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Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty.$ Rees proved that for all $n\gg0$, $\lambda(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$…

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and…

Let A be a commutative noetherian ring. Call a functor <<commutative A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When…

For any finitely generated module $M$ with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant $h(M)$ was introduced and studied in the author's previous work "Partial Trace Ideals and Berger's…

Let $(R,\fm)$ be a commutative Noetherian local ring. Suppose that $M$ and $N$ are finitely generated modules over $R$ such that $M$ has finite projective dimension and such that $\Tor^R_i(M,N)=0$ for all $i>0$. The main result of this note…

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the…

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking…

Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products…

We study rings over which an analogue of the Weierstrass preparation theorem holds for power series. We show that a commutative ring $R$ admits a factorization of every power series in $R[[x]]$ as the product of a polynomial and a unit if…

We study how Betti numbers of ideals in a local ring change under small perturbations. Given $p\in\mathbb N$ and given an ideal $I$ of a Noetherian local ring $(R,\mathfrak m)$, our main result states that there exists $N>0$ such that if…

Let $R$ be a Cohen-Macaulay local ring with a canonical module $\omega_R$. Let $I$ be an $\m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $n\longmapsto \ell (\Hom_R(M,{\omega_R}/{I^{n+1}…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of prime characteristic $p$ and $Q$ be an $\mathfrak{m}$-primary parameter ideal. We give criteria for F-rationality of $R$ using the tight Hilbert function $H^*_Q(n)=\ell(R/(Q^n)^*$ and the…

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the…

Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. It is shown that if $\Ext^i_R(R/\fa,M)$ is minimax for all $i\leq \dim M$, then the $R$-module $\Ext^i_R(N,M)$ is minimax for all $i\geq 0$ and for…

The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the…

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given…

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we study the set…

The main object of this note is to study the conormal module $M$ and the computation of the second symbolic power $\bar I^{(2)}$ of an ideal $\bar I$ in the residue ring $R/H$ of a polynomial ring $R$ over a field of characteristic zero.…

Exploiting symmetry in Groebner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…