Mathematics

Let R=k[ x 1 ,?? x n ] be a polynomial ring over a field k and let I?�R be a monomial ideal preserved by the natural action of the symmetric group S n on R . We give a combinatorial method to determine the S n -module structure of Tor i (I,k) . Our formula shows that Tor i (I,k) is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an S n -equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of S n -invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or >n ) we compute the S n -invariant part of Tor i (I,k) in terms of Tor groups of the unsymmetrization of I .

For any commutative ring A we introduce a generalization of S -noetherian rings using a hereditary torsion theory σ instead of a multiplicatively closed subset S⊆A . It is proved that if A is a totally σ -noetherian ring, then σ is of finite type, and that totally σ -noetherian is a local property.

For any commutative ring A we introduce a generalization of S --artinian rings using a hereditary torsion theory ? instead of a multiplicative closed subset S?�A . It is proved that if A is a totally ? --artinian ring, then ? must be of finite type, and A is totally ? --noetherian.

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be obtained by means of a simple computer algebra procedure once the integral closure of each row ideal is known. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

The almost purity theorem is central to the geometry of perfectoid spaces and has numerous applications in algebra and geometry. This result is known to have several different proofs in the case that the base ring is a perfectoid valuation ring. We give a new proof by exploiting the behavior of Faltings' normalized length under the Frobenius map.

The goal of the article is to get a satisfactory theory of cosupport in the derived category D(R) , this is done by introducing another versions of the "big" and "small" cosupport for complexes. We provide some properties for cosupport that are similar--or rather dual--to those of support for complexes. By examples we show that these versions are differ from the cosupport in [J. Reine Angew. Math. 673 (2012) 161--207]. We also study some relations between the "big" and "small" cosupport and give some computations and comparisons of the "small" support and "small" cosupport.

Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.

A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.

We give examples showing that the usual Artin Approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach algebra. In particular we construct an example of a commutative integral Banach algebra R such that the ring of formal power series over R is not flat over the ring of convergent power series over R .

Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R -module. In this paper we proved that if Supp F i a (M) is finite for all i<t , then so is Ass( F t a (M)) .