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The Notion of Tight Closure in Equal Characteristic Zero1 THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO by Melvin Hochster Introduction Our ob jective is to explain how the results of tight closure theory in characteristic p can b e extended to equal characteristic zero The results describ ed in this pap er and not otherwise attributed are joint work of the author and Craig Huneke The detailed treatment of the theory we sketch here is given in HH One do es in fact get over an arbitrary No etherian ring containing the rational numbers Q a closure op eration dened on submo dules of nitely generated mo dules The op eration has the same kind of p ersistence prop erties as in the characteristic p case For regular rings every ideal and every submo dule of every nitely generated mo dule is tightly closed The tight closure of an ideal is contained in the integral closure and is usually much smaller One has the same kind of coloncapturing prop erties as in characteristic p and more generally one has an analogous phantom homology theory In consequence one has a theory that yields equal characteristic versions of what has b een done in characteristic p one gets a very short pro of that direct summands of regular rings are CohenMacaulay and more they are Fregular improved versions of the socalled lo cal homological conjectures these conjectures are now theorems for the most part and a tight closure version of the BrianconSkoda theorem In HH several notions of tight closure in equal characteristic are developed There is one for each eld of equal characteristic valid for algebras over that eld and a notion big equational tight closure that yields a larger tight closure than any of the theories linked to a sp ecic eld However here for simplicity we shall fo cus on the notion asso ciated to the eld of rational numbers also called equational tight closure in HH and denoted eq Q there by either or Since we shall b e dealing with only one notion we denote it and we omit the adjective equational This is the smallest of our tight closure notions It is not known whether the various characteristic zero notions of tight closure actually yield diering results It is p ossible that they all agree whenever they are dened We should note that some of the denitions here are simpler than those in HH although equivalent in the sp ecial case that we are studying 1 This is an expanded and contracted version of two talks given by the author at the NSFCBMS Conference on Tight Closure Big CohenMacaulay Algebras and Uniform ArtinRees Theorems held at North Dakota State University June 2 The author was supp orted in part by a grant from the National Science Foundation 3 The author wishes to thank Craig Huneke b oth for his enormous contributions to the development of tight closure theory and for the monumental eort and care he put into preparation for this conference Typeset by A ST X M E MELVIN HOCHSTER We shall use HH and Hu as our basic references for characteristic p tight closure theory We conclude this section by recalling some of the alternative characterizations of tight closure available in characteristic p Before immersing ourselves in the technique of reduc tion to characteristic p we want to stress the p oint that a variant of one of these other characterizations may give a go o d theory in equal characteristic zero or even p ossibly in mixed characteristic Having a dierent approach to the equal characteristic zero case would undoubtedly b e very valuable Recall that for go o d rings of characteristic p and in particular for rings p ossessing a completely stable test element one may test tight closure by passing to the complete lo cal domains of R ie to the rings obtained by lo calizing R at a prime completing and killing a minimal prime It turns out that u I if and only if the image of u is in IB for every complete lo cal domain B of R Thus the central problem is to characterize tight closure in a complete lo cal domain R m K Part a of the Theorem b elow is phrased in terms of Hilb ertKunz functions e p when I is primary to m RI as a function of e is called the HilbertKunz function cf Ku Mo HaMo Here denotes length By a result of Mo if dim R d e q d d there is a p ositive real number such that for all q p RI q O q If I I dim R then is a p ositive integer In general is conjectured to b e rational but this I I is an op en question even in dimension two The b ehavior of the Hilb ertKunz functions P is quite surprising For example if R ZZx x x x G where G x i i e e e Cf HaMo then Rm Recall that the absolute integral closure of the domain R cf Ar which we denote R is the integral closure of R in an algebraic closure of its fraction eld and is unique up to nonunique isomorphism When R is a domain an Rmo dule M is called solid if it p ossesses a nonzero Rmo dule map to R When R m K is a complete lo cal domain of dimension d this is equivalent d M Cf Ho to the condition that the lo cal cohomology mo dule H m Theorem Let R m K be a complete local domain of characteristic p Let d dim R Let I be an ideal of R and let u R Each of the fol lowing statements some of which include a supplementary hypothesis gives a characterization of when u is in I a Suppose that I is mprimary Let J I Ru Then u I if and only if J I More generally if I J m are mprimary then J I if and only if J I This result characterizes tight closure in complete local domains of characteristic p since an element is in the tight closure of I m if and only if it is in the tight closure of al l mprimary ideals containing I b Fix a discrete Zvaluation nonnegative on R and positive on m and extend it to a valuation v of R to Q Then u I if and only if there exist elements R fg with v arbitrarily smal l such that u IR c K E Smith Sm Assume that I is generated by part of a system of parameters Then u I if and only if u IR d u I if and only if there exists a big CohenMacaulay algebra S for R such that u IS THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO e u I if and only if there exists a solid Ralgebra S such that u IS see the discussion just prior to the statement of this theorem Proof By Theorem of HH when I J are mprimary ideals we have that J I q q if and only if lim J I which by Monskys result is equivalent to the e!1 d q condition that yielding a For b see HH Part c is proved in Sm The I J results of d and e are obtained in Ho from the p ersp ective of solid closure Since R is a big CohenMacaulay algebra in the characteristic p case cf HH and since big CohenMacaulay algebras are solid conditions c d and e are closely related Whether tight closure in lo cally excellent domains of characteristic p is in general simply the contracted expansion from R is an imp ortant op en question It reduces to the case of complete lo cal domains The answer is not known even in dimension It is known however that R is never a big CohenMacaulay algebra in equal charac teristic if dim R Moreover the main result of Ro shows that solid closure the notion dened by attempting to extend the characterization of tight closure given in part e do es not b ehave well in equal characteristic ideals of regular rings of dimension fail to b e solidly closed in general But d might give a go o d notion in equal characteristic see x See also Ho and HH Comparison of b ers and a simple example It will turn out that if K is any eld of characteristic then in the ring R K X Y Z X Y Z K x y z the element z is in the tight closure of I x y R although it is not in I Before giving the explanation we want to recall that if A R is a ring homomorphism P is any prime ideal of A and P A P A then the ber of A R over P is dened as the P algebra P P P R If P is a maximal ideal of A then A and the b er is simply RR A b ers over maximal ideals are referred to as closed b ers On the other hand if A is a domain and P then P is the fraction eld of A and the b er over may b e identied with A R and is referred to as the generic b er When R is nitely generated over a No etherian domain A of characteristic zero and one is studying some go o d prop erty P of the b ers esp ecially a geometric prop erty it is often true that the generic b er has prop erty P if and only if there is a dense op en subset U of the maximal sp ectrum of A such that the closed b ers over the p oints of U have prop erty P we say that almost al l closed b ers have P in this case It is true that the generic b er has P if and only if almost all closed b ers have P for the following prop erties smo othness normality b eing reduced the CohenMacaulay prop erty the Gorenstein prop erty equidimensionality Let x x R let u R and I x x R It is also true that the image n n of u is nonzero or not in the expansion of I in the generic b er if and only if this is true for almost all closed b ers and that if x x generate a prop er ideal of height n MELVIN HOCHSTER n in the generic b er then the images of the xs generate a prop er ideal of height n in almost all closed b ers which will b e reduced and equidimensional but not necessarily domains Consider Zx y x y whether the b er mo dulo p is a domain dep ends on the residue of p mo dulo This kind of b ehavior can b e remedied by adjoining a square ro ot i of to Z after which one has nondomain b ehavior for all b ers over Z i There is a detailed and essentially selfcontained exp osition of results on comparison of b ers in HH x The idea b ehind our tight closure theory in equal
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