We can think of the "tangent category" of a category $\mathcal{C}$ at an object $A$ as being the abelian group objects in the overcategory $\mathcal{C}/A$ (with whatever conditions …

$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is eqidimensional and $M$ does not have any embedded prime. Given $x\in I$ where $I$ is an ideal of $A$ an …

$\underline{\textbf{Embedded associated prime}}$ I am reading the book "Joins and Intersections". In the proof of Rees theorem I have some doubt. Let $\mathbf M$ be a finitely ge …

This question is just a curiosity, but I'm really interested in the answer. It was originally posted on math.stackexchange (http://math.stackexchange.com/questions/368897/inverse-p …

If $A$ is a commutative ring we have the estimate $$ \dim (A)+1 \le \dim (A[x])\le 2\dim (A)+1 $$ for the Krull dimension, with $\dim (A)+1 = \dim (A[x])$ for Noetherian rings. I …

I hope this question is not unreasonable. We all know how to take products of numbers, this generalises to a huge amount of different types of products in mathematics. In a certa …

I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R …

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$, and index set $\langle n\rangle = \{ 1, \dots, n \}$, and the submatrix $A(\alpha)$ with the columns and rows of $A$ wit …

Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordina …

I am currently reading the paper "A Numerical characterization of reduction ideals" by Hubert Flenner and Mirella Manaresi. In this paper they have quoted two results from "Joins …

EDIT : I edited the question according to Prof. Rickard's suggestions Let $Y$ be an affine variety over $\mathbb{C}$ and $A$ and $B$ be $2$ algebras with finite homological dimens …

Does there exist an uncountable (possibly commutative, unital) ring with a countably infinite spectrum? More generally, given two cardinalities $\kappa$ and $\lambda$, does there …

Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.) It is not difficult to show that i …

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution o …

In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$ $$D \to E \to F,$$ we have th …