Journal of Algebra 323 (2010) Contents lists available at ScienceDirect. Journal of Algebra.

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1 Joural of Algebra 323 (2010) Cotets lsts avalable at SceceDrect Joural of Algebra Fotae rgs ad local cohomology Paul C. Roberts Uversty of Utah, Departmet of Mathematcs, Salt Lake Cty, UT, Uted States artcle fo abstract Artcle hstory: Receved 31 July 2009 Avalableole13February2010 Commucated by Kazuhko Kurao Keywords: Local cohomology Almost zero modules Cohe Macaulay algebras We use a costructo due to Fotae to costruct rgs wth almost vashg local cohomology from rgs of mxed characterstc ad dscuss the questo of usg ths method to costruct almost Cohe Macaulay algebras over the orgal rg. We also show that the exstece of almost Cohe Macaulay algebras mples the Moomal Cojecture ad gve a example to show how ths procedure ca be carred out a otrval case Elsever Ic. All rghts reserved. 1. Itroducto I ths paper we use Fotae rgs to study propertes of local cohomology for rgs of mxed characterstc. Fotae rgs, whch we defe ad dscuss more detal Secto 3, gve a method of costructg a rg of postve characterstc from oe of mxed characterstc such a way that, uder certa crcumstaces, the orgal rg ca be recostructed up to p-adc completo. The dea s to use the Frobeus map o the rg of postve characterstc ad the to deduce results o the orgal rg. We dscuss the extet to whch ths program ca be carred out ad problems that arse. Essetally, we beg wth a certa Noethera subrg of the Fotae rg of a rg R ad look for extesos T of ths rg satsfyg two propertes. The frst property s that a gve quotet of the rg of Wtt vectors o T s almost Cohe Macaulay. Almost Cohe Macaulay algebras are algebras for whch certa local cohomology groups are almost zero; these cocepts wll be defed Secto 2. We wll also expla ther coecto wth some of the homologcal cojectures. The secod property s that relatos that are mapped to zero R are also mapped to zero ths quotet. It s possble to costruct a exteso satsfyg the frst property, but ths paper we cocetrate o the secod oe ad costruct a mmal exteso T such that the requred relatos are mapped to zero. It s ot clear whether or ot ths rg s almost Cohe Macaulay. I the last secto we preset a example to show that, at least some otrval cases, the mage of the relevat local cohomology modules Ths research was supported part by a grat from the Natoal Scece Foudato. E-mal address: roberts@math.utah.edu /$ see frot matter 2010 Elsever Ic. All rghts reserved. do: /j.jalgebra

2 2258 P.C. Roberts / Joural of Algebra 323 (2010) of the orgal rg s almost zero the local cohomology of ths exteso, gvg some evdece that ths rg mght be almost Cohe Macaulay. 2. Almost vashg of local cohomology Let R 0 be a complete local Noethera doma of mxed characterstc. We let R + 0 be the absolute tegral closure of R 0 ; that s, R + 0 s the tegral closure of R 0 the algebrac closure of ts quotet feld. Throughout ths paper R wll deote a rg betwee R 0 ad R + 0. Although we do ot assume that R s Noethera, R s a uo of local Noethera domas that are tegral over R 0.Asaresult, we ca defe a system of parameters R to be a sequece of elemets x 1,...,x d of R that s a system of parameters a local Noethera subrg of R cotag R 0. Let R be a rg as above; we ext take a valuato v o R wth values the ordered abela group R of real umbers. The v s a fucto from R to R { } satsfyg (1) v(rs) = v(r) + v(s) for r, s R. (2) v(r + s) m{v(r), v(s)} for r, s R. (3) v(r) = f ad oly f r = 0. We wll assume also that v(r) 0forr R ad that v(r)>0forr the maxmal deal of R. The exstece of such a valuato follows from stadard facts o extesos of valuatos, see for example Zarsk ad Samuel [15, Chapter VI]. We say that a R-module M s almost zero wth respect to v f for all m M ad for all ɛ > 0, there exsts a r R wth v(r)<ɛ ad rm = 0. We wll also use aother verso of almost zero modules f c s a elemet of R, we wll say that M s almost zero for c f for every elemet m of M we have c 1/ m = 0 for arbtrarly large tegers. If v s a valuato o R ad v(c)<, ths codto mples that M s almost zero wth respect to v. Whle we wll be cocered ths paper wth the two classes of almost zero modules defed above, we wll also meto that more geeral deftos ca be used. Defto 1. AclassC of modules s a class of almost zero modules f t satsfes the followg two codtos. (1) If we have a short exact sequece 0 M M M 0, the M s C f ad oly f M ad M are C. (2) C s closed uder drect lmts. It s easy to see that both of the classes we have defed satsfy these codtos. Gve a class C satsfyg these codtos we say that a module M s almost zero wth respect to C f M C. Forthe remader of ths secto we wll assume that we have fxed such a class C ad almost zero wll mea wth respect to C. We ow recall some facts about local cohomology ad defe almost Cohe Macaulay rgs. Let x 1,...,x d be a system of parameters for R, ad let H(x) (R) deote the local cohomology of R wth support (x 1,...,x d ). More precsely, H(x) (R) s the cohomology degree of the Čech complex 0 R R x < j R x x j R x1 x 2 x d 0, where R has degree 0 ad R x1 x 2 x d has degree d.

3 P.C. Roberts / Joural of Algebra 323 (2010) Defto 2. A R-algebra A s almost Cohe Macaulay f (1) H(x) (A) s almost zero for = 0,...,d 1. (2) A/(x 1,...,x d )A s ot almost zero. A alteratve defto of almost Cohe Macaulay ca be obtaed by defg a sequece x 1,...,x d to be almost regular f {r rx (x 1,...,x 1 )}/(x 1,...,x 1 ) s almost zero for = 1,...,d ad defg R to be almost Cohe Macaulay f a system of parameters s almost regular (together wth codto (2)). Stadard methods show that ths defto mples the former oe (see for example Matsumura [10, Theorem 16.5()]). I ths paper we wll oly use the weaker property defed above. The mportace of the exstece of Cohe Macaulay algebras has bee kow for may years see for stace Hochster [7]. Recetly Hetma [5] has show that the weaker codto of havg a almost Cohe Macaulay algebra the sese defed here has some of the same cosequeces; partcular, he showed that the Drect Summad Cojecture dmeso 3 follows from the exstece of a almost Cohe Macaulay algebra. He also showed that R + 0 s a almost Cohe Macaulay algebra dmeso 3; he was usg a form of the secod defto, although dmeso 3 t does ot really matter whch oe you use. We ext show that ay dmeso the exstece of a almost Cohe Macaulay algebra the weaker sese we are usg here mples the Moomal Cojecture, whch s equvalet to the Drect Summad Cojecture (see for stace Hochster [6] for a dscusso of these cojectures). The Moomal Cojecture states that f x 1,...,x d sasystemofparametersfor alocalrgr 0,the for ay t 0. x t 1 xt 2 xt d / ( ) x t+1 1,...,x t+1 d Proposto 1. Let R 0 be a Noethera local rg of mxed characterstc, ad assume that there exsts a almost Cohe Macaulay algebra A over R 0. The the Moomal Cojecture holds for R 0. Proof. Let d be the dmeso of R 0. Mel Hochster [6] has prove that t suffces to prove the Moomal Cojecture for a system of parameters x 1,...,x d for whch x 1 = p the mxed characterstc case. We ca also assume that R 0 s complete. Usg these facts, we let Q be a regular subrg of R 0 cotag x 1 (= p), x 2,...,x d as a regular system of parameters. Let F be a mmal free resoluto of Q /(x t 1 xt 2 xt d, xt+1 1, x t+1 2,...,x t+1 ) over Q. Ths resoluto ca be chose so that F d 0 = Q, F 1 = Q d+1 wth the map d F 1 from F 1 to F 0 gve by the row of matrx (x t 1 xt 2 xt d xt+1 1 x t+1 2 x t+1 ) d ad F 2 = Q d+(d 2), where the map d F 2 from F 2 to F 1 s gve by the matrx x 1 x 2 x d 0 0 x 2 x d 0 0 Koszul 0 x 1 x 3 x d 0 relatos... o x t+1 1,...,x t+1. d 0 0 x 1 x d 1 Sce Q s a regular local rg of dmeso d ad Q /(x t 1 xt 2 xt d, xt+1 1,...,x t+1 ) has fte legth, d F s a complex of legth d. Let ths complex tesored wth A be deoted F A.SceF becomes exact whe ay of the x s verted, the same holds for F A.Let C = 0 R R x < j R x x j R x1 x 2 x d 0

4 2260 P.C. Roberts / Joural of Algebra 323 (2010) be the complex defg the local cohomology of R wth support (x 1,...,x d ). I ths complex we let R have degree 0 ad R x1 x d have degree d. We ext cosder the total tesor product complex F A C over R. I ths tesor product we gve F C j degree j. Lemma 1. (1) The complex F A C s quas-somorphc to F A. (2) The cohomology of F A C s almost zero postve degrees (ad hece the same holds for F A ). Proof. The two parts of the proof follow from the two spectral sequeces assocated wth the tesor product F A C. For the frst statemet, we cosder the spectral sequece obtaed by frst takg the homology of F A C for fxed. For = 0, C 0 = R, sof A C 0 = F A ad ths gves F A.For > 0, F A C s a product of localzatos of F A obtaed by vertg products of the x,sof A C s exact. Thus the projecto of F A C oto F A C 0 = F A s a quas-somorphsm, provg (1). For the secod statemet, we cosder the homology of F A geerated free module over A, F A C for fxed. SceF A s a ftely C s a fte drect sum of copes of A C, ad sce A s almost Cohe Macaulay, the homology of A C s almost zero degrees less tha d. To prove (2), t wll suffce to show that the homology of F A C s almost zero at F A C j wheever j > 0. The oly value of j for whch the homology of F A C j mght ot be almost zero s j = d. If j > 0, the > j, ad f j = d, thewemusthave > d. However, the complex F A has legth d, so ths meas that F A C j = 0. Ths completes the proof of the secod statemet. It follows from the lemma partcular that H 1 (F A ) s almost zero. We ow cosder the projecto of F A 1 = A d+1 oto the frst factor; ths gves a map to A. The mage of the kerel of d F 1 uder ths map s the set of a A for whch there s a -tuple a, a 1,...,a d wth ax t 1 xt 2 xt d + a 1x t a d x t+1 d = 0. I other words, the mage s {a A ax t 1 xt 2 xt d (xt+1 1,...,x t+1 )}. Wedeotethsdeala. Thus the d projecto of F A 1 oto A duces a map from Ker df 1 oto a. The mage of df 2 maps to the deal I = (x 1,...,x d ). Hece ths defes a map from H 1 (F A ) oto a/i. SceH 1 (F A ) s almost zero, t follows that a/i s almost zero. O the other had, oe of the codtos for A to be almost Cohe Macaulay s that A/(x 1,...,x d ) s ot almost zero. Hece 1 s ot a, sox t 1 xt 2 xt s ot (xt+1 d 1,...,x t+1 ), d ad the Moomal Cojecture holds. 3. Fotae rgs ad Wtt vectors Let Q be a rg of mxed characterstc p. TheFotae rg of Q,deotedE(Q), sdefedto be the verse lmt over of Q as rages over the ordered set of oegatve tegers, where each Q s Q /pq ad the map from Q +1 to Q s the Frobeus map for all. Ths costructo s due to Fotae [3] ad has bee exteded by Wteberger [14] ad Adreatta [1]. The otato ad termology that we use are take from Gabber ad Ramero [4, Secto 8.2] ( the most recet verso); however, we use the otato E(Q ) rather tha E(Q ) + to avod cofuso wth the absolute tegral closure. A elemet of E(Q ) ca be represeted by a sequece (q 0, q 1,...)= (q ), where each q s a elemet of Q take modulo pq ad q p q 1 modulo pq for > 0. E(Q ) s a perfect rg of characterstc p. Let R 0 be a complete local doma of mxed characterstc as above; we assume addto that the resdue feld k of R 0 s perfect. Let D 0 be a uramfed DVR cotaed R 0 wth the same resdue feld as R 0 such that there exsts a homomorphsm from a power seres rg S 0 = D 0 [[ y 2, y 3,...,y t ]] oto R 0. Such a homomorphsm exsts by the Cohe structure theorem (see for example Matsumura [10, Secto 29]). Let x be the mage of y for each, ad assume that the elemets are chose such a way that p, x 2,...,x d formasystemofparametersforr 0.

5 P.C. Roberts / Joural of Algebra 323 (2010) We let S = S 0[p 1/p, y 1/p 2,...,y 1/p t ], adjog p th roots of p ad the y for each. Smlarly, adjo p th roots of the elemets p ad x of R 0 to form a rg R = R 0[p 1/p, x 1/p 2,...,x 1/p t ]. We ca the exted the map from S 0 to R 0 to gve a surjectve homomorphsm from S to R. Sce pth roots are uque oly up to a root of uty, there s a choce adjog the roots; we choose the roots so that (p 1/p+1 ) p = p 1/p for each ad smlarly for the x ad y.wecachoosethe surjecto from S to R so that t takes p 1/p to p 1/p ad y 1/p to x 1/p for each ad. We ow take the Fotae rgs E(R) ad E(S). WeletP deote (p, p 1/p,...), Y = (y, y 1/p,...), ad X = (x, x 1/p,...) for our gve choces of p th roots. Next, we take the rgs of Wtt vectors W (E(S)) ad W (E(R)) of the rgs we have defed. We refer to Bourbak [2] ad Serre [13] for geeral facts about Wtt vectors ad Gabber ad Ramero [4] for coectos wth Fotae rgs. Let ˆR be the p-adc completo of R; that s, the verse lmt over of R/p R.Wehaveamap φ R, or smply φ, frome(r) to ˆR defed by lettg φ R ((r )) = lm r p. It s show [4] that ths sequece coverges. The map φ preserves multplcato but ot addto. We have φ(p) = p ad φ(x ) = x for each. Themap φ duces a rg homomorphsm φ from E(R) to R/pR, ad t exteds to a rg homomorphsm ψ from W (E(R)) to ˆR that also takes P to p ad X to x, where elemets e of E(R) are detfed wth the correspodg elemets (e, 0, 0,...) the rg of Wtt vectors. The reducto of ψ modulo p s φ. We refer aga to Gabber ad Ramero [4, Secto 8.2] for detals. As outled the troducto, the am of ths paper s to vestgate the possblty of usg ths costructo to costruct almost Cohe Macaulay algebras mxed characterstc. We frst revew the stuato for rgs of postve characterstc. 4. The case of postve characterstc We assume ow that R s a Noethera tegral doma of postve characterstc p ad defe the perfect closure of R, deotedr, to be the rg obtaed from R by adjog all p th roots of elemets of R. Alteratvely, R ca be defed as the drect lmt of R for 0, where R = R for all ad the map from R to R +1 s the Frobeus map. Theorem 1. Let R be a complete Noethera local doma of postve characterstc, ad let x 1,...,x d be a system of parameters for R. The there s a ozero elemet c R such that c 1/p η = 0 for all η H (x j ) (R ) for = 0,...,d 1 ad for all tegers 0. Proof. We wll use the fact that there s a ozero elemet c R whch ahlates the local cohomology H (x j )(R) for = 0,...,d 1; ths ca be foud Roberts [11] or Hochster ad Hueke [8, Secto 3]. We clam, fact, that ths elemet wll satsfy the statemet the theorem. To see ths, let be a teger wth 0 d 1, ad let η be a elemet of H (x j ) (R ).LetF deote the map duced by Frobeus map o H (x j ) (R ). Sce every elemet of R has ts p k th power R for large eough k, there s some k for whch F k (η) H (x )(R). Letm be a teger such that m + k. Thec 1/p s a multple of c 1/pm+k, so t suffces to show that c 1/pm+k η = 0. We have F m+k( c 1/pm+k η ) = cf m+k (η) = cf m( F k (η) ) = 0, sce F m (F k (η)) s H (x j ) (R). SceR s perfect, F s a somorphsm o R so duces a somorphsm o H (x j ) (R ). Hece the fact that F m+k (c 1/pm+k η) = 0 mples that c 1/pm+k η = 0.

6 2262 P.C. Roberts / Joural of Algebra 323 (2010) The above theorem mples that R s a almost Cohe Macaulay algebra for R ether of the seses troduced Secto 2. As outled the troducto, the am of ths paper s to attempt to apply ths kd of argumet to a rg costructed va the Fotae rg ad to deduce results mxed characterstc by usg the rg of Wtt vectors. I fact, for ay rg R 0 t appears that oe ca deed fd a almost Cohe Macaulay rg by ths method. The procedure s to frst fd a system of geerators {x } for R 0 for whch the Fotae rg E(R) of the rg R defed by adjog p th roots as above cotas a Noethera rg E 0 wth a deal J 0 wth the followg propertes. (1) P s a ozero-dvsor E 0 / J 0. (2) P, X 2,...,X d form a system of parameters for E 0 / J 0. If we the let E be the perfect closure of E 0 ad let J be the radcal of the deal geerated by J 0 E, the we ca defe a rg T ( J) by adjog all elemets of the form j/p to E for all j J ad for all tegers 0. The rg W (T (I))/(P p) s the a almost Cohe Macaulay rg. However, there s geeral o way of mappg R 0 to ths rg, so t does ot gve a almost Cohe Macaulay algebra for R 0, ad we do ot pursue ths dea further here. I the ext secto we descrbe a somewhat more complcated costructo for whch the requred map from R 0 does exst. I the fal secto we gve aotrvalexampler 0 for whch we ca verfy that the mage of the otrval local cohomology of R 0 s almost zero the local cohomology of ths algebra. However, we do ot kow whether ths holds geeral. 5. A algebra defed from the Fotae rg We ow retur to the stuato whch R s obtaed by adjog p th roots of certa elemets to a complete local doma of mxed characterstc R 0. More precsely, we have a power seres rg S 0 = V [[ y 2,...,y t ]] over a DVR wth perfect resdue feld k together wth a surjectve map from S 0 to R 0 that takes y to x for each. We assume also that p, x 2,...,x d s a system of parameters for R 0. R s the obtaed from R 0 by adjog p th roots of p ad the x, ad S s obtaed smlarly from S 0. We the take the Fotae rgs E(S) ad E(R) together wth ther rgs of Wtt vectors W (E(S)) ad W (E(R)). We would ow lke to pass to W (E(R))/(P p) ad show that t s a almost Cohe Macaulay algebra for R 0. However, there are two problems. Frst, E(R) s ot the perfect closure of a Noethera rg, so we caot drectly apply the method of the prevous secto. We address ths questo frst. The rg E(S) cotas a power seres rg P, Y 2,...,Y t over k ad we defe E 0 to be the subrg of E(R) whch s the mage of ths power seres rg uder the duced map from E(S) to E(R). TheE 0 s a Noethera rg of postve characterstc, ad we let E 0 deote ts perfect closure, asubrgofe(r). The secod problem s that the kerel of the map ψ from W (E(R)) to ˆR doesotgotozero W (E(R))/(P p), so that there s o map duced from R 0. Oe way aroud ths problem s to embed R to a larger rg C(R) ={s R p s p R for some }. It s show Roberts [12] that the kerel of the map from W (E(C(R))) to Ĉ(R) s geerated by P p. What we do here s fd a smaller rg T cotag E 0 for whch there exsts a map from R 0 to W (T )/(P p). We wll detfy E(R) as a subset of the rg W (E(R)) of Wtt vectors by assocatg e E(R) wth the Techmüller elemet (e, 0, 0...) of W (E(R)). Uder ths detfcato, let W 0 be the completo of the subrg of W (E(R)) geerated by P, X 2,...,X t over the dscrete valuato rg V.LetI 0 be the kerel of the map from W 0 to ˆR duced by ψ; weotethatw0 maps oto R 0 uder ths map. The f T s a rg cotag E 0 such that every elemet I 0 maps to (P p)w (T ) uder the cluso from W (E) to W (T ), wewllhaveaducedmapfromr 0 to W (T )/(P p)w (T ). Sce P p = P(1 p(1/p)) ad the rg of Wtt vectors W (E 0 ) s p-adcally complete, P p s autw ((E 0 ) P ), where (E 0 ) P deotes the rg obtaed from E 0 by vertg P.Heceforay elemet a = (a 0, a 1,...,a,...) of W 0 we ca fd a uque elemet z = (z 0, z 1,...) of W ((E 0 ) P )

7 P.C. Roberts / Joural of Algebra 323 (2010) such that a = (P p)z. WedefeT 0 to be the subrg of (E 0 ) P geerated by all such z for all elemets a that belog to the deal I 0, ad we defe T to be the perfect closure of T 0. We ow go to more detal as to how the rg T 0 ca be computed. Let a = (a 0, a 1,...,a,...) be a elemet of I 0.Theweeedtofdz = (z 0, z 1,...,z,...) such that a = (P p)z = Pz pz. Usg the stadard formulas for Wtt vectors (see for stace Bourbak [2]) we have Pz= ( Pz 0, P p z 1,...,P p z,... ) ad pz = ( 0, z p 0, zp 1,...,zp 1,...). Thus, aga usg the rules for computato the rg of Wtt vectors, to fd the z we must solve recursvely the followg formulas for z : P p z p 0 + ppp z p p P p z = a p 0 + pap p a + pz p 0 + p2 z p p z p 1. We remarked above that the z are (E 0 ) P ; fact, t s clear that whe we solve these equatos the values wll be (E 0 ) P. More precsely, t ca be show that z wll be of the form f (P, X )/P (+1)p, where f (P, X ) s E 0. We also ote that t s eough to take the rg geerated by the z where the (a ) ru over a set of geerators for I 0. To see ths we must show, for stace, that f a ad b are I 0 ad a/(p p) = (u ), b/(p p) = (v ), ad (a + b)/(p p) = (z ), the the z are the rg geerated over E 0 by the u ad v. Ths follows from the fact that (a + b)/(p p) = a/(p p) + b/(p p) ad that the etres a sum of Wtt vectors are polyomals the etres of the summads. Smlarly, oe shows that f a I 0 ad e E 0, ad f a/(p p) = (u ), e/(p p) = (v ), ad (ea)/(p p) = (z ), the the z are the rg geerated over E 0 by the u ad v. As stated above, we let T be the perfect closure of T 0. It s ot clear whether W (T )/(P p) s a almost Cohe Macaulay algebra or ot. We show ext that f T s ay exteso of E 0 such that the local cohomology of T s almost zero the sese descrbed below, the the local cohomology of W (T )/(P p) wll be almost zero as well. Theorem 2. Let T be a rg cotag E 0, ad suppose that P s ot a zero-dvsor T ad that there exsts a elemet c T such that c 1/p ahlates H (P,X 2,...,X d )(T ) for = 0,...,d 1 for all 0. Let c 1 be the mage of c W (T ) or W (T )/(P p). Thec 1/p 1 ahlates H (p,x 2,...,X d )(W (T )/(P p)) for = 0,...,d 1 for all 0. Proof. We ote frst that sce the map from T to W (T )/(P p) preserves multplcato, c 1/p wll map to a p th root of c 1. We wll use the term almost zero to descrbe the property of beg ahlated by c 1/p for all for T -modules or by c 1/p 1 for all for W (T )-modules. We frst use the log exact sequeces of local cohomology assocated to the short exact sequeces 0 W (T )/pw(t ) W (T )/p W (T ) W (T )/p 1 W (T ) 0 ad ducto to show that the local cohomology of W (T )/p W (T ) wth support (P, X 2,...,X d ) s almost zero degrees 0,...,d 1; the case = 1 s the hypothess. Note that local cohomology of W (T )/p W (T ) wth support (P, X 2,...,X d ) s the same as local cohomology wth support

8 2264 P.C. Roberts / Joural of Algebra 323 (2010) (p, P, X 2,...,X d ) sce W (T )/p W (T ) s ahlated by a power of p. We the cosder the log exact sequece assocated to 0 W (T ) p W (T ) W (T )/p W (T ) 0. Ths log exact sequece produces, for each, a exact sequece H 1 (p,p,x 2,...,X d )( W (T )/p W (T ) ) ( ) H p ) (p,p,x 2,...,X d ) W (T ) H (p,p,x 2,...,X d )( W (T ). From ths we deduce that the submodule of H (p,p,x 2,...,X d ) (W (T )) ahlated by p s almost zero for = 0,...,d. SceH (p,p,x 2,...,X d ) (W (T )) s the uo of these submodules, H (p,p,x 2,...,X d )(W (T )) s almost zero for = 0,...,d. Fally, sce P s ot a zero-dvsor T, P p s ot a zero-dvsor W (T ). Wetheusethe log exact sequece comg from the short exact sequece 0 W (T ) P p W (T ) W (T )/(P p) 0 to coclude that H (p,x 2,...,X d )(W (T )/(P p)) s almost zero for = 0,...,d 1. We ote that f we have a map from R 0 to W (T )/(P p) that takes x to X for all, the the local cohomology modules H (p,x 2,...,X d )(W (T )/(P p)) are the same as the local cohomology modules H (p,x 2,...,x d )(W (T )/(P p)). Thus the cocluso of Theorem 2 meas that W (T )/(P p) satsfes the frst codto of Defto 2 to be a almost Cohe Macaulay algebra for the rg R A example We ow gve a example of how ths costructo works practce. The example s a o-cohe Macaulay ormal doma R 0 of dmeso three, ad we show that the mage of the local cohomology of R 0 the algebra descrbed the prevous secto s almost zero. I would lke to thak Aurag Sgh for brgg ths example to my atteto ad explag may of ts propertes. Let p be a prme umber greater tha 3, ad let V 0 be a complete DVR wth maxmal deal geerated by p. LetR 0 be the power seres rg V 0 [[x, y, u, v, w]] modulo the deal geerated by, frst, the 2 by 2 mors of the matrx ( ) p x y, u v w ad, secod, the elemets p 3 + x 3 + y 3, p 2 u + x 2 v + y 2 w, pu 2 + xv 2 + yw 2, u 3 + v 3 + w 3. The rg R 0 has the followg propertes. (1) R 0 s a 3-dmesoal ormal doma. (2) A system of parameters for R 0 s p, v, x + u. (3) R 0 s ot Cohe Macaulay, ad ts local cohomology degree 2 s geerated by the elemet comg from the relato (x + u)(yw) = xyw + uyw = vy 2 + pw 2.

9 P.C. Roberts / Joural of Algebra 323 (2010) We wll ot prove these facts here, but we remark that the correspodg facts the aalogous stuato whch p s replaced by aother varable over a feld ca be deduced from the fact that the rg s a completo of a Segre product (see for example [9]) ad our case ca be deduced from that case. We ow compute what happes whe we adjo p th roots of the geerators of R 0. We frst adjo p th roots of p for each to form a (o-dscrete) valuato rg V.Letπ, x, y, u, v, w be elemets of R + 0 such that π p = π 1 for all ad smlarly for the other varables. By choosg v ad w approprately we ca esure also that the 2 by 2 mors of the matrx ( ) π x y u v w ( ) are zero. We clam that the oly other relatos o these elemets are the 3p + 1 polyomals π 3p + x 3p + y 3p, π 3p 1 u + x 3p 1 v + y 3p 1 w,..., u 3p + v 3p + w 3p. ( ) We frst ote that these elemets are zero; the frst oe s the frst of the orgal cubc relatos, ad the others ca be show to be zero usg the determatal relatos amog π, x, y, u, v, w. To see that they geerate the deal of relatos, t suffces to show that the deal geerated by the elemets ( ) ad ( ) the power seres rg V [[x, y, u, v, w ]] s prme, where V s a dscrete valuato rg wth maxmal deal geerated by π. Let U deote the quotet of the power seres rg by the determatal deal ( ) ad localze U by vertg oe of π, x, y, u, v, or w. The deal geerated by the above polyomals ths localzato s geerated by ether π 3p + x 3p + y 3p or u 3p + v 3p + w 3p. We assume that t s geer- ; the other case s smlar. The localzato of the determatal rg ated by π 3p + x 3p s regular, ad π 3p π 3p + x 3p + y 3p + y 3p + x 3p = y 3p s a product of dstct prme elemets. It follows that the elemet + (π 3p + x 3p ) s prme. Thus the deal s prme after localzato at ay of the sx geerators, ad to fsh the proof t suffces to kow that the depth s at least two, whch ca be carred out by reducto to the case of a Segre product as outled above. Thus the quotet obtaed by dvdg by these polyomals s a tegral doma ad s somorphc to the exteso obtaed by adjog the p th roots of the geerators of our rg. Ulke the case of R 0, ths exteso s ot ormal. We ow vestgate what happes whe we take the Fotae rg of R. We have elemets E(R) correspodg to p ad the varables R 0 that we wll deote, as above, by captals: P, X, Y, U, V, W. As the prevous secto, they geerate a Noethera subrg E 0 (up to completo), ad we let E 0 be ts perfect closure. We have relatos gve by the determats of the matrx ( P 1/p X 1/p Y 1/p ) U 1/p V 1/p W 1/p for each, ad we clam that these geerate the relatos amog these elemets E 0. Sce E 0 s the perfect closure of E 0, t suffces to show that the kerel of the map duced from k[[ P, X, Y, U, V, W ]] to E(R) s geerated by PV XU, PW YU, ad XW YV. Let f (P, X, Y, U, V, W ) be a elemet of ths kerel. If we represet f by ( f 0, f 1,...) E(R), the compoet f degree s gve by the same power seres f wth the coeffcets ad varables replaced by ther p th roots modulo p R. From the above descrpto of the relatos betwee these elemets R, we deduce that ths compoet s the deal geerated by π v x u, π w y u, ad x w y v ad the relatos ( ) modulo p. Therelatos( ) are cotaed the deal ge-

10 2266 P.C. Roberts / Joural of Algebra 323 (2010) erated by p, x, y, u, v, w, so f s the deal geerated by π v x u, π w y u, ad x w y v ad p, x, y, u, v, w. Wrte f = a (π v x u ) + b (π w y u ) + c (x w y v ) + d, where a, b, ad c are R ad d s the deal of R geerated by p, x, y, u, v, w. If we kew that the d were zero ad that the a satsfed a p = a 1 R/pR ad smlarly for the b ad c, we could coclude that f was the deal geerated by PV XU, PW YU,ad XW YV. By otg that f = f j j for all j ad usg the equato f j = a j (π j v j x j u j ) + b j (π j w j y j u j ) + c j (x j w j y j v j ) + d j we ca deduce that f s the deal geerated by π v x u, π w y u, ad x w y v modulo the deal geerated by the p j th power of (x, y, u, v, w) (modulo p) for all j, so f s the deal geerated by π v x u, π w y u, ad x w y v. Fally, ay relato betwee these three geerators of ths deal ca be lfted to a relato betwee the correspodg geerators degree + 1 (usg that the relatos are gve by the rows of ( )), so we ca adjust a, b, ad c step by step to make them compatble ad coclude that f s the deal geerated by PV XU, PW YU, ad XW YV. Thus the quotet has dmeso 4 ad s a determatal rg. The elemets P, V, X + U are ot part of a system of parameters; fact, they geerate a deal of heght 2. We ow let I 0 deote the kerel of the map from W 0 to ˆR as Secto 5. Let T be the perfect closure of the exteso of E 0 as defed there. There s a elemet of local cohomology of T 0 wth support (P, V, X + U) that maps to the geerator of the local cohomology of E 0 defed above; ths elemet s defed by the relato (X + U)(YW) = XYW + UYW = VY 2 + PW 2. We wsh to show that the mage of ths elemet the local cohomology of W (T )/(P p) s almost zero. To acheve ths we wll use Theorem 2 ad show that there s a ozero c T such that c 1/p ahlates the correspodg elemet the local cohomology of T. I fact, we wll show that t s ahlated by arbtrarly small powers of each of the geerators of the rg E 0. To see that ths elemet of local cohomology s ahlated by small powers of the geerators we eed to go back ad compute some relatos the rg T.WeotefrstthatI 0 s geerated by the elemets P 3 + X 3 + Y 3, P 2 U + X 2 V + Y 2 W, PU 2 + XV 2 + YW 2, ad U 3 + V 3 + W 3, all computed W (E(R)) (as well as P p). As stated the prevous secto, f a deotes oe of these elemets ad we let z be the elemet wth a = (P p)z, thewehave z = f (P, X )/P (+1)p, where f (P, X ) s E 0. We wll compute more precsely what the z look lke, but frst we prove a smple lemma o Wtt vectors over graded rgs. Lemma 2. Let A be a graded rg, ad let f (x t ) be a polyomal wth coeffcets Z of degree k wth etres A. Let τ (x t ) deote the elemet (x t, 0, 0,...)of W (A) for each t, ad let f (τ (x t )) = (a 0, a 1,...) W (A). The a s homogeeous of degree kp for each. Proof. We prove ths by ducto o. For = 0wehavethata 0 = f (x t ), whch has degree k = kp 0, so the lemma s true ths case.

11 P.C. Roberts / Joural of Algebra 323 (2010) Now let > 0, ad assume that the lemma holds for all j wth 0 j <. Wehave a p 0 + pap p j a p j j + + p a = ( ) f x p t. Sce f (x t ) s homogeeous of degree k, f (x p t ) s homogeeous of degree kp.also,foreach j <, a j s homogeeous of degree kp j by ducto, so a p j s homogeeous of degree kp.hecea j s also homogeeous of degree kp. I our example we use two gradgs, oe whch P, X, Y have degree 1 ad U, V, W have degree 0, ad oe whch P, X, Y have degree 0 ad U, V, W have degree 1. Ths gves a bdegree to each of the geerators of I 0. The lemma mples, for example, that whe PU 2 + XV 2 + YW 2 s expaded as the Wtt vector (a 0, a 1,...), a wll have degree p the frst gradg ad 2p the secod gradg. The ext lemma gves a descrpto of the quotet whe dvded by P p. Lemma 3. Let A be a graded rg as above, ad let (a 0, a 1,...) be a elemet of W (A) such that a s homogeeous of degree kp for each. Let (a 0, a 1,...)= (P p)(z 0, z 1,...). The for each 0 we ca wrte z = ap 0 + P j b j P (+1)p where each j s a postve teger ad b j s a homogeeous elemet of degree kp. Proof. Aga we prove ths by ducto o. We ca wrte the equato defg the z as (a 0, a 1, a 2,...)= ( Pz 0, P p z 1, P p2 z 2,... ) ( 0, z p 0, zp 1,...). Thus for = 0wehavez 0 = a 0 /P, ad sce P = P (0+1)p0, ths s the correct form (here all the other terms are zero). We ow assume that the result holds for j < ad prove that t holds for. The defg equato for z s a p 0 + pap p j a p j + + p a j = (Pz 0 ) p + ( ) p P p p 1 ( ) z p j P p j p j z j + + p ( ) P p z p ( z p 0) p 1 p j ( z p j 1 ) p j p ( z p 1). Hece z s a combato of the other terms the above expresso dvded by p P p. The factor p wll dvde the other terms ths expresso after the formulas for the z j for j < are substtuted from the geeral theory of Wtt vectors; the factor we have to cosder s P p. Thus to complete the proof we must show that each term the above equato other tha p (P p z ) s a sum of terms that ca be wrtte the form P a/p p wth a homogeeous of degree kp ad that the oly term for whch = 0sa p 0. Each of the terms p j a p j s homogeeous of degree kp ad we ca take = p,sotheseterms j clearly satsfy the requred codto.

12 2268 P.C. Roberts / Joural of Algebra 323 (2010) We ext cosder a elemet of the form p j (P p j z j ) p j.byducto,z j s a sum of terms P k m b m dvded by P ( j+1)p j wth b m homogeeous of degree kp j ad exactly oe k m = 0, for whch b m = a p j 0. Whe ths sum s multpled by P p j ad rased to the p j th power we obta a sum of teger multples of terms of the form (P p j ) p j P r k m b r m (P ( j+1)p j ) p j. ( ) I ths product the sum of the r s p j,thek m are postve except for oe term (comg from a p j 0 ) whch we compute below, ad the b m are homogeeous of degree kp j. It follows that the product of the b r m s homogeeous of degree ( r )kp j = p j kp j = kp.deotgthsproductb, ad lettg k = r k m, we ca wrte ths term the form P p +k b P ( j+1)p. Sce the deomator of ths term s (P ( j+1)p j ) p j = P ( j+1)p,f j < 1, the ay term of ths form ca be wrtte the desred form wth postve power of P.If j =, there s oly oe term where the power s zero. That term has to come from a product of b m expresso ( ) whch every k m s zero, ad the oly such term s (a p 0 )p = a p+1 0.Hecez has the stated form, so ths completes the proof. We wll ow show that for η equal to each of the varables P, X, Y, U, V, W ad for ay 1, we have η 1/p YW (P, V )T. Choose such a, ad let m be a postve teger such that 4/p m < 1/p. Let a be oe of the four geerators of I 0 as above, ad let (z ) be the Wtt vector a/(p p). We cosder z for = p k 1. By Lemma 3 we kow that z s the quotet whose umerator s a polyomal P wth costat term a p 0 ad coeffcets homogeeous of the degree of a 0 tmes p ad whose deomator s P (+1)p = P pk p = P pk+.weowtakethep k+ th root of ths elemet. It s ow of the form β/p, where β s a polyomal P wth fractoal expoets wth costat term a 1/pk 0.The coeffcets are homogeeous of the degree of a 0 dvded by p k.wehaveβ = P(β/P) (P, V )T,so β s a multple of P T so s the deal (P, V ). Thus we are reduced to showg that f we set the elemets descrbed the prevous paragraph to zero for each of the four geerators of I 0 ad for k ragg betwee 0 ad m, we ca show that η 1/p YW s the deal (P, V ). To llustrate the method, we outle oe step detal. Lettg k = 1, the procedure above appled to the elemet PU 2 + XV 2 + YW 2 gves a elemet Y 1/p W 2/p ( P 1/p U 2/p X 1/p V 2/p + P a γ ), where a s a postve ratoal umber ad γ s a polyomal each of whose terms has degree P, X, Y at least 1/p ad degree U, V, W at least 2/p. Thusfwe substtute the expresso paretheses for Y 1/p W 2/p we wll decrease the total degree of η 1/p YW Y ad W ad add terms that are multples of P a. We wll show that f the degree Y ad W of a moomal of the same bdegree as η 1/p YW s small eough, the the moomal s (P, V ). Thus ths process wll evetually crease the (ratoal) power of P that dvdes our elemet, so that t wll evetually be a multple of P. It s mportat that we are fxg a boud m o the expoets that we use, so that ths s fact a fte process ad wll evetually termate. Let m = P a U b X c V d Y e W f be a moomal satsfyg the equaltes o the degrees satsfed by η 1/p YW; that s, the degrees P, X, Y ad U, V, W are at least 1 ad the total degree s at least 2 + 1/p.Ifa + b 1, sce a + c + e 1, we ca use the relatos X 1/pk U 1/pk = P 1/pk V 1/pk ad Y 1/pk U 1/pk = P 1/pk W 1/pk to rase the expoet of P to 1, so the term s a multple of P. Smlarly, f c + d 1 we ca show that the term s a multple of V. If both are less tha oe, the e + f > 1/p, sce the total degree s at least 2 + 1/p.Sce4/p m < 1/p, we ca fd oegatve tegers ad

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