Local Cohomology and the Homological Conjecturesin Commutative Algebra
During the past few decades a considerable amount of effort has been spent on a setof problems in Commutative Algebra known as the “Homological Conjectures”. These
questions originated from Serre’s homological theory of intersection multiplicities definedin terms of Euler characteristics of Tor functors. Numerous conjectures arose from thiswork concerning intersection multiplicities as well as related properties of modules of finite
projective dimension and Cohen-Macaulay rings and modules. In these talks we describe
some of the history of these problems, their current status, and some recent developmentsin this area. We will concentrate on progress related to local cohomology.
We first state the Monomial Conjecture of Mel Hochster; this conjecture implies several
other ones and is at present still not known for rings of mixed characteristic.
Conjecture: Let A be a Noetherian local ring of Krull dimension d, and let x1 , . . . , xd
be a system of parameters for A (so that the quotient A/(x1 , . . . , xd ) is an Artinian ring).
Then for all integers t > 0, we havet+1
t+1
xt1 xt2 · · · xtd 6∈ (xt+11 , x2 , . . . , xd ),
t+1
t+1
t+1
where (xt+1.
1 , x2 , . . . , xd ) denotes the ideal generated by the elements xi
This conjecture is rather trivial in characteristic zero, but it is not trivial in positiveor mixed characterisitic, and, as mentioned above, has consequences for other questions in
those cases.
We recall that a ring is Cohen-Macaulay if a system of parameters forms a regularsequence, so that if x1 . . . , xd is a system of parameters, then if axi ∈ (x1 , . . . , xi−1 ), thena ∈ (x1 , . . . , xi−1 ) for i = 1, . . . , d.
If the elements x1 , . . . , xd form a regular sequence, then it is a straighforward exercise
t+1
t+1
to show that if xt1 xt2 · · · xtd ∈ (xt+11 , x2 , . . . , xd ), then 1 ∈ (x1 , . . . , xd ). Thus if the ring isCohen-Macaulay, the Monomial Conjecture is clear. By extending these ideas, it is easy toshow that the existence of Cohen-Macaulay modules and algebras implies this conjecture.
A related approach to these questions is via local cohomology. If I is the ideal generated by a system of parameters x1 , . . . , xd , the local cohomology with support in I is
the same as the local cohomology with support in the maximal ideal. Thus the local
cohomology with support in m is the direct limit of the homology of Koszul complexes
on powers of the xi , and the ring is Cohen-Macaulay if and only if the local cohomologyiHm(A) vanishes for i < d. We also mention that even in the non-Cohen-Macaulay casethere are close relations bewteen the annihilators of local cohomolgy and the annihilatorsof homology of a Koszul complex on a system of parameters.
The most significant recent advance on the Monomial Conjecture was its proof forrings of mixed characteristic of dimension three by Ray Heitmann in 2002. The proof
was based on a lemma that said that for a complete Noetherian domain A of dimension
2
3 of mixed characteristic p, the local cohomology Hm(A+ ) is annihilated by p1/n for alln > 0. Here A+ denotes the absolute integral closure of A; that is, the integral closureof A in the algebraic closure of its quotient field. The fact that A+ is integrally closed

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implies that Hm(A+ ) and Hm(A+ ) are zero, so, while this lemma does not imply that A+
is Cohen-Macaulay, it comes close, and we say that A+ is “almost Cohen-Macaulay” (wegive a more general definition of this term below). It is still an open question whether A+is Cohen-Macaulay in this situation. Heitmann proved, however, that it suffices to provethat A+ is almost Cohen-Macaulay to prove the Monomial Conjecture for the ring A.
We next give a precise definition of “almost Cohen-Macaulay”. In the remainder of
the abstract, A will denote a Noetherian integrally closed domain of dimension d, and A+
will denote its absolute integral closure.
Definition: We say that A+ is almost Cohen-Macaulay if there is a valuation v on+iA which is positive on m such that for every i < d, for every η ∈ Hm(A+ ) for i < d, andfor every ² > 0, there is an element x in A+ with v(x) < ² and xη = 0.
The theorem of Heitmann implies that this condition holds for rings of mixed characteristic of dimension 3, where the element x can be taken to be p1/n for sufficiently largen. Also, Heitmann showed that this condition suffices to prove the Monomial Conjecture.
It is an open question whether A+ is almost Cohen-Macaulay for all complete integrally
closed domains A. In these talks we will discuss this question in three cases, depending on
the characteristic of A.

1. A has positive characteristic. In this case it is easy to show, using classical techniques with the Frobenius map, that A+ is almost Cohen-Macaulay. However, Hochsterand Huneke have proven the much deeper result that A+ is actually Cohen-Macaulay inthis case.
2. A contains a field of characteristic zero. In this case A+ is never CohenMacaulay if its dimension is at least 3. It is also true that the relevant homologicalconjectures are known in this case; however, this question appears to be interesting in
itself, and it would give a proof of some of these conjectures that does not involve reduction
to positive characteristic.
2In this case there are some partial results for Hm(A+ ) done in joint work with Anurag
Singh and V. Srinivas. First, for several examples in dimension 3 we were able to showthat annihilators of small valuation exist, and these examples seem to indicate that, unlikein the mixed characteristic case, the annihilator x depends strongly on the element η oflocal cohomology. The most general theorem was found by Srinivas, who showed thatfor standard graded rings defining a normal scheme, with the valuation v defined by thegrading, such elements x can be found by mapping to the Albanese variety and pullingback the map defining multiplication by an integer N . We discuss this method and thefacts on abelian varieties that are used.
iFor more general rings, and for Hm(A+ ) with i > 2, almost nothing is known on thisquestion.
3. A has mixed characteristic. This is the most interesting case, since the Monomial Conjecture is open in this case. In these talks we will say a little about Heitmann’s
result and describe a new approach to the question using the “Fontaine ring” of a ring of
mixed characteristic. This construction has been used in Arithmetic Geometry to reduce
questions to the case of positive characteristic, but has not been studied in the context
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of classical problems in Commutative Algebra. We discuss the questions that arise inattempting to use this method to investigate properties of local cohomology.

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