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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

14 questions from the last 30 days
Newest Active Bountied Unanswered
2 votes
0 answers
37 views

Let $E \to F$ be a map of vector bundles on a scheme $X$ of ranks $e, f$ (actually, I hope $X$ may be a stack here). Suppose $e \leq f$. I want to describe the locus $D \subseteq X$ where $E \to F$ ...
Leo Herr's user avatar
  • 1,154
0 votes
0 answers
30 views

I've identified the generating functions for the tangent Chern numbers of the complex projective spaces $CP^n$ given in "Algebraic topology of the Lagrange inversion" by Victor Buchstaber ...
Tom Copeland's user avatar
  • 11.2k
7 votes
3 answers
292 views

A regular subobject classifier in a category with finite limits is a morphism $1 \to \Omega$ such that every regular monomorphism $Y \hookrightarrow X$ is the pullback of $1 \to \Omega$ along some ...
Martin Brandenburg's user avatar
  • 66.2k
11 votes
1 answer
328 views

Fix some finite field $F$. Is there a (commutative) noetherian ring $R$ having infinitely many residue fields isomorphic to $F$? By replacing $R$ with $R/pR$, with $p$ being the characteristic of $F$, ...
Uriya First's user avatar
  • 3,328
14 votes
2 answers
978 views

As part of a course in commutative algebra or algebraic geometry, one will generally learn that, for a Noetherian local ring: regular $\Rightarrow$ complete intersection $\Rightarrow$ Gorenstein $\...
Gro-Tsen's user avatar
  • 40.2k
2 votes
0 answers
162 views

Let $R=\mathbb{F_p}[[x_1,\ldots,x_n]]$ be the ring of power series over a field of $p$-elements. Notice that $R$ is graded by $\mathbb{Z}^n$, that is, the homogenous elements are monomials and by $\...
Yiftach Barnea's user avatar
  • 5,996
0 votes
0 answers
106 views

Are there algorithmic procedures for constructing finitely-generated $\mathbb{Z}$-graded $\mathbb{C}$-algebras $R$, given knowledge of the Hilbert function? That is, say I have a sequence $n_1, n_2, \...
sheride's user avatar
  • 43
5 votes
1 answer
201 views

Let $R$ be a ring and $M$ be an $R$-module. Suppose $a_1, \dots, a_k$ is an $R$-sequence such that the ideal $I = (a_1, \dots, a_k)$ has $\text{grade(I, M)} = k$. Can we then say that the sequence $(...
feynhat's user avatar
  • 191
7 votes
1 answer
518 views

$\require{AMScd} \newcommand{\sheaf}[1]{\mathcal{#1}} \newcommand{\tensor}{\otimes} \newcommand{\Ohol}{{\mathcal O}}$I recently posted the following to math.stackexchange: Let $A$ be a noetherian ring ...
Jürgen Böhm's user avatar
  • 820
1 vote
0 answers
92 views

While looking at an ideal $I$ of $m$ forms of degree $d$ in $R:=\mathbb{C}[x_0,\dots,x_n]$ with just one common zero, say $(1:0:\dots :0)$, one can add $x_0$ to $I$, obtaining an ideal $I'$ without ...
Dima Pasechnik's user avatar
  • 15k
8 votes
1 answer
535 views

The starting point for this question is the fact that the power series ring in one variable $t$ can be expressed as an inverse limit $$\mathbb{Z}[[t]] = \lim_{\longleftarrow} \Big(\mathbb{Z}[t]/(t) \...
Vidit Nanda's user avatar
  • 16k
3 votes
0 answers
275 views

$\DeclareMathOperator\Perm{Perm}$ Below I show (in my original posting it was a conjecture) that the power series corresponding to the multiplicative inverse, or reciprocal, of a power series with ...
Tom Copeland's user avatar
  • 11.2k
7 votes
1 answer
415 views

Let $f=(f_1,\dots,f_n)$, with $$ f_i(x_1,\dots,x_n)=a_{ii}T_i(x_i)+\sum_{j\not=i}a_{ij}x_j $$ where $$ T_i(x_i)=x_i-d_i-\sum_{l}\frac{|b_{i,l}|}{x_i-c_{i,l}}. $$ and $A=(a_{ij})$ is positive-definite....
user1728960's user avatar
  • 173
2 votes
0 answers
48 views

Consider the ring $R$ of Laurent polynomials in $n$ variables over the finite field of two elements, $R= \mathbb F_2[x_0,x_0^{-1},x_1,x_1^{-1},\ldots]$, with the standard involution by inverting ...
Andi Bauer's user avatar
  • 3,125