# Tagged Questions

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

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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 r-...
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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 ...
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### grade of ideals in non-noetherian rings

Let $R$ be a commutative ring with unity, and $M$ an $R$-module. Assume that $I$ and $J$ are finitely generated ideals and $K$ another ideal of $R$. Let $\textbf{x}$ be a sequence of generators of $I$...
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### Ring of invariants for the regular representation

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
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### Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...
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### Algebraic independance of exponentials

First of all, a happy new year. Be it better than 2015, healthy, wealthy, fruitful and cross-fertilizing for you, familly and friends. In order to cope with families of solutions of evolution ...
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### Base change for non-flat coherent sheaves and affine maps

Let $A$ be a finitely generated $k$-algebra, where $k$ is a field, let $I$ be an ideal in $A$, let $M$ be a finitely generated $A/I$-module, and let $M^{\prime}$ denote $M$ considered as an $A$-module....
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### Division and multiplication that preserve Euclidean norms

I am looking for ways to define $$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$ where $x,y\in \mathbb{R}^n$ such that \left\|\frac{1}{x}\right\|=\frac{1}{\|...
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### Generalizing Dedekind's Factorization Theorem

A classical theorem due to Dedekind states the following: Let $O_{K}$ be the ring of integers of a number field $K$, and assume $K$ is generated by adjoining the algebraic integer $\alpha$ to ...
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### When does 'Zariski tangent space derivative' vanishes everywhere imply that a section is constant?

Consider an abelian algebra, $R$, over the field $K$ with the properties that every residue field of $R$ is (canonically) isomorphic to $K$ (I'm not sure but I think this is necessary, otherwise we ...
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### What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
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### Looking for a reference in commutative algebra

I need "I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43." in my research, but it seems to be very old and rare. Does anyone know a site for ...
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### Reference request for $R$-index

Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
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### Algebraic independence criterion

Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
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### Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$

I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...
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### Cubic, divisor of rational function $x/z$? [closed]

Let $k$ be a field, and let $a \neq 0$, $1 \in k$. Let $C = V(y^2z - x(x-z)(x - az))$. What is the divisor of the rational function $\psi([x, y, z]) = x/z \in k(C)$?
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### Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$). Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
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### Theorem 16 , Chapter 5 of Northcott 's, Finite Free Resolutions: p.grade

Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has: and in Theorem 16 of Chapter 5 proves that: $p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...
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### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here. Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...
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### Examples of Noetherian overkill

I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite ...
### Assuming $depth M\ge depth N$, what can one say about $depth M_p$ and $depth N_p$?
Let $(R,m)$ be a Noetherian local ring􀀀, $M$ and $N$ finite $R$-modules, $p$ a prime ideal,􀀀 and $I$ an ideal such that $IM\neq M$. Definition: The common length of the maximal $M$-sequences in $I$...