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# 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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### Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...
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### What generalizes symmetric polynomials to other finite groups?

Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...
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### Automorphisms of rational functions of two variables

Let $k$ be a field. In 1941, Jung showed that all polynomial $k$-algebra automorphisms of the rational (polynomial) functions in two variables, denoted by $k(x,y)$ can be written as compositions of ...
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### Eventually non vanishing tors

Let $A$ be a commutative $k$-algebra, for $k$ a field of characteristic $0$. Let $Perf_{A}$ denote the dg category of cohomologically graded $A$-modules and let $M\in Perf_{A}$ be a classical perfect ...
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Let $f:A \to B$ be a finite flat local homomorphism of noetherian local rings. Are there some nice conditions on $A$ and $B$ which guarantee that the dimension of the Zariski tangent space of $... 1answer 125 views ### Condition for a monomial to belong to a particular ideal Consider the polynomial ring$R[x_1,x_2,\ldots,x_n]$, where$R$be an algebraically closed field (preferably$\mathbb{C}$) and the ideal$J=\langle m_1, m_2,\ldots,m_n\rangle$generated by monomials . ... 0answers 65 views ### Surjective maps between power series rings Suppose that$A$is a complete neotherian local ring and that we're given a surjective homomorphism$f: A[[x_{1}, \ldots, x_{n}]] \rightarrow A[[t_{1}, \ldots, t_{m}]]$. Can we always find a ... 0answers 58 views ### Infinitely generated modules with weirdly jumping ranks Assume we have a commutative Noetherian ring$R$with a unit and a connected spectrum and a module$M$over it. The following is known:$\mathrm{Spec}(R)\$ has a finite stratification (in the sense of ...

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