Intersection multiplicity, canonical element conjecture and the syzygy problem
This paper consists of independent work by the two authors, but published in a single document. Both authors' work pertains to the direct summand conjecture (DSC) and monomial conjecture (MC), and related homological conjectures, all of which are stated for the reader's convenience. The first author's work focuses on the question (Q) of when \(\chi^A(M,N) > \chi^A_2(M,N)\), where \(A\) is regular local ring and \(M\) and \(N\) are finitely generated \(A\)-modules with length\((M \otimes_A N) < \infty\). The first result states that, under the additional assumption that \(A\) is equicharacteristic or unramified, the answer to (Q) is negative when dim \(M\) + dim \(N <\) dim \(A\) and depth \(M\) + depth \(N <\) dim \(A-1\), and positive when both inequalities are replaced with equalities. Although the answer to (Q) is unknown when one of the modules, say \(N\), has the form \(N=A/I\), where \(I\) is an ideal of \(A\) generated by an \(A\)-sequence of length \(n\) such that length\((M/IM) < \infty\) and dim \(M\) + dim \(A/I =\) dim \(A\), the author shows that, in this case, there exists a minimal set of generators \(x_1, \dots, x_n\) of \(I\) and \(s >> 0\) for which (Q) has a positive answer if \(N = A/(x_1, \dots, x_{n-1}, x_n^s)\). The proof uses superficial elements. Next, the author relates the validity of MC to pairs of ideals \(I\) and \(J\) in \(A\) such that \(I\) is a complete intersection, \(J\) is an almost complete intersection, ht \(I\) + ht \(J\) = dim \(A\), \(I+J\) is primary to the maximal ideal, and \(I+J\) is not a complete intersection ideal in \(A\). One consequence of the results in this section is that for a complete local domain \(A\) with a system of parameters \(x_1, \dots, x_n\), there exist \(y_1, \dots, y_n\) in \((x_1, \dots, x_n)\) such that \((y_1, \dots, y_{n-1}, x_n) = (x_1, \dots, x_n)\) and \((y_1, \dots, y_{n-1}, x_n^s)\) satisfies MC for \(s>>0\). Additionally, the author advances some of his earlier work [Math Z. 216, 379--388 (1994; Zbl 0806.13002); J. Lond. Math. Soc., II. Ser. 56, No. 1, 49--63 (1997; Zbl 0892.13004)] involving MC and the canonical element conjecture, and shows that if \(A\) is an almost complete intersection ring with system of parameters \(\underline{x}\), then the system satisfies MC if any of the following hold: \(H_1(\underline{x}; A)\) is cyclic or decomposable; \(H_1(\underline{x}; A)^{\vee}\) is not cyclic; \(A\) is a domain. In the course of these results, the author also proves that MC is valid for all local rings if and only if it holds for all local almost complete intersections. The second author first focuses on Koh's proof of the validity of DSC for a ring extension of a regular local ring \(R\) of mixed characteristic \(p\) to the integral closure \(C\) of a \(p\)-th root extension of \(R\). The author aims to give a more conceptual proof of Koh's result by providing a criteria for a finite extension of normal domains \(R \hookrightarrow A\) to be \(R\)-split and constructs a canonical free \(R\)-subalgebra \(S\) of \(A\) such that \(pA \subseteq S\). The argument uses Galois theory and the linear algebra of eigenvalues. In particular, the author constructs Galois and Kummer extensions of the field of fractions \(K\) of \(R\) and \(K[\zeta_p]\), where \(\zeta_p\) is a primitive \(p\)-th root of unity, and uses an eigenspace decomposition of the field of fractions of \(C\). In the third section, this author synthesizes \textit{R.~Heitmann}'s result [Ann. Math. 156, No. 2, 695--712 (2002; Zbl 1076.13511)] on DSC for local rings in dimension less than or equal to three, the improved new intersection theorem (INIT), and the original syzygy theorem by \textit{E.~G.~Evans} and \textit{P.~Griffith} [Ann. Math. 114, 323--333 (1981; Zbl 0497.13013)], to obtain results on the syzygy conjecture in dimension less than or equal to five, in mixed characteristic. First of all, the author proves the order ideal theorem for syzygy modules of finite projective dimension using a special form of INIT. Then the author shows that for a local ring \(A\) of dimension \(n\), a non-free \((n-2)\)th syzygy of finite projective dimension over \(A\) must have rank at least \(n-2\), in any characteristic. This follows from a technical result involving syzygies of finite projective dimension, as well as the application of Heitmann's work and \textit{M.~Hochster}'s result [J. Algebra 84, 503--553 (1983; Zbl 0562.13012)] that DSC implies INIT. Finally, the author shows that the syzygy theorem holds for all regular local rings of dimension at most five.
scientific article; zbMATH DE number 5604529
scientific article