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
de ned 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 colon capturing 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 Cohen Macaulay and more
they are F regular improved versions of the so called lo cal homological conjectures
these conjectures are now theorems for the most part and a tight closure version of the
Brian con Skoda 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 eci c 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 di ering results It is p ossible that they all agree whenever they are de ned We
should note that some of the de nitions 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 NSF CBMS
Conference on Tight Closure Big Cohen Macaulay Algebras and Uniform Artin Rees 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 e ort and care he put into preparation for this conference
Typeset by A S T 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 di erent 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 i e 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 ert Kunz functions
e
p
when I is primary to m R I as a function of e is called the Hilbert Kunz 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 R I 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 ert Kunz functions
P
is quite surprising For example if R Z Z x x x x G where G x
i
i
e
e e
Cf HaMo then R m
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 non unique isomorphism
When R is a domain an R mo dule M is called solid if it p ossesses a nonzero R mo 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 m primary Let J I Ru Then u I if and only if
J I
More generally if I J m are m primary 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 m primary ideals containing I
b Fix a discrete Z valuation 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 f g
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 Cohen Macaulay 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 R algebra 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 m primary ideals we have that J I
q q
if and only if lim J I which by Monsky s 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 Cohen Macaulay algebra in the characteristic p case cf HH and
since big Cohen Macaulay 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 Cohen Macaulay algebra in equal charac
teristic if dim R Moreover the main result of Ro shows that solid closure the
notion de ned 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 de ned as the P algebra
P P
P R If P is a maximal ideal of A then A and the b er is simply R R
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
identi ed 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 Cohen Macaulay 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 x s 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 Z x 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 non domain b ehavior for all b ers over Z i There
is a detailed and essentially self contained exp osition of results on comparison of b ers in
HH x
The idea b ehind our tight closure theory in equal characteristic zero is to formulate the
de nition so that comparison of b ers works a priori by virtue of the de nition
In the example that we are studying it should turn out that z is in the tight closure
of x y R b ecause roughly sp eaking when one drops down or descends to the nitely
generated Z subalgebra of R call it R generated by x y and z over Z here R
Z Z
Z X Y Z X Y Z then for almost every closed b er of Z R i e mo dulo
Z
all but nitely many p ositive prime integers p the image of z mo dulo p is in the tight
closure of IR pR in R pR Fix p The rest of this parenthetical remark is
Z Z Z Z
carried through entirely mo dulo p but we continue to write x y and z for the images of
q q q q e
the variables It is enough to see that x z I x y for all q p Let Z pZ
and let the subscript indicate the result of tensoring with We can write q h r
where h is a nonnegative integer and r Note that z z is a free basis for R
over x y T so that every element has a unique representation as z z with
q h r h r q q
the T Then x z b ecomes x z z x x y z and this will b e in x y
i
provided that for all choices of nonnegative i j with i j h we have that at least one of
i j
the exp onents in x y is at least q But if b oth exp onents are at most q we have
that i j q or h q h r contradicting r
We leave it as an exercise for the reader to verify that z is not in the tight closure of
x y R and that z is not in the integral closure of x y R Also show that the ideal
generated by x and y is tightly closed in the ring Z pZ X Y Z X Y Z precisely
for those primes p such that p
A general de nition for tight closure Let S b e a lo cally excellent No etherian
ring containing the rationals Keeping the example of the preceding section in mind we
want to give the general de nition of when an element u of S is in the tight closure of an
ideal J of S
De nition The element u is in the tight closure J of J if there exists a nitely
generated Z subalgebra R of S containing u such that with I J R one has that
Z Z Z
for all but at most nitely many closed b ers R of Z R the image u of u in R is
Z
in the characteristic p tight closure of the image of I in R
We should make several remarks here Once one has a choice of subalgebra R that
Z
enables one to see that u is in the tight closure of J every larger choice works as well this
is a consequence of p ersistence prop erties for tight closure in characteristic p Thus in the
ab ove de nition instead of saying that there exists a nitely generated Z subalgebra
we could have said for every su ciently large nitely generated Z subalgebra For
THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
simplicity we have given the de nition only for ideals But there is no di culty extending
the de nition to the case of mo dules As in the p ositive characteristic case the de nitions
are set up so that an element u of the No etherian mo dule M is in the tight closure of
N M if and only if the image of u in M N is in the tight closure of in M N If we
h
map a nitely generated free R mo dule G R onto M and replace N by its inverse image
in G and u by an element of G that maps to u then we reduce the problem of de ning
h
tight closure to the case where the ambient mo dule is G R The de nition in this case
is essentially the same as in the case of ideals
It is also worth noting in dealing with nitely generated Z algebras in this context that
after lo calizing at one nonzero integer a one may assume that all of the mo dules algebras
etc that one is considering are free over the lo calized base A Z This is a sp ecial case
a
of the lemma of generic freeness Even when one has a cokernel of a map of Z algebras
it is p ossible to lo calize and assume that cokernel is free This means that for almost all
closed b ers an injection of nitely generated Z algebras stays injective when one passes
to the b er i e works mo dulo the corresp onding prime integer p The following strong
form of generic freeness developed in HR Lemma p su ces for our needs
here
Lemma generic freeness Let A be a Noetherian domain let R be a nitely
generated A algebra let S be a nitely generated R algebra let W be a nitely generated
S module let M be a nitely generated R submodule of W and let N be a nitely generated
A submodule of W Let V W M N Then there exists an element a A f g such
that V is free over A
a a
The very imp ortant sp ecial case where R S and M N are b oth has b een known
much longer this case may b e found in Mat x In most cases where one uses generic
freeness the atness of the mo dule algebra etc under consideration would su ce
Note that for example since one may lo calize to make R I at it follows that
Z Z
I R is an injection for almost all b ers
i to indicate the image of a mo dule under Basic prop erties We shall use h
a map which map should b e obvious from the context The most frequent use of this
0 0
notation is when N M are S mo dules S is an S algebra and hS N i denotes the
S
0 0
image of S N in S M
S S
For the intrepid reader we comment brie y on the other notions of tight closure in the ideal case If
K
one is considering algebras over a eld K of characteristic in the de nition of one replaces R by
Z
an a ne Z algebra R R where A is a suitably large a ne Z subalgebra of K and then one considers
A
what happ ens for almost all closed b ers of A R The largest notion big equational tight closure
A
EQ
denoted is governed by what happ ens in rings of the form C that have lo cal maps to the complete
Q
lo cal domains of R where C is an a ne Z algebra and Q is a prime ideal meeting Zin For every prime
integer p let B p b e the lo calization of C pC at the multiplicative system of all nonzero divisors on
C Q pC For a ring of the form C an element u C and an ideal I C one de nes u to b e in
Q Q Q
EQ
I if for almost all prime integers p the image of u in B p is in the characteristic p tight closure of
K L
IB p in B p If R is an L algebra for some eld L and K L is a sub eld then I I I which
Q EQ K
is I is contained in I and if R is a K algebra then I is b etween them Mo dules are treated in an entirely similar way
MELVIN HOCHSTER
The following result states many of the basic prop erties of tight closure in equal char
acteristic zero The pro of is given in x of HH The idea is generally to pass to
consideration of one or more nitely generated Z algebras and then deduce the result
from a corresp onding result in characteristic p by considering what happ ens for almost all
b ers
Theorem basic prop erties of tight closure in characteristic zero Let S be
0
a locally excellent Noetherian algebra containing Q Let N N M be nitely generated
S modules Let u M and let v be the image of u in M N Let I be an ideal of S Unless
otherwise indicated indicates tight closure in M
a N is a submodule of M containing N
b u N if and only if v
M
M N
0 0
0
c If N N M then N N and N N
M N M
M
d N N
0 0
e N N N N
0 0
f N N N N
g IN I N
R M
M
M
0 0
h N I N I respectively N N N N Hence if N N then
M M M S S
0 0
N I N I respectively N N N N
M M S S
i If N M are nitely many nitely generated S modules and we identify N N
i i i i
with its image in M M then the obvious injection N M maps N
i i i M i M
i i
i i
isomorphically onto N
M
0
j Persistence of tight closure Let S S be a ring homomorphism Let u N
M
0 0
Then u hS N i over S
0
S
S M
S
0
k Persistence of tight closure second version Let S S be as in j let u N
M
0
and let V W be nitely generated S modules Suppose also that there is an S
homomorphism M W such that N V Then u V
W
l Irrelevance of nilpotents If J is the nilradical of S then J and so J I
N denotes the image of for al l ideals I of S Consequently JM N Moreover if
N which N in M J M then N is the inverse image in M of the tight closure
M J M
may be computed either over S or over S S J
red
s i i i
m Let p p be the minimal primes of S and let S R p Let M
i i i i i
S M and let N be the image of S N in M Let u be the image of u
S S
i i i i i
in M Then u N if and only if u N in M over S i s
h
n If R R is a nite product and M M and N N are the corresponding
i i i i i
i
product decompositions of M N respectively then u u u M is in N
h M
over R if and only if for al l i i h u N
i i
M
i
The next result which is essentially Theorem of HH shows that the notion of
characteristic zero tight closure we are using can b e tested after passing to the complete
lo cal domains of R Some discussion of the ideas in the pro of is given following the
statement of the theorem
Theorem Let S be a locally excellent Noetherian algebra containing Q and let
N M be nitely generated S modules Then the fol lowing three conditions on an element
u M are equivalent
THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
u N
M
For every maximal ideal m of S if B is the quotient of the completion of S by a
m
minimal prime then the image of u in B M is in the tight closure of hB N i
S S
working over B
For every prime ideal P of S if B is the quotient of the completion of S by a minimal
P
prime then the image of u in B M is in the tight closure of hB N i working
S S
over B
The pro of of this theorem dep ends on the fact that the Henselization of an excellent
lo cal ring of equal characteristic is an approximation ring i e that whenever a nite
system of p olynomial equations over the ring has a solution in the completion it has a
solution in the Henselization and hence in a suitable etale extension of the original ring
This result which generalizes Ar can b e deduced from general N erondesingularization
which asserts that an arbitrary regular meaning at with geometrically regular b ers
homomorphism of No etherian rings is a ltered inductive limit of smo oth homomorphisms
i e smo oth of nite type This means that if R S is regular and factors R R S
with R of nite type over R then R S factors R S S where S is smo oth
over R General N eron desingularization has a somewhat complicated history and there
has b een some disagreement concerning the correctness of what has b een published We
refer the reader to Po Og And and Sp for further discussion The sp ecial cases
that we need here have b een established indep endently in Rot and ArR In particular
a self contained argument that excellent Henselian rings of equal characteristic zero are
approximation rings is given in Rot
To prove the most interesting implication in Theorem namely that implies
one uses the approximation result for the lo calized completion at every maximal ideal and
then a kind of compactness argument to show that the element is in the tight closure after
descending to a faithfully at etaleextension of a nitely generated over Q subalgebra of
the original ring Characteristic p results then imply that the faithfully at etaleextension
is unnecessary
Remark The de nition of tight closure given in HH for the case where the ring is not
necessarily lo cally excellent forces the conclusion of Theorem to hold e g in the
ideal case an element u of the ring is de ned to b e in the tight closure of I if it is in the
tight closure in the sense that we have already de ned of IB for every complete lo cal
domain B of R
Descent from the complete case and the Artin Rotthaus theorem There
is a general strategy for proving results ab out tight closure one rst passes to the complete
lo cal case often the case of a complete lo cal domain In the second step one passes to
the study of the nitely generated Q and then Z subalgebras of that complete lo cal
domain The third step is to pass to closed b ers using results comparing what happ ens
at the generic b er with what happ ens for almost all closed b ers
To prove results on capturing colons and phantom homology which we shall treat in
the next two sections one needs at the second step to pass to an a ne Z subalgebra of
MELVIN HOCHSTER
a complete lo cal ring while preserving heights The following result which is essentially
Theorem of HH p ermits one to do this
Theorem Let K be a eld of characteristic zero and let S m L be a complete
local ring that is a K algebra Assume that S is equidimensional and unmixed
Suppose that R is a subring of S that is nitely generated as a K algebra and that we
i
are also given nitely many sequences of elements fz g in R each of which is part of a
t
system of parameters for S
Then there is a nitely generated K algebra R such that the homomorphism R S
factors R R S and such that the fol lowing conditions are satis ed
R is biequidimensional
i
The image of each sequence fz g in R consists of parameters in the fol lowing sense
t
t
each subsequence consisting of say elements generates an ideal that has height
modulo every minimal prime
If m is the contraction of m to R then dim R depth R dim S depth S In
m m
particular R is Cohen Macaulay i S is Cohen Macaulay
m
If S is reduced respectively a domain then so is R
N B In general dim R is substantial ly bigger than dim S
m
Of course K may b e Q Having passed to a nitely generated Q subalgebra it is then
easy to descend further to a nitely generated Z subalgebra
We shall not say very much ab out the pro of of Theorem except that one thinks of
the ring S as mo dule nite over a complete regular lo cal ring and reduces the problem of
descending from S to that of descending from the complete regular lo cal ring The pro of
is then completed by using the following result which is another sp ecial case of general
N erondesingularization A self contained pro of is given by Artin and Rotthaus in ArR
Theorem Let K be a eld or an excellent DVR Let T K x x be the
n
formal power series ring in n variables over K Then every K algebra homomorphism of
a nitely generated K algebra R to T factors R S T where the maps are K algebra
homomorphisms and S is smooth of nite type over K x x Thus S is regular
n
and x x is a regular sequence in S
n
Further prop erties of tight closure We are now ready to establish a number
of imp ortant prop erties of characteristic tight closure that parallel the prop erties of tight
closure in characteristic p
Theorem Let S be a regular Noetherian ring containing Q and let N M be
nitely generated S modules Then N N
M
Sketch of the proof One reduces to the case where S m L is a complete lo cal domain
If u N N one can preserve this while replacing N by a submo dule maximal with
resp ect to b eing disjoint from u Passing to M N we may assume that N that M
is an essential extension of L and the the so cle generator u is in the tight closure of
Then the injective hull of M is the same as the injective hull of the residue eld and so
THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
if x x is a regular system of parameters for S we see that we may assume that M
n
t t
embeds in S x x S for t su ciently large Hence we may take u to b e the image
n
t
of x x since this element generates the so cle Thus it will su ce to show that
n
t t t
u x x is not in the tight closure of x x S in S
n
n
By Theorem if there is an a ne Q algebra containing elements X that map to
i
t
t t t
the x and such that X X is in the tight closure of X X there is also
i
n n
one that is smo oth over Q and in which the X s form a regular sequence One can descend
to a Z subalgebra where a is a nonzero integer One now gets a contradiction from the
a
fact that the closed b ers are all regular rings
Let N M b e nitely generated mo dules over a No etherian ring S We shall say that
reg
u M is in the regular closure N of N in M if for every regular ring T to which S
M
maps u hN i in M This is slightly di erent from the notion considered in HH
T T T
and HH where it was required that S map into T where the sup erscript indicates
the set of elements of the ring not in any minimal prime This regular closure is a priori
smaller than the one considered in HH and HH although we do not know an example
where it is actually strictly smaller This makes the following Corollary slightly stronger
than if it were stated for the notion of HH and HH
Corollary Let S be a Noetherian ring containing Q Let N M be nitely
reg
generated S modules Then N N
M
M
and supp ose that S maps to a regular No etherian ring T By the Proof Let u N
M
p ersistence of tight closure after tensoring with T one has that the image of u is in
reg
hT N i hT N i since T is regular Thus u N
S S M
T M
S
Corollary Let S be a Noetherian ring containing Q Let I be any ideal of S Then
I I the integral closure of I Hence al l radical ideals of S and in particular al l prime
ideals of S are tightly closed
Proof An element is in I if and only if it is in IV for all maps of R to discrete valuation
reg
rings V which shows that I I and we may apply
We refer to HH and Hu for the background of the Brian con Skoda theorem
Theorem tight closure Brian con Skoda theorem Let S be a Noetherian
ring of equal characteristic zero and let I be an ideal of S generated by at most n elements
k
n k n
I I In particular I I Then for every k N
Proof Fix generators of I say I u u It is clear that if an element z is in
n
n k
u u then this remains true when S is replaced by a suitable a ne Z subalgebra
n
containing z and u u The equation that demonstrates integral dep endence here
n
will continue to do so when we pass to closed b ers The result is now immediate from
the de nition of tight closure and the fact that the tight closure Brian con Skoda theorem
holds for all the closed b ers The nal statement is the case where k
MELVIN HOCHSTER
Corollary Let S be a Noetherian ring containing Q and let I be a principal ideal
of S Then I I
Proof I I by Corollary and the other inclusion follows from the generalized
Brian con Skoda theorem in the case where n and k
The following very imp ortant result is representative of a large class of results that assert
that manipulations of ideals generated by monomials in a xed system of parameters yield
the same result as if the parameters formed a regular sequence or were indeterminates
up to tight closure
Theorem tight closure captures colons Let S be a locally excellent Noether
ian ring containing Q Let x x generate an ideal of height at least n modulo every
n
minimal prime of S Then x x x S x x
n S n n
Sketch of the proof Supp ose that x u x x We must show that u
n n
x x The hypothesis is preserved mo dulo minimal primes and also by lo
n
calization Thus we may assume that the ring is an excellent lo cal domain We may
then complete and we are in the reduced equidimensional case the height condition will
still hold mo dulo every minimal prime and so we may assume that S is a complete lo cal
domain The case where one of the x is not in the maximal ideal is easy and so we may
i
assume that the x are part of a system of parameters
i
Since x u x x we know that there is an a ne Q subalgebra R of S
n n
By Theorem containing x x x and u such that x u x x R
n n n n
we can give a Q algebra factorization R R S of R S such that R is
a domain nitely generated over Q and such that the images of x x generate a
n
prop er ideal of height n in R We can then pass to an a ne Z subalgebra with the
same prop erties Almost all closed b ers will b e reduced equidimensional and have the
prop erty that the images of the x generate an ideal of height n with the image of ux still
i n
in the characteristic p tight closure of the ideal generated by the images of x x
n
and the result now follows from colon capturing in characteristic p
We now de ne a No etherian ring containing Q to b e weakly F regular if every ideal is
tightly closed and F rational if it is a pro duct of domains in which every ideal of height n
generated by n elements is tightly closed These notions are considered in greater general
ity in HH Thus weakly F regular implies F rational Exactly as in the characteristic
p case over a weakly F regular ring the tight closure of every submo dule of a nitely gen
erated mo dule is again tightly closed The pro of of the following imp ortant result is quite
similar to the characteristic p case and is omitted
Corollary An F rational ring is normal and if it is locally excellent then it is
Cohen Macaulay In particular a weakly F regular ring is normal and if it is locally
excellent then it is Cohen Macaulay
De nition and discussion purity We say that a map of R mo dules N M
is pure if W N W M is injective for every R mo dule W Since W may b e equal
R R
to R this implies that N M is injective If M N is nitely presented then N M
THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
is pure if and only if it splits It follows that if R is No etherian N M is pure if and
0 0 0
only if N M splits for all M M containing the image of N such that M N is
nitely generated Cf HR x We shall very often b e interested in the condition that a
ring homomorphism R S b e pure over R in which case we shall say that R is a pure
subring of S We are particularly interested in this condition when R is a No etherian ring
The condition that a ring homomorphism R S b e pure implies that every ideal of R is
contracted from S
We now prove a considerable strengthening of the main result of HR on the Cohen
Macaulay prop erty for rings of invariants of linearly reductive groups G acting on regular
rings the key p oint is that in the situations describ ed in HR S is regular and the xed
G
ring S is a pure subring of S The situation just b elow is therefore much more general
Theorem generalized Ho chster Roberts theorem Every pure subring of
an equicharacteristic regular ring is a Cohen Macaulay ring and normal in fact the
completion of each of its local rings is normal
Proof As in the pro of for the characteristic p case see for example x of HR one can
reduce to the case where R is complete The result now follows from and
b elow the pro of of is left as an exercise
Prop osition If S is weakly F regular and R is pure in S then R is weakly F
regular
Boutot has shown Bou that if an a ne algebra S over a eld of characteristic has
rational singularities and R is pure in S then R has rational singularities There is
a great deal of evidence that the prop erty of having rational singularities is connected
with F rationality For example in Sm it is shown that if R has the prop erty that
Z
almost all b ers are F rational R is then said to have F rational type there is a similar
de nition for weakly F regular type then K R has rational singularities for any eld
Z
K of characteristic It may b e p ossible to characterize rational singularities along these
lines
Phantom homology and homological theorems There has b een a great
deal of work done on a family of problems that used to b e known as the lo cal homological
conjectures Many are now theorems cf PS Ho Ro and Du In this
section we consider some very general tight closure results that recover many of these
results often in a greatly strengthened form
Throughout this section let R b e a lo cally excellent No etherian ring which for simplicity
we assume is reduced and lo cally equidimensional The results stated here are valid more
generally one wants the conditions we imp ose to hold mo dulo every minimal prime of R
b b
d 0
Let G denote a nite free complex over R say R R let denote
i
a matrix of the i th map i d let r b e the determinantal rank of and let I b e
i i i
the ideal generated by the size r minors of and r Recall that G
i i d d
satis es the standard conditions on rank and height if for i d b r r and
i i i
ht I i The complex G is called phantom acyclic if the cycles in G are contained in
i i
MELVIN HOCHSTER
the tight closure of the b oundaries in G for i We refer to HH HH and Hu
i
for more detail We then get the following analogues of characteristic p results detailed
pro ofs are given in HH although we shall make some comments on the pro ofs b elow
Theorem phantom acyclici ty criterion With R and G as above if G
satis es the standard conditions on rank and height then G is phantom acyclic
Sketch of the proof The argument is very much like the pro of of Theorem One
reduces to the case where the ring is a complete lo cal domain and then descends to an
a ne Q algebra while preserving height conditions using Theorem From there it is
easy to pass to an a ne Z subalgebra and then one uses the fact that the hypothesis holds
for almost all closed b ers and that the theorem holds in characteristic p to complete the
pro of
This yields at once a very p owerful result
Theorem vanishing theorem for maps of Tor Let R be an equicharacteristic
regular ring let S a module nite extension of R that is torsion free as an R module e g
a domain and let S T be any homomorphism to a regular ring Then for every nitely
R R
generated R module M the map Tor M S Tor M T is for al l i
i i
Sketch of the proof One reduces to the case where T is complete lo cal regular and then
to the case where R is complete and regular as well A minimal free resolution G of M
satis es the standard conditions by the usual acyclicity criterion and this is preserved
when when we pass to S G which by is then phantom acyclic any given cycle
R
R
M S is in the tight closure of in degree i representing a typical element of Tor
i
the b oundaries This is preserved when we tensor further and pass to T G Since T
R
is regular it is weakly F regular and the result follows
The mixed characteristic version of remains op en If it is true it implies that reg
ular rings are direct summands of their mo dule nite extensions and that pure subrings of
regular rings are Cohen Macaulay b oth of which are op en questions in mixed characteris
tic These issues are explored in HH x where is proved by in equal characteristic
by a di erent metho d Finally we mention the following analogue of a characteristic p
result from HH
Theorem phantom intersection theorem Let R and G be as above with
G of length d and suppose that the standard conditions on rank and height hold for G
Let z M H G be any element whose annihilator in R has height d the length of
G Then z In consequence
M
if R m K is local z cannot be a minimal generator of M
the image of z is in H S G for any regular or weakly F regular ring S to
R
which R maps
Sketch of the proof In proving that z one rst reduces to the case where R is a
M
complete lo cal domain and then uses to descend rst to a Q algebra and then to a
Z algebra while preserving the rank and height conditions The result then follows from
the characteristic p version by passage to closed b ers Part then follows b ecause mM
THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
is tightly closed in M M mM is a direct sum of copies of R m and m is tightly closed
in R while part is obvious
Theorem is a strengthening of the improved new intersection theorem discussed
for example in Ho
Questions There are many op en questions concerning the b ehavior of tight
closure we shall touch brie y on only a few of them here excluding those that dep end
primarily on further progress in characteristic p One obvious question is whether the
Q K EQ
various notions of tight closure and see the fo otnote on p agree when
they are de ned
We next ask whether an element of a complete equicharacteristic lo cal domain R is
in the tight closure of an ideal I if and only if it is in IS for some big Cohen Macaulay
algebra S for R This is correct in characteristic p and the only if part is correct in
EQ
equal characteristic by Theorem of Ho in fact this part holds for See
also Theorem here
For a ne Q algebras we ask whether b eing weakly F regular and having weakly F
regular type are the same and whether b eing F rational is the same as having F rational
type It is an op en question for a ne Q algebras whether F rational type is equivalent to
having rational singularities and there is an analogous question for which we have not
made the necessary de nitions for a ne algebras over an arbitrary eld Cf Sm
In all the known examples where an ideal I of an a ne Q algebra is tightly closed it
actually turns out that one can nd I R such that I is tightly closed in R for almost
Z Z
all b ers We do not know whether this is always the case We also do not know whether
the image of an element can b e in the tight closure of I for a dense in the case of Z this
simply means in nite set of closed b ers without b eing in the tight closure for almost all
closed b ers
Last but certainly not least is the problem of whether there is a theory with similar
prop erties to tight closure that is valid in mixed characteristic go o d enough for example
to prove the vanishing conjecture for maps of Tor itself an imp ortant op en question
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Department of Mathematics
University of Michigan
Ann Arbor MI
USA
E mail ho chster math lsa umich edu