KUMMER COVERINGS AND SPECIALISATION

We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

Local cohomology modules and their properties

UDC 512.5 Let be a complete Noetherian local ring and let be a generalized Cohen-Macaulay -module of dimension We show thatwhere and is the ideal transform functor. Also, assuming that is a proper ideal of a local ring , we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

Reducibility of parameter ideals in low powers of the maximal ideal

A commutative noetherian local ring ( R , m ) (R,\mathfrak {m}) is Gorenstein if and only if every parameter ideal of R R is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer ℓ \ell (depending on R R ) such that R R is Gorenstein if and only if there exists an irreducible parameter ideal contained in m ℓ \mathfrak {m}^\ell . We give upper bounds for ℓ \ell that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.

Little dimension and the improved new intersection theorem

Let $R$ be a commutative noetherian local ring. We define a new invariant for $R$-modules which we call the little dimension. Using it, we extend the improved new intersection theorem.

Annihilator of local cohomology of homogeneous parts of a graded module

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [Formula: see text].

POWERS OF THE MAXIMAL IDEAL AND VANISHING OF (CO)HOMOLOGY

We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

Some homological properties of generalized local cohomology modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text], [Formula: see text] be two finitely generated [Formula: see text]-modules. In this paper, it is shown that [Formula: see text] and [Formula: see text] for each [Formula: see text] and each integer [Formula: see text]. In particular, if [Formula: see text] then [Formula: see text]. Moreover, some applications of these results will be included.

The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.

A short note on annihilators of local cohomology modules

Let [Formula: see text] be a commutative Noetherian domain, [Formula: see text] a nonzero [Formula: see text]-module of finite injective dimension, and [Formula: see text] be a nonzero ideal of [Formula: see text]. In this paper, we prove that whenever [Formula: see text], then the annihilator of [Formula: see text] is zero. Also, we calculate the annihilator of [Formula: see text] for finitely generated [Formula: see text]-modules [Formula: see text] and [Formula: see text] with conditions [Formula: see text] and [Formula: see text]. Moreover, if [Formula: see text] is a regular Noetherian local ring and [Formula: see text] such that [Formula: see text], then we show that there exists an ideal [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text].