Around general neron desingularization

It gives some new forms of General Neron Desingularization and new applications.Comment: It simplifies the proof of Theorem 1

Simple General Neron Desingularization in local Q{\bf Q}-algebras

In this form will appear in Communications in Algebra.Comment: Proposition 5 was extended and a sketh proof is include

Stanley depth of monomial ideals

Let $I\supsetneq J$ be two monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . We study when the Stanley Conjecture holds for $I/J$ using the recent result of \cite{IKM} concerning the polarization

Stanley conjecture on intersections of four monomial prime ideals

We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra $S$ over a field and for an arbitrary intersection of monomial prime ideals $(P_i)_{i\in [s]}$ of $S$ such that $P_i\not\subset \Sigma_{1=j\not =i}^s P_j$ for all $i\in [s]$.Comment: accepted to Communications in Algebr

Groups actions, and D equivalences of categories of coherent sheaves of symplectic resolutions

Let k be an algebraically closed field of characteristic p>>0. Let $X\rightarrow Y$ be a symplectic resolution. There are two questions which motivates this work. One question is a construction of an action of a group on the category $\mathcal{C}:=D^b(Coh(X))$ - The bounded derived category of coherent sheaves of the symplectic resolution X. Second question is understanding equivalence functors between derived categories of coherent sheaves for different symplectic resolutions of Y. Let G/k be a reductive group. In this paper, we construct a local system on a topological space called $V^0_{\mathbb{C}}$ with value the category $D^b(Coh(T^*G/P))$ for a parabolic subgroup P. This induces an action of $\pi_1 V^0_{\mathbb{C}}$ on the category. In another paper we further explain how a refinement of this local system construction, gives an answer to the second question, showing that these equivalence functors, are parametrized by homotopy classes of maps between certain points in the base space. We also lift the result to characteristic zero

A Parameterization of D equivalences of coherent sheaves of symplectic resolutions of a given symplectic singularity

Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical equivalence. By refining a construction from a previous paper, we construct a local system of categories over a topological space V^0_C, where these categories are assigned to different points in V^0_C. Natural equivalence functors between these categories are parameterized by homotopy classes of paths between the corresponding points

Nested Artin approximation

A short proof of the linear nested Artin approximation property of the algebraic power series rings is given here.Comment: We simplified the statements of proposition

On the generalized Riemann hypothesis II

Some of my previous publications were incomplete in the sense that non trivial zeros belonging to a particular type of fundamental domain have been inadvertently ignored. Due to this fact, I was brought to believe that computations done by some authors in order to show counterexamples to RH were affected of approximation errors. In this paper I illustrate graphically the correctness of those computations and I fill the gaps in my publications.Comment: 12 pages, 2 figure

On the Bass-Quillen Conjecture and Swan's question

We present a question which implies a complete positive answer for the Bass-Quillen Conjecture.Comment: In this form will appear in Combinatorial Structures in Algebra and Geometry, Editors Dumitru Stamate, Thomasz Szemberg, Springer Proceedings in Mathematics and Statistics Series, 202