# Tagged Questions

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

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### Witt-vector vectors

I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ...
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### Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero. When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...
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### Tensor products of simple modules over algebras [closed]

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...
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### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
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### Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question. I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...
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### ideals linked to an almost complete intersection

Is a grade $3$ type $3$ perfect ideal in a Noetherian ring linked to a grade $3$ almost complete intersection? I know that grade $3$ type $2$ perfect ideals are (by a work of Anne Brown).
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### non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian ...
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### If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
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### Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
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### Ring with Cohen-Macaulay canonical module

Let $(R,m)$ be Noetherian local ring which is an imagine of a Gorenstein ring $(S,n)$. Set $$K_R:= Ext_S^{s-d}(R,S),$$ where $d=\dim R$, $s=\dim S$. If $K_R$ is Cohen-Macaulay (i.e. $R$ is a ...
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### How to associate the following two kinds of real polynomials?

Suppose the following real polynomial of $n$ variables $$f(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}a_IX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$$ is easy or familiar to us, but I need to deal with ...
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that ...