser. It shows that, when R s local and of prme characterstc p, a T-torson left R[T, f]-module whch s Artnan (that s, cofnte n the termnology of Hartshorne and Speser as an R-module exhbts a certan unformty of behavour. Theorem 3.9 (G. Lyubeznk [13, Proposton 4.4]. (Compare Hartshorne Speser [6, Proposton 1.11]. Suppose that (R, m s local and of prme characterstc p, and let G be a left R[T, f]-module whch s Artnan as an R-module. Then there exsts e N such that T e Γ T (G 0. Hartshorne and Speser frst proved ths result n the case where R s local and contans ts resdue feld whch s perfect. Lyubeznk appled hs theory of F-modules to obtan the result wthout restrcton on the local rng R of characterstc p. A short proof of the theorem, n the generalty acheved by Lyubeznk, s provded n [20]. Lemma 3.10 (Katzman Sharp [10, Lemma 3.5]. Suppose that R has prme characterstc p. ( Let n N + and let U be a trangular subset of R n. Then the module of generalzed fractons U n R has a structure as left R[T, f]-module wth ( r r p T (u 1,...,u n (u p 1,..., for all r R and (u 1,...,u n U. up n ( It follows easly that the complex C(U(x, R of modules of generalzed fractons of 2.6 s a complex of left R[T, f]-modules and R[T, f]-homomorphsms; hence all ts cohomology modules H (C(U(x, R ( N have natural structures as left R[T, f]-modules. 4. Preparatory results Most of the results n ths secton concern the case where R has prme characterstc p, but the frst two do not. Lemma 4.1. Suppose that (R, m s local, and consder the complex of modules of generalzed fractons C(U(x, R of 2.6. Let r be an nteger such that 0 r < l. (In the case where r 0, a generalzed fracton such as h/(x 1,...,x r (where h R s to be nterpreted smply as h. ( If h (( r 1 x R : x r+1, then h (x 1,..., x r Ker er, so that [ ] h (x 1,...,x r (The notaton [ ] s explaned n 2.6. H r (C(U(x, R.

13 12 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO ( Let n N + and h R. Then f and only f h ( r 1 xn R lm. h Im (x n 1,..., x n er 1 r Proof. When r 0, all the clams are easy. We therefore omt the proofs n that case and assume that 1 r < l. ( Snce x r+1 h r 1 x R, t s mmedate from 2.2 and 2.4(v that ( e r h h (x 1,...,x r (x 1,...,x r, 1 x r+1 h (x 1,..., x r, x r+1 0 U(x r 1 r+1 R. ( ( Assume that h ( r 1 xn R lm. Thus there exsts m N such that x m [1,r] h r 1 xn+m R. Thus we can wrte x m [1,r] h r 1 s x n+m for some s 1,..., s r R. Then, n U(x r r R, we have h (x n 1,...,x n r 1, x n r x m [1,r] h r (x n+m 1,..., x n+m r 1, x n+m r 1 s x n+m (x n+m 1,...,x n+m r 1, x n+m r s r x n+m r (x n+m 1,..., x n+m (on use of 2.4(v r 1, x n+m r s r (x n+m 1,..., x n+m r 1 {, 1 e r 1 (s r f r 1, ( e r 1 s r f r 2. (x n+m 1,...,x n+m r 1 ( By 2.2, we can wrte, n U(x r r ( R, h (x n 1,...,x n r g er 1 (x m+n 1, x m+n 2,...,x m+n for some m N and g R. Thus x m [1,r] h xm+n r g (x m+n 1, x m+n 2,...,x m+n r r 1 0 n U(x r r R. x m+n r g (x m+n 1, x m+n 2,...,x m+n r By the defnton of modules of generalzed fractons (see 2.1, there exst u N and H D r (R such that H[x m+n 1,...,x m+n r ] T [x m+n+u 1,...,x m+n+u r and det(h(x m [1,r] h xm+n r g r 1 R. Snce 1 xm+n+u dag(x u 1,...,x u r[x m+n 1,...,x m+n r ] T [x m+n+u 1,...,x m+n+u r ] T, we can use the method of 2.4( to see that x m+n+u [1,r] Consequently, x 2m+n+2u [1,r] h r 1 as requred. h (( r x u [1,r](x m [1,r]h x m+n r g r 1 1 x2m+2n+2u R. 1 x2m+2n+2u 1 x2m+2n+2u R : x 2m+n+2u [1,r] R + x 2m+2n+2u r ] T R; ths mples that ( r 1 xn R lm,

14 FROBENIUS TEST EXPONENTS 13 It s an mmedate consequence of Corollary 3.3 that a generalzed Cohen Macaulay local rng has a system of parameters that s an uncondtoned strong d-sequence. We now use modules of generalzed fractons to establsh a converse of ths. Theorem 4.2. Suppose that (R, m s local; recall that dm(r t. Then the followng statements are equvalent: ( R s generalzed Cohen Macaulay; ( there exsts h N + such that y k 1,...,yk t s an undcondtoned strong d- sequence, for every system of parameters y 1,..., y t of R and every k h; ( there exsts a system of parameters of R whch s an uncondtoned strong d-sequence. Proof. ( ( Ths s mmedate from Corollary 3.3 and the defnton of generalzed Cohen Macaulay local rng: just choose h so that m h t 1 0 Ann R(Hm(R. ( ( Ths s clear. ( ( Take l t and x 1,...,x t to be a system of parameters of R that s an uncondtoned strong d-sequence. By 3.1(, every permutaton of x 1,...,x t s an m-flter regular sequence. In order to show that R s generalzed Cohen Macaulay, t s enough, by symmetry, to show that x +1 Hm(R 0 for all 0,..., t 1. We now apply [24, Corollary 2.3 and Theorem 2.4] to the complex C(U(x, R of 2.6. Note that, for all N +, every element of U(x s an m-flter regular sequence. The cted results from [24] therefore show that H (C(U(x, R Ker e / Ime 1 H m (R for all 0,..., t 1. It s therefore enough for us to show that, for an {0,..., t 1}, we have x +1 Ker e Im e 1. We shall deal here wth the case where > 0, and leave to the reader the (easy modfcaton for the case where 0. Let α Ker e. By 2.2, we can wrte r α (x n 1,...,x n for some r R and n N +. Therefore x n +1 r (x n 1,...,x n, xn +1 r (x n 1,...,x n, 1 e (α 0. By 2.1 and 2.2, ths means that there exst v N + and H D +1 (R such that H[x n 1,...,x n +1] T [x n+v 1,...,x n+v +1 ]T and det(hx n +1r j1 xn+v j R. Snce dag(x v 1,...,xv +1 [xn 1,...,xn +1 ]T [x n+v 1,...,x n+v +1 ]T, we can use the technque of 2.4( to see that x n+2v [1,+1] xn +1 r j1 x2n+2v j R. Therefore, snce x 1,...,x t s an uncondtoned strong d-sequence, Hence x +1 r x n+2v [1,] r ( by Lemma 4.1(. ( j1 x2n+2v j ( R : x 2n+2v +1 j1 x2n+2v ( j R : x +1. lm, j1 x2n+2v j R : x n+2v [1,] j1 xn j R so that r x +1 α x +1 (x n 1,...,x n x +1 r Im (x n 1,...,x n e 1

15 14 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO It s well known that, when R has prme characterstc p, each local cohomology module Ha (R, where N, has a natural structure as a left R[T, f]-module. A detaled explanaton s gven n [10, 2.1], and the argument there can easly be modfed to show that, f M s an arbtrary left R[T, f]-module, then Ha (M (formed by regardng M as an R-module by restrcton of scalars nherts a natural structure as a left R[T, f]-module. However, n ths paper, we are gong to use the followng rather stronger statement. Proposton 4.3. Suppose that R has prme characterstc p. Then ( Ha s a N negatve strongly connected sequence of covarant functors from R[T,f] Mod to tself. Note. We dentfy H 0 a wth the a-torson functor Γ a n the natural way. If M s a left R[T, f]-module, then Γ a (M s an R[T, f]-submodule of M. It should be noted from the proof below that ths R[T, f]-module structure on Γ a (M s exactly the same as the natural left R[T, f]-module structure on H 0 a (M Γ a(m provded by the proposton. Proof. As ths proof reles on the Independence Theorem for local cohomology (see [2, 4.2.1], we shall use notaton smlar to that employed n [2, 4.2]. Let : Mod R Mod R denote the functor obtaned from restrcton of scalars usng the Frobenus homomorphsm f: thus, f Y s an R-module, then Y denotes Y consdered as an R-module va f. Let M be a left R[T, f]-module. The map τ M : M M defned by τ M (m Tm s an R-module homomorphsm. Consequently, for each N, there s an nduced R-homomorphsm H a (τ M : H a (M H a (M. Let Θ (θ N : ( H a ( N ( H a [p] ( N be the somorphsm of negatve (strongly connected sequences of covarant functors (from Mod R to Mod R that s nverse to the one gven n the Independence Theorem for local cohomology, n the form n whch t s stated n [2, 4.2.1]. Thus θ 0 s the dentty natural equvalence. Snce a and a [p] have the same radcal, Ha and H are a [p] the same functor, for each N. Consder the Z-endomorphsm ξm : θ M H a(τ M : Ha(M Ha(M. We now modfy the argument of [10, 2.1] and use [10, Lemma 1.3] to show that Ha (M has a natural structure as a left R[T, f]-module n whch Tγ ξm (γ for all γ H a (M. Furthermore, f β : M N s an R[T, f]-homomorphsm of left R[T, f]-modules, then M τ M M β β N τ N N

16 FROBENIUS TEST EXPONENTS 15 s a commutatve dagram of R-homomorphsms, so that, for N, the dagram θ M Ha (M H a(τ M H a (M Ha (M H a(β H a(β H a(β Ha (N H a (τ N H θ N a (N Ha (N also commutes. Ths means that, when Ha(M and Ha(N are gven ther natural structures as left R[T, f]-modules, as n the precedng paragraph, then the R- homomorphsm Ha (β : H a (M H a (N s an R[T, f]-homomorphsm. In ths way, Ha becomes a functor from R[T,f]Mod to tself. Next, whenever 0 L α M β N 0 s an exact sequence of left R[T, f]- modules and R[T, f]-homomorphsms, the dagram α 0 L M N 0 β τ L τ M τ N α β 0 L M N 0 of R-modules and R-homomorphsms commutes, and so the vertcal maps nduce a morphsm of the long exact sequence of local cohomology modules of the upper sequence nto that for the lower sequence. It follows from ths (and propertes of the somorphsm Θ of connected sequences that the connectng R-homomorphsms Ha (N H+1 a (L ( N are all homomorphsms of left R[T, f]-modules. Hence the long exact sequence of local cohomology R-modules nduced by 0 L α M β N 0 s actually a long exact sequence of left R[T, f]-modules and R[T, f]-homomorphsms. Everythng else needed for completon of the proof s now straghtforward. Our next result can be vewed as a strengthenng, n the partcular case where R has prme characterstc p, of specal cases of [24, Theorem (2.4] and of results of K. Khashyarmanesh, Sh. Salaran and H. Zaker n [12, Theorem 1.2 and Consequences 1.3(] (whch refer for proof to the proof of [24, Theorem (2.4]. Theorem 4.4. Suppose that (R, m s local and of prme characterstc p, and that x x 1,..., x l s an m-flter regular sequence of elements of m. Consder the complex C(U(x, R of modules of generalzed fractons of 2.6, and note that, by 3.10(, ths s a complex of left R[T, f]-modules and R[T, f]-homomorphsms. Then there are somorphsms of R[T, f]-modules H (C(U(x, R H m(r for all 0,...,l 1, where the Hm (R are consdered as left R[T, f]-modules n the natural way descrbed n 4.3. ( Proof. Frst, t follows from [21, 3.2] and [22, 2.2] that Hm j U(x R 0 for all 1,...,l and all j 0. Second, one can use 1.2( and the Exactness Theorem

17 16 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO for complexes of modules of generalzed fractons (see 2.5 (n conjuncton wth [5, Proposton 2.1] to see that Supp (H (C(U(x, R {m} for all 0. Wth these observatons, the theorem can be proved by an obvous modfcaton of the argument used to prove [24, Theorem (2.4], provded one notes that all the sequences 0 Ker e 0 R Im e 0 0, and 0 Kere U(x R Im e 0 (1 l, 0 Im e 1 U(x R Coker e 1 0 (1 l 0 Im e 1 Ker e Ker e / Im e 1 0 (1 l are exact sequences of left R[T, f]-modules and R[T, f]-homomorphsms, so that, by Proposton 4.3, all the somorphsms of local cohomology modules that they nduce are R[T, f]-somorphsms. Corollary 4.5. Suppose that (R, m s local and of prme characterstc p; recall that dm(r t. Snce the local cohomology modules Hm (R are left R[T, f]-modules that are Artnan as R-modules, t follows from the Hartshorne Speser Lyubeznk Theorem 3.9 that there exsts e 1 N such that T e 1 Γ T (Hm (R 0 for all 0,...,t 1. Set Q 1 : p e 1. Then Q 1 has the followng property: whenever x x 1, x 2,...,x t s a system of parameters of R that s also an m-flter regular sequence, and whenever r s an nteger wth 0 r < t and h ( r 1 x R (x-unm s such that h q ( r 1 xq R lm for some q, then h Q 1 ( r 1 xq 1 R lm. Proof. In the case where r 0, a generalzed fracton such as h/(x 1,...,x r (where h R s to be nterpreted smply as h. Let x x 1, x 2,...,x t, r and h be as n the statement of the corollary. Consder the complex of modules of generalzed fractons C(U(x, R of 2.6. Snce h ( r 1 x R (x-unm (( r 1 x R : x r+1 by 1.1(v, t follows from Lemma 4.1( that h (x 1,...,x r Kerer, so that [ h (x 1,...,x r Snce h q ( r 1 xq R lm, t follows from Lemma 4.1( that [ ] h Γ T (H r (C(U(x, R. (x 1,...,x r ] H r (C(U(x, R. Now H r (C(U(x, R Hm r (R as left R[T, f]-modules, by Theorem 4.4. Therefore [ ] [ ] h Q 1 (x Q 1 1,.. T e h h Q 1 1 0, so that.,xq 1 r (x 1,...,x r (x Q 1 1,...,xQ 1 r Im er 1. Therefore h Q 1 ( r 1 xq 1 R lm by Lemma 4.1(.

18 FROBENIUS TEST EXPONENTS 17 Proposton 4.6. Suppose that (R, m s local and of prme characterstc p; recall that dm(r t. Then there exsts Q 2 such that, for each system of parameters x x 1,..., x t of R, we have (( t 1 x R F [Q2 ] ( t 1 xq 2 R F ( t 1 xq 2 R lm. (Note that t 1 xq 2 R ( t 1 x R [Q 2 ]. Proof. Our ntenton s to apply the Hartshorne Speser Lyubeznk Theorem 3.9 to the top local cohomology module Hm(R t of R. Recall that Hm(R t can be realzed as the t-th cohomology module of the Cech complex of R wth respect to x 1,...,x t. Thus Hm t (R can be represented as the resdue class module of R x 1 x t modulo the mage, under the Cech dfferentaton map, of t 1 R x 1 x 1 x +1 x t. See [2, 5.1]. We use [ ] to denote natural mages of elements of R x1 x t n ths resdue class module. Recall also (from, for example, [10, 2.3] that the natural left R[T, f]-module structure on Hm(R t s such that [ T r (x 1 x t n ] [ r p (x 1 x t np ] for all r R and n N. Snce Hm t (R s an Artnan R-module, t follows from the Hartshorne Speser Lyubeznk Theorem 3.9 that there exsts e 2 N such that T e 2 Γ T (Hm t (R 0. Set Q 2 p e 2. Let a ( t 1 x R F, so that there exsts Q p e such that a Q ( t 1 x R [Q] t 1 xq R. Thus, n Ht m (R, we have [ ] [ ] [ ] [ ] T e a a Q a Q 2 0, so that T e a 2 0. x 1 x t (x 1 x t Q (x 1 x t Q 2 x 1 x t By [10, (2.3(], ths means that a Q 2 ( t 1 xq 2 R lm. It s easy to check that (( t 1 x R F [Q2 ] ( t 1 xq 2 R F, and so the proof s complete. Lemma 4.7. Suppose that (R, m s local and of prme characterstc p. If the deal a of R satsfes ((a R F [Q] (a R [Q], then (a F [Q] a [Q]. Proof. Let r a F ; then r (a R F n the rng R. Thus r Q (a R [Q] R (a [Q] R R, and the latter deal s just a [Q] because R s a fathfully flat extenson of R. Lemma 4.8. Suppose that R s of prme characterstc p. If the deals a and n of R satsfy n [Q ] 0 and (((a+n/n F [ Q] ((a+n/n [ Q] (n R/n, then (a F [Q Q] a [Q Q]. Proof. Let r a F ; then r + n ((a + n/n F n R/n. Thus (r + n Q ((a + n/n [ Q] ; that s, r Q a [ Q] + n. Consequently, r Q Q a [Q Q]. 5. The man results Throughout ths secton, we assume that R has prme characterstc p. Proposton 5.1. Suppose that R s of prme characterstc p. Assume that R s sem-local or that the ntegral closure of R/ 0 n ts total rng of fractons s modulefnte over R/ 0 (ths would be the case f R was excellent. Then there exsts Q 3 such that ((xr F [Q 3] (xr [Q 3] for all x R : R \ p mn(r p.

19 18 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO Proof. In case R s sem-local, as everythng nvolved commutes wth localzaton at the fntely many maxmal deals of R, we can assume that (R, m s local. Then, by Lemma 4.7, we can further assume that (R, m s complete and hence excellent. Thus, also by Lemma 4.8, we can assume that R s reduced and that R s modulefnte over R, where R s the ntegral closure of R n ts total fracton rng (R 1 R. Consder ( R R 1/q q1, whch forms an ascendng chan of R-submodules of R. As R s module-fnte over R, there exsts Q such that R R 1/q R R 1/Q for all q Q. For any x R and any y (xr F, there exsts q such that y q ax q for some a R. Ths means that (y/x q a/1 n (R 1 R, and ths mples that y/x R R 1/q. By our choce of Q, we get y/x R R 1/Q. Thus (y/x Q b/1 for some b R and hence y Q bx Q (xr [Q]. The next theorem s the man result of ths paper. Recall from Theorem 4.2 that a local rng s generalzed Cohen Macaulay f and only f t has a system of parameters that s an uncondtoned strong d-sequence. Theorem 5.2. Suppose that (R, m s a generalzed Cohen Macaulay local rng of prme characterstc p; recall that dm(r t > 0. Then there exsts Q such that (( j 1 x R F [Q] ( j 1 x R [Q] for all subsystems of parameters x 1,..., x j of R. Proof. In vew of Proposton 5.1, we can assume that t 2. In the frst part of the proof, we are gong to show that there exsts Q 0 such that (( t 1 x R F [Q0 ] ( t 1 x R [Q 0 ] for all systems of parameters x x 1,...,x t of R that are uncondtoned strong d-sequences. Let Q 1 be as n Corollary 4.5. Also, by Proposton 4.6, there exsts Q 2 such that (( t 1 x R F [Q2 ] ( t 1 xq 2 R F ( t 1 xq 2 R lm for all systems of parameters x x 1,...,x t of R. Set Q 0 pq 1 Q 2. We are gong to show that (( t 1 x R F [Q0 ] ( t 1 x R [Q 0 ] for all systems of parameters x x 1,...,x t of R that are uncondtoned strong d-sequences. Notce that (( t 1 x R F [pq1 Q 2 ] ((( t 1 x R F [Q2 ] [pq1 ] (( t 1 xq 2 R F ( t 1 xq 2 R lm [pq1 ] by Proposton 4.6. Therefore, t suffces to prove that (( t 1 xq 2 R F ( t 1 xq 2 R lm [pq1 ] ( t 1 xq 2 R [pq 1 ] for each system of parameters x x 1,...,x t such that x Q 2 1,...,xQ 2 t s an uncondtoned strong d-sequence, and so t s enough to prove that (( t 1 x R F ( t 1 x R lm [pq1 ] ( t 1 x R [pq 1 ] for all systems of parameters x x 1,..., x t of R that are uncondtoned strong d- sequences. We therefore fx a typcal such x x 1,...,x t. Notce that x s also an m-flter regular sequence n any order, by Remark 3.1(. Let y ( t 1 x R F ( t 1 x R lm. Then there exsts q pq such that y q ( t 1 x R [q ], and wthout loss of generalty we can assume that q max{p, Q1 }. We see, from Corollary 3.7(, that y p (( t 1 x R lm [p] ( t 1 xp R lm

20 Λ [1,t] xp 1 Λ FROBENIUS TEST EXPONENTS 19 ( Λ x R (x-unm. We can therefore wrte y p Λ [1,t] xp 1 Λ h Λ wth h Λ ( Λ x R (x-unm for all Λ [1, t]. Consequently, we have ( 0 Λ [1,t] xpq q Λ h q Λ ypq ( t 1 x R [pq], n whch h q Λ (( Λ x R (x-unm [q] ( Λ xq R (x-unm for all Λ [1, t] (n vew of 3.1(. The mmedate goal s to show that y pq 1 ( t 1 x R [pq 1 ]. To ths end, as y pq 1 Λ [1,t] xpq 1 Q 1, t s enough for us to prove that Λ h Q 1 Λ ( x pq 1 Q 1 Λ h Q 1 Λ Λ xpq 1 R for all Λ such that Λ [1, t]. We now prove ( by nducton on Λ, the cardnalty of Λ. When Λ 0, we have Λ ; we recall our conventons that xpq 1 R (0 and x 1. When we consder R as a left R[T, f]-module as n 1.3, the R-submodule Γ m (R s actually a T-torson R[T, f]-submodule. Snce ( x R (x-unm ( 0 : t 1 x R, we have h Γ m (R, so that h Q 1 0. Now suppose that 1 r < t, and assume that ( has been proved for Λ < r. That assumpton and ( 0 mean that ( r Λ [1,t], Λ r xpq q Λ h q Λ ( t 1 x R [pq]. To prove ( for Λ [1, t] wth Λ r, there s no loss of generalty n our assumng that Λ [1, r]. For every Λ [1, t] wth Λ r but Λ [1, r], we have x pq q Λ h q Λ R. Therefore, by ( r, we have t r+1 xpq q x pq q [1,r] hq [1,r] t 1 xpq R + t r+1 xpq q Snce h q [1,r] ( r 1 xq R (x-unm (( r ths mples that R r 1 xpq R + t r+1 xpq q R. 1 xq R : x t, t follows that x pq q [1,r] hq [1,r] x ( t x pq q r [1,r] 1 xq R r 1 xpq R; x pq q [1,r] hq [1,r] (( r 1 xpq R : x t ( r 1 xpq R (x-unm. Thus x pq q [1,r] hq [1,r] ( r 1 xpq R (x-unm ( r 1 xpq R + t r+1 xpq q R, and ths s equal to r 1 xpq R by Theorem 3.8(; therefore h q [1,r] ( r 1 xq R lm. Therefore h Q 1 [1,r] ( r 1 xq 1 R lm by 4.5, so that x pq 1 Q 1 [1,r] h Q 1 [1,r] r 1 xpq 1 R by Corollary 3.7(. Ths concludes the nductve step n the proof of ( and so t follows that y pq 1 ( t 1 x R [pq 1 ]. Ths s enough to complete the proof that (( t 1 x R F [Q0 ] ( t 1 x R [Q 0 ] for all systems of parameters x x 1,...,x t of R that are uncondtoned strong d-sequences. Now let h be the nteger of 4.2( and let Q 4 be a power of p wth Q 4 h. Also set Q Q 4 Q 0. Let y 1,...,y t be an arbtrary system of parameters of R. By Theorem 4.2, the system of parameters y Q 4 1,...,y Q 4 t s an uncondtoned strong d-sequence.

21 20 CRAIG HUNEKE, MORDECHAI KATZMAN, RODNEY Y. SHARP, AND YONGWEI YAO Therefore, by the frst part of the proof, (( t 1 y R F [Q4 Q 0 ] ((( t 1 y R F [Q4 ] [Q0 ] (( t 1 yq 4 R F [Q0 ] ( t 1 yq 4 R [Q 0 ] ( t 1 y R [Q 4 Q 0 ]. Thus we have shown that (( t 1 x R F [Q] ( t 1 x R [Q] for all systems of parameters x 1,...,x t of R. Fnally, let j {0,..., t 1}. For every n N +, we can apply what we have just proved to the system of parameters x 1,...x j, x n j+1,...,xn t. Thus (( j 1 x R F [Q] (( j n N + 1 x R + t j+1 xn R F [Q] n N + ( j 1 x R + t j+1 xn R [Q] n N + ( j 1 xq R + t j+1 xnq by Krull s Intersecton Theorem. Now the proof s complete. R ( j 1 x R [Q] Corollary 5.3. Suppose that (R, m s a formally catenary local rng all of whose formal fbres are Cohen Macaulay. (These hypotheses would be satsfed f R was an excellent local rng. Assume further that R s equdmensonal, of prme characterstc p and of dmenson 2. Then there exsts Q such that (( l 1 x R F [Q] ( l 1 x R [Q] for all subsystems of parameters x x 1,...,x l (where l 2, of course of R. Proof. The hypotheses about R are all nherted by R/ 0, and so, n vew of Lemma 4.8, we can assume that R s reduced. But then R s Cohen Macaulay on the punctured spectrum, and so s a generalzed Cohen Macaulay local rng by 3.4. The result now follows from Theorem 5.2(. References [1] H. Brenner, Bounds for test exponents, Composto Mathematca, to appear (arxv math.ac/ [2] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebrac ntroducton wth geometrc applcatons, Cambrdge Studes n Advanced Mathematcs 60, Cambrdge Unversty Press, MR (99h:13020 [3] S. Goto and T. Ogawa, A note on rngs wth fnte local cohomology, Tokyo J. Math. 6 (1983, MR (85j:13020 [4] S. Goto and K. Yamagsh, The theory of uncondtoned strong d-sequences and modules of fnte local cohomology, preprnt. [5] M. A. Hameh and R. Y. Sharp, Krull dmenson and generalzed fractons, Proc. Ednburgh Math. Soc. 28 (1985, MR (87g:13013 [6] R. Hartshorne and R. Speser, Local cohomologcal dmenson n characterstc p, Ann. of Math. 105 (1977, MR (56 #353 [7] M. Hochster and C. Huneke, F-regularty, test elements, and smooth base change, Transactons Amer. Math. Soc. 346 ( MR (95d:13007 [8] M. Hochster and C. Huneke, Localzaton and test exponents for tght closure, Mchgan Math. J. 48 ( MR (2002a:13001 [9] C. Huneke, The theory of d-sequences and powers of deals, Adv. n Math. 46 (1982, MR (84g:13021 [10] M. Katzman and R. Y. Sharp, Unform behavour of the Frobenus closures of deals generated by regular sequences, J. Algebra 295 ( MR

22 FROBENIUS TEST EXPONENTS 21 [11] T. Kawasak, On Macaulayfcaton of Noetheran schemes, Transactons Amer. Math. Soc. 352 (2000, MR (2000j:14077 [12] K. Khashyarmanesh, Sh. Salaran, H. Zaker, Characterzatons of flter regular sequences and uncondtoned strong d-sequences, Nagoya Math. J. 151 (1998, MR (99h:13030 [13] G. Lyubeznk, F-modules: applcatons to local cohomology and D-modules n characterstc p > 0, J. rene angew. Math. 491 (1997, MR (99c:13005 [14] H. Matsumura, Commutatve rng theory, Cambrdge Studes n Advanced Mathematcs 8, Cambrdge Unversty Press, MR (88h:13001 [15] L. O Carroll, On the generalzed fractons of Sharp and Zaker, J. London Math. Soc. (2 28 (1983, MR (85e:13025 [16] P. Schenzel, Enge Anwendungen der lokalen Dualtät und verallgemenerte Cohen Macaulay- Moduln, Math. Nachr. 69 (1975, MR (53 #2940 [17] P. Schenzel, Dualserende Komplexe n der lokalen Algebra und Buchsbaum-Rnge, Lecture Notes n Math. 907, Sprnger-Verlag, Berln, MR (83:13013 [18] P. Schenzel, N. V. Trung and N. T. Cuòng, Verallgemenerte Cohen Macaulay-Moduln, Math. Nachr. 85 (1978, MR (80:13008 [19] R. Y. Sharp, Tght closure test exponents for certan parameter deals, Mchgan Math. J., to appear (arxv math.ac/ [20] R. Y. Sharp, On the Hartshorne Speser Lyubeznk Theorem about Artnan modules wth a Frobenus acton, Proc. Amer. Math. Soc., to appear (arxv math.ac/ [21] R. Y. Sharp and H. Zaker, Modules of generalzed fractons, Mathematka 29 (1982, MR (84a:13008 [22] R. Y. Sharp and H. Zaker, Local cohomology and modules of generalzed fractons, Mathematka 29 (1982, MR (85b:13035 [23] R. Y. Sharp and H. Zaker, Modules of generalzed fractons and balanced bg Cohen-Macaulay modules, n: R. Y. Sharp (Ed., Commutatve Algebra: Durham 1981, London Mathematcal Socety Lecture Notes 72, Cambrdge Unversty Press, 1982, pp MR (84j:13021 [24] R. Y. Sharp and H. Zaker, Generalzed fractons, Buchsbaum modules and generalzed Cohen Macaulay modules, Math. Proc. Cambrdge Phlos. Soc. 98 (1985, MR (86m:13026 Department of Mathematcs, Unversty of Kansas, Lawrence, KS 66045, USA, Fax number: E-mal address: huneke@math.ku.edu Department of Pure Mathematcs, Unversty of Sheffeld, Hcks Buldng, Sheffeld S3 7RH, Unted Kngdom, Fax number: E-mal address: M.Katzman@sheffeld.ac.uk Department of Pure Mathematcs, Unversty of Sheffeld, Hcks Buldng, Sheffeld S3 7RH, Unted Kngdom, Fax number: E-mal address: R.Y.Sharp@sheffeld.ac.uk Department of Mathematcs, Unversty of Mchgan, Ann Arbor, MI 48109, USA, Fax number: E-mal address: ywyao@umch.edu