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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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      5
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      0answers
      92 views

      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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      1answer
      153 views

      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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      73 views

      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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      64 views

      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 ...
      1
      vote
      1answer
      66 views

      Jacobson radical of a derived $I$-complete ring

      Let $A$ be a commutative ring and $I \subseteq A$ a finitely generated ideal (I am not assuming that $A$ is Noetherian). Assume that $A$ is derived $I$-complete, meaning, let's say, that $\mathrm{...
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      0answers
      100 views

      Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

      All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
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      109 views

      When is a maximal ideal isolated?

      Let $R$ be a commutative ring with 1, and let $0\not=a$ be an element of $R$ such that there exists an ideal $I$ of $R$ with the property that $a\not\in I$ and if $J$ is an ideal with $a\not\in J$ ...
      3
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      0answers
      141 views

      $\mathcal{C}$-filtering of modules inherited by submodules

      I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology. DEFINITION: Let $\mathcal{C}$ be a ...
      2
      votes
      0answers
      145 views

      Induced morphism of completions of local rings

      Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced ...
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      0answers
      79 views

      Nice forms for the Weierstrass equation

      The motivation for this question comes from the study of elliptic curves but the question itself is about choosing convenient algebraic transformations. Fix a base ring (commutative, with a ...
      3
      votes
      1answer
      121 views

      A local ring with a unique minimal ideal

      Let $R$ be a commutative ring with 1 such that it is local with a unique non-idempotent minimal ideal, that is, there are two ideals $I$ and $m$ of $R$ such that for each ideal $K$ of $R$ with $0\not=...
      6
      votes
      1answer
      235 views

      Flat maps and Zariski tangent spaces

      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 $...
      4
      votes
      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 . ...
      3
      votes
      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 ...
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      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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