arXiv subjects

Mel Hochster

Publications and source records attributed to Mel Hochster.

5 recordsLinked to original sources

arXiv2026

Parameters and lim CM sequences: the Q-sequence conjecture and Lech's formula

We prove that if a local ring has a weakly lim Cohen-Macaulay sequence of modules, then it satisfies the conjecture that every system of parameters is a Q-sequence. We also give a generalization of Lech's formula for the multiplicity of a local ring $(R,m)$ with respect to a system of parameters that allows a very large class of sequences of $m$-primary ideals generated by monomials in the parameters to be used in the calculation. The cases where the sequence of ideals consists of the powers of the ideal generated by parameters, or where the ideals in the sequence are generated by varying powers of the individual parameters all of which tend to infinity, which is Lech's original formula, are included.

math.AC↗

arXiv2026

Stanley-Reisner Theory in Mixed Characteristic

We discuss a class of mixed characteristic rings defined analogously to Stanley-Reisner rings by replacing one variable with a uniformizing parameter for a discrete valuation ring. We adapt Hochster's formula for Tor and Ext modules, Hochster's formula for local cohomology, and Terai's criterion for Serre conditions to this new setting; one of the tools needed is cellular sheaf cohomology, for which we give a brief treatment.

math.AC↗

arXiv2026

Filtration theorems, Hilbert functions, and generic freeness

Let $M$ be a finitely generated module over a ring $R$ that is a finitely generated algebra over a Noetherian ring $A$ by $r$ elements. We prove that there are finitely many primes $P_1, \ldots, P_k$ of $A$ such that $M$ has a countable ascending filtration by finitely generated $A$-modules with the property that every factor $F_n$ has a filtration by cyclic modules of the form $A/P_i$, $1 \leq i \leq k$ such that the number of copies $H_i(n)$ of $A/P_i$ that occur in the filtration of $F_n$ is an eventually polynomial function of $n$. We prove a number of bounds on the degrees of these eventually polynomial functions: $r-1$ is always a bound, and the dimension of $\dim(M_P)-1$ is a bound under mild hypotheses. One application is an immediate proof Grothedieck's classic theorem on generic freeness. Several other applications are given.

math.AC↗

arXiv2026

Lim Cohen-Macaulay sequences of modules

We introduce the notion of a lim Cohen-Macaulay sequence of modules. We prove the existence of such sequences in positive characteristic, and show that their existence in mixed characteristic implies the long open conjecture about positivity of Serre intersection multiplicities for all regular local rings, as well as a new proof of the existence of big Cohen-Macaulay modules. We describe how such a sequence leads to a notion of closure for submodules of finitely generated modules: this family of closure operations includes the usual notion of tight closure in characteristic $p>0$, and all of them have the property of capturing colon ideals. In fact they satisfy axioms formulated by G.~Dietz from which it follows that if a local ring $R$ has a lim Cohen-Macaulay sequence then it has a big Cohen-Macaulay module. We also prove the existence of lim Cohen-Macaulay sequences for certain rings of mixed characteristic.

math.AC↗

arXiv2016

Content of Local Cohomology, Parameter Ideals, and Robust Algebras

This paper continues the investigation of quasilength, of content of local cohomology with respect to generators of the support ideal, and of robust algebras begun in joint work of Hochster and Huneke. We settle several questions raised by Hochster and Huneke. In particular, we give a family of examples of top local cohomology modules both in equal characteristic 0 and in positive prime characteristic that are nonzero but have content 0. We use the notion of a robust forcing algebra (the condition turns out to be strictly stronger than the notion of a solid forcing algebra in, for example, equal characteristic 0) to define a new closure operation on ideals. We prove that this new notion of closure coincides with tight closure for ideals in complete local domains of positive characteristic, which requires proving that forcing algebras for instances of tight closure are robust, and study several related problems. This gives, in effect, a new characterization of tight closure in complete local domains of positive characteristic. As a byproduct, we also answer a question of Lyubeznik in the negative.

math.AC↗