Free resolutions of parameter ideals over some rings with finite local cohomology

Let $R$ be a noetherian local ring. We consider the following quastion: Does there exist an integer $n$ such that all idelas generated by a system of parameters contained in the $n$-th power of the maximal ideal have the same Betti numbers? We obtain a positive answer for some rings with finite local cohomology.Comment: 8 Page

Contracting Endomorphisms and Gorenstein Modules

We characterize Gorenstein modules over those local rings that admit a finite contracting endomorphism.Comment: 7 page

Pretty kk-clean monomial ideals and kk-decomposable multicomplexes

We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $\Gamma$ is $k$-decomposable if and only if its associated monomial ideal $I(\Gamma)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author.Comment: 15 page

Variation of Mixed Hodge Structure and Primitive elements

We study the asymptotic behaviour of polarization form in the variation of mixed Hodge structure associated to isolated hypersurface singularities. The contribution characterizes a modification of Grothendieck residue as the polarization on the extended fiber in this case. We also provide a discussion on primitive elements to explain conjugation operator in these variations, already existed in the literature.Comment: This article is a brief of my other article On the mixed Hodge structure associated to isolated hypersurface singularities. Its removal is for its content already existed in the aforementioned paper, with more detail

Mixed Hodge Structure on Theta divisor I

We explain the mixed Hodge structure (MHS) on the complement of the generalized theta divisor of a curve in its generalized jacobians. Our approach considers the generalized jacobian via degeneration of smooth jacobians as a determinantal varieties. We make the question if the MHS can be explained by graphs or their degenerations

Toric Structure on Mumford-Tate domains and Characteristic cohomology

We explain a higher structure on Kato-Usui compactification of Mumford-Tate domains as toric stacks. As a motivation the universal characteristic cohomology of Hodge domains can be described as cohomology of stacks which have better behaviour in general.Comment: arXiv admin note: text overlap with arXiv:1107.1906 by other author

On the mixed Hodge structure associated to hypersurface singularities

Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of $f$ can be extended over the degenerate point $0$ of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of $f$ denoted $\Omega_f$. A mixed Hodge structure can be defined on $\Omega_f$ in this way. The polarization on $\mathcal{H}$ deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces in this process. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the Hodge signature of Grothendieck pairing

Higher residue pairing on Crystalline local systems

We explain a generalization of the K. Saito higher residue pairing for local system of $p$-adic isocrystals

A Hodge index for Grothendieck residue pairing

In this text we apply the methods of Hodge theory for isolated hypersurface singularities to define a signature for the Grothendieck residue pairing of these singularities

Cohen-Macaulay-ness in codimension for simplicial complexes and expansion functor

In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $t\geq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we prove that it can be obtained from a Buchsbaum complex.Comment: 8 page