Related papers: On unmixed and equi-dimensional associated graded …

We had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $\ell(R/I) <\infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK…

We study how solid closure in mixed characteristic behaves after taking ultraproducts. The ultraproduct will be chosen so that we land in equal characteristic, and therefore can make a comparison with tight closure. As a corollary we get an…

This paper is devoted to the study of graded associative algebras that satisfy a graded polynomial identity of degree $2$. % Let $\mathsf{G}$ be a finite abelian group, $\mathbb{F}$ a field of characteristic zero and $\mathfrak{A}$ a…

Let R be an excellent local ring, m its maximal ideal and I an ideal. Then there exists a positive integer c such that for all integers n, the integral closure of (I + m^n) is contained in m^(n/c) + the integral closure of I. In the proof,…

Given a local noetherian ring $R$ whose formal completion is integral, we introduce and study the $p$-radical closure $R^\text{prc}$. Roughly speaking, this is the largest purely inseparable $R$-subalgebra inside the formal completion…

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local…

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on…

For an ideal $\mathcal{I}$ in a $\sigma$-complete Boolean algebra $\mathcal{A}$, we show that if the Boolean algebra $\mathcal{A}\langle\mathcal{I}\rangle$ generated by $\mathcal{I}$ does not have the Nikodym property, then it does not have…

This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient…

In this paper we consider a class of connected closed $G$-manifolds with a non-empty finite fixed point set, each $M$ of which is totally non-homologous to zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With the help of…

Let R be a commutative Noetherian ring, a a proper ideal of R and M a finite R-module. It is shown that, if (R;m) is a complete local ring, then under certain conditions a contains a regular element on DR(Hc a(M)), where c = cd(a;M). A…

We define a notion of compressed local Artinian ring that does not require the ring to contain a field. Let $(R,\mathfrak m)$ be a compressed local Artinian ring with odd top socle degree $s$, at least five, and $\operatorname{socle}(R)\cap…

Let $R$ be a commutative Noetherian ring, $N$ a finitely generated $R$-module and $I$ an ideal of $R$. The set $\bar{Q^*}(I, N)$, the quintasymptotic primes of $I$ with respect to $N$, was originally introduced by McAdam \cite{Mc2}. Also,…

Let $\mathbf{k}$ be a field which is either finite or algebraically closed and let $R = \mathbf{k}[x_1,\ldots,x_n].$ We prove that any $g_1,\ldots,g_s\in R$ homogeneous of positive degrees $\le d$ are contained in an ideal generated by an…

Given an ideal $I$ in a regular local ring $A$, the cohomological dimension of $I$ in $A$ is the index of the highest non-vanishing local cohomology of $A$ supported at $I$. Determining effective upper bounds on the cohomological dimension…

Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional…

Given a trivially graded polynomial ring $A=K[a_1,\dots,a_m]$ over a field $K$ and a positively graded polynomial ring $P=A[x_1,\dots,x_k]$, we study graded rings $R=P/I$, where $I$ is a homogeneous ideal in $P$ such that $I\cap A = \{0\}$.…

Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ \wrt \ $I$ and let $\R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with…

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ and let $I=(f_1,...,f_s)$ be an ideal of $R.$ We prove that every associated prime $P$ of $H^i_I(R)$ satisfies $\text{dim}R/P\geqslant…