Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

‎Let $M$ be a non-zero finitely generated module over a commutative Noetherian local ring $(R,mathfrak{m})$ with $dim_R(M)=t$‎. ‎Let $I$ be an ideal of $R$ with $grade(I,M)=c$‎. ‎In this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎The main purpose of this article is to investigate when the natural homomorphisms $gamma‎: ‎Tor^{R}_c(k,H^c_I(M))to kotim...

We investigate the rational powers of ideals. find that in case monomial ideals, canonical indexing leads to a characterization yielding symbolic squarefree ideals are indeed themselves. Using connection with techniques, we use splittings show convergence depths and normalized Castelnuovo–Mumford regularities. Stanley for powers, as consequence this, before-now unknown integral closure powers. ...

The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A, m) in terms of local cohomology modules of Rees algebras ...

We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the loc...

The aim of this work is to describe some operations in the category of regular holonomic D-modules with support a normal crossing and variation zero introduced in [2]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers.

We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include hypersurfaces of characteristic zero with rational singularities, as well as F-regular hypersurfaces of positive characteristic. As a consequence, we answer a question on the associated primes of certain families of ideals which arose from the localization problem in tight clos...

In this article, we study certain local cohomology modules over F-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula computing these invariants when ring is standard graded and, by its means, provide new examples tables. associated primes Ext-modules, showing that they are all compatible ideals. Finally, focus on properties numbers detect glo...