The Direct Summand Conjecture in Dimension Three

Abstract

In [Ho1], Hochster proved the following results:Theorem (Hochster). If R is a regular Noetherian ring which contains a field and S ⊃ R is a module-finite R-algebra, then R is a direct summand of S as an R-module.Theorem (Hochster). If S is any local ring which contains a field and x 1 , . . . , x n is a system of parameters for S, then for every integer k ≥ 0, (The mixed characteristic case of these results is easy for dim R ≤ 2 but relatively little is known for dim R > 2. The general statements, which are equivalent, became known as the direct summand and monomial conjectures. The principal advance in this subject occurred in Hochster's 1983 article [Ho2] in which he introduced the canonical element conjecture. He demonstrated that this new conjecture was equivalent to the other two both overall and in the case of fixed dimension. He also showed that it was sufficiently strong to imply the validity of a number of other homological conjectures which had been shown to follow from the big Cohen-Macaulay modules conjecture.In this article, we prove a result (Theorem 3.7) which implies the three dimensional case of the direct summand conjecture and so also the three dimensional cases of each of the conjectures that follow from it. As it happens, most of these results also follow from the New Intersection Theorem which was proved by Roberts in the mixed characteristic case [R1]. Two exceptions are the improved new intersection conjecture and the mixed characteristic version of the Evans-Griffith Syzygy Theorem [EG]. These two are now theorems in the three dimensional case. It should also be noted that Hochster [Ho3] has just proved the existence of balanced big Cohen-Macaulay algebras in dimension three -using (3.7) as a key ingredient of the proof.The main result of this article is of independent interest. In [H2], the author introduced several forms of an extended plus closure. These were intended to fill the void due to the absence of a tight closure analog in mixed characteristic. A key property of the tight closure is the colon-capturing property. Theorem 3.7 tells us that the extended plus closures have the colon-capturing property in dimension three, at least for excellent rings. It was shown in [H2] that if any of these new closures had the colon-capturing property, it would follow that the closures would also have a second very desirable property -that all ideals in regular local rings are closed. There is an additional implication concerning the Briançon-Skoda Theorem, proved in the general case by Lipman and Sathaye [LS]. In [H2, p.818], mimicking the tight closure treatment, a generalization of this result was proved for the extended plus closures. Unfortunately, this result is not a true generalization in that it would only imply the original result if ideals in regular rings were closed. This result now does imply the original in the three dimensional case. So Theorem 3.7 goes a long way towards establishing a good closure operation for three dimensional mixed characteristic rings.

Closure operations that induce big Cohen–Macaulay modules and classification of singularities

“…We do not know whether Heitmann's mixed characteristic plus closure, full extended plus closure, and full rank one closure [Hei02] are Dietz closures, even in dimension 3. To discuss this question, we first extend the definition of full extended plus closure (epf) to finitely generated modules.…”

“…While they are known to exist over rings of equal characteristic [Hoc75] and rings of mixed characteristic and dimension at most 3 [Hei02,Hoc02], it is not known whether they exist over mixed characteristic rings of higher dimension. The existence of big Cohen-Macaulay modules (or algebras) is also sufficient to imply a large group of equivalent conjectures, including the Direct Summand Conjecture [Hoc73], Monomial Conjecture [Hoc73], and Canonical Element Conjecture [Hoc83].…”

Local cohomology of Segre product type rings

“…(1) We have (u) s: (2) For all > 0, there is an integer n such that if (v) s + , then v is in the ideal generated by X 1 ; : : : ; X i 1 in C n =I n .…”

“…(1) If R is generated by x 1 ; : : : ; x n and S is generated by y 1 ; : : : ; y m in degree one, then R#S is generated in degree one by the x i #y j as i runs from 1 to n and j runs from 1 to m. (2) If the dimension of R is d and the dimension of S is d 0 , then the dimension of R#S is d + d 0 1.…”

Free resolutions of parameter ideals for some rings with finite local cohomology

“…In [3], R. Heitmann shows that the Direct Summand Conjecture, hence the Canonical Element Conjecture, holds for every 3-dimensional ring.…”