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**The direct summand conjecture in dimension three.**
*(English)*
Zbl 1076.13511

From the introduction: M. Hochster [Nagoya Math. J. 51, 25–43 (1973; Zbl 0245.13012)] proved the following results:

Theorem. If \(R\) is a regular noetherian ring which contains a field and \(S\supset R\) is a module-finite \(R\)-algebra, then \(R\) is a direct summand of \(S\) as an \(R\)-module.

Theorem. If \(S\) is any local ring which contains a field and \(x_1,\dots, x_n\) is a system of parameters for \(S\), then for every integer \(k\geq 0\), \((x_1\cdots x_n)^k\not\in (x_1^{k+1},\dots, x_n^{k+1})S\).

The mixed characteristic case of these results is easy for \(\dim R\geq 2\) 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 M. Hochster’s 1983 article [J. Algebra 84, 503–533 (1983; Zbl 0562.13012)] in which he introduced the canonical element conjecture.

In this article, we prove a result 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.

The main result of this article is of independent interest. In J. Algebra 238, No. 2, 801–826 (2001; Zbl 1036.13007), R. C. Heitmann introduced several forms of an extended plus closure. These are 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. The main theorem of the paper under review tells us that the extended plus closures have the colon-capturing property in dimension three, at least for excellent rings. The author (loc. cit.) showed 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. The author (loc. cit.), mimicking the tight closure treatment, proved a generalization of this result 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 our result goes a long way towards establishing a good closure operation for three-dimensional mixed characteristic rings.

For the direct summand conjecture, the property that we in fact need is that ideals in regular rings are closed under the extended plus closure. In turn this implies that such ideals are closed under the plus closure. This latter property is equivalent to the direct summand conjecture.

Theorem. If \(R\) is a regular noetherian ring which contains a field and \(S\supset R\) is a module-finite \(R\)-algebra, then \(R\) is a direct summand of \(S\) as an \(R\)-module.

Theorem. If \(S\) is any local ring which contains a field and \(x_1,\dots, x_n\) is a system of parameters for \(S\), then for every integer \(k\geq 0\), \((x_1\cdots x_n)^k\not\in (x_1^{k+1},\dots, x_n^{k+1})S\).

The mixed characteristic case of these results is easy for \(\dim R\geq 2\) 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 M. Hochster’s 1983 article [J. Algebra 84, 503–533 (1983; Zbl 0562.13012)] in which he introduced the canonical element conjecture.

In this article, we prove a result 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.

The main result of this article is of independent interest. In J. Algebra 238, No. 2, 801–826 (2001; Zbl 1036.13007), R. C. Heitmann introduced several forms of an extended plus closure. These are 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. The main theorem of the paper under review tells us that the extended plus closures have the colon-capturing property in dimension three, at least for excellent rings. The author (loc. cit.) showed 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. The author (loc. cit.), mimicking the tight closure treatment, proved a generalization of this result 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 our result goes a long way towards establishing a good closure operation for three-dimensional mixed characteristic rings.

For the direct summand conjecture, the property that we in fact need is that ideals in regular rings are closed under the extended plus closure. In turn this implies that such ideals are closed under the plus closure. This latter property is equivalent to the direct summand conjecture.

### MSC:

13D22 | Homological conjectures (intersection theorems) in commutative ring theory |

13C05 | Structure, classification theorems for modules and ideals in commutative rings |

13B22 | Integral closure of commutative rings and ideals |