# Questions tagged [ac.commutative-algebra]

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

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### Algorithm to compute the largest submodule of two given modules, containing a common submodule

Let $M_2\supset N\subset M_1$ be finitely generated modules, say over a polynomial ring. Here the inclusion just means an injection. There exists a maximal $K$ extending $N$, contained in both $M_1$ ...
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### Origin of the relations of Leavitt path algebras

I know the formal definition of Leavitt path algebras, but I want know why the relations defining Leavitt Path Algebras are defined in that way? what is special of this relations? My real hidden ...
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### Finite generation of kernel of derivations

Let $A$ be a finitely generated regular $k$-algebra, $k$ algebraically closed of characteristic zero, elements $x_1,\dots,x_n\in A$, such that $dx_1,\dots,dx_n$ give rise to a trivialization of the ...
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### Infinitely many initial ideals for non-Artinian monomial orders?

Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...
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### Lifts of smooth algebras

Let $(R, I)$ be a Henselian pair, with $I$ a finitely generated ideal. We know that for any smooth $R/I$-algebra $A_0$, there exists a smooth $R$-algebra $A$ such that $A/I\simeq A_0$. We also know ...
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### On the definition of regular (non-noetherian, commutative) rings

All rings are commutative with unit. A ring $R$ is called regular if it satisfies (Reg) Every finitely generated ideal of $R$ has finite projective dimension. Clearly this gives the usual ...
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### tensor and hom complex of complexes of graded modules

If $R$ is a commutative ring, $^\cdot,Y^\cdot$ are complexes of $R$ modules,then it is natural to define $X^\cdot \otimes ^\cdot Y^\cdot$ by the n-th component $\coprod_{i+j=n}X^i\otimes_R Y^j$ with ...
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### Contractible affine surfaces of log Kodaira dimension 2

The first examples of contractible smooth affine algebraic surfaces (over the complex numbers) of log Kodaira dimension 2 were constructed in a famous paper of Ramanujam https://www.jstor.org/stable/...
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### Ultrapower of a field is purely transcendental

Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$? According to Chapter VII, Exercise 3.6 from Barnes, Mack "...
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### Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
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### For a holonomic $D_X$-module $M$, can $\operatorname{gr}M$ have embedded primes?

Let $M$ be a holonomic $D_X$-module. This means that the minimal primes in $\sqrt{\operatorname{Ann}(\operatorname{gr}M)}$ are $n=\dim X$ dimensional, for some (and any) good filtration on $M$. But ...
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### Noether’s “set theoretic foundations” of algebra. Reference

In [C Mclarty] we read [Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra ...
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### If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a finite-...
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### Reference request: J-P Serre, “Groupes finis d'automorphismes d'anneaux locaux réguliers”

Does anyone have, or know a link to, a copy of the paper named in the title? It is published in Colloq. d'Alg. École Norm. de Jeunes Filles, Paris (1967), 1-11. I do not have ready access to Serre's ...
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### Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$. Let $L \subset \hat{K}$ be a separable finite ...
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### Fine tuning the growth rate of the degrees of polynomials

Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$. Suppose that \...
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### Factorizations of etale morphisms

Let $f:X \rightarrow Y$ be a finitely presented separated etale morphism, with $Y$ quasicompact and quasiseparated. By Zariski’s main theorem, we can factor $f$ as $f= g \circ j$ with $j$ an open ...
Let $k$ be an algebraically closed field of characteristic p. Let $Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in \$k[...