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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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      141 views

      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 "...
      1
      vote
      1answer
      104 views

      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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      votes
      1answer
      81 views

      Are integral extensions of a catenary ring still catenary?

      A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\...
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      votes
      1answer
      127 views

      Complete local rings, automorphisms and approximation

      Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective. Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...
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      votes
      0answers
      99 views

      Does a surjection between polynomial rings lose one Krull dimension per generator quotiented?

      Let $k$ be a Noetherian commutative ring with unity (I am happy adding hypotheses, as I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a ...
      2
      votes
      1answer
      107 views

      Are the ring of power series and the ring of germs of holomorphic functions catenary?

      A (commutative unitary) finite dimensional Noetherian ring $R$ is said to be catenary if for any prime ideal $\mathfrak{p}$ of $R$, one has $\dim R =ht(\mathfrak{p})+\dim(R/\mathfrak{p})$. I am ...
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      votes
      0answers
      139 views

      What is wrong with this argument that $ \mathbb{A}^{2}_{k} $ is not cancellative in positive characteristic?

      I have a question about a result of Abyankar, Heinzer, and Eakin, and a similar result in Russell. One of the results in the first paper is that if $ Y $ is a variety such that $ \mathbb{A}^{1}_{k} \...
      3
      votes
      0answers
      45 views

      Ideals generalizing maximal ideals and ideals generated by regular sequences

      Let $R$ be a local commutative Noetherian ring with maximal ideal $m$. My questions concern ideals $I \subseteq m$ of $R$ such that for any non-zero number $n \in \mathbb{N}$ the $R/I$-module $I^n/I^...
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      votes
      0answers
      150 views

      Reconstructing almost known polynomial from a system of polynomials with common roots

      We have $n$ algebraically independent degree $2$ homogeneous system of polynomials with $\mathbb Z$ coefficients in $n$ variables with exactly $t$ primitive (gcd of coefficients is $1$) integer roots ...
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      votes
      2answers
      458 views

      Relations between homogeneous polynomials

      Edit: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation: Let $\mathbb{P}$ be a projective space, and $V$ a general linear subspace ...
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      vote
      0answers
      125 views

      Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

      Let $R$ be a domain and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
      7
      votes
      1answer
      173 views

      Equivalence of definitions of Cohen-Macaulay type

      I know that the Cohen-Macaulay type has this two definitions: Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^...
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      votes
      0answers
      87 views

      Invariants of linear endomorphisms of tensor products

      Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero. Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
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      votes
      0answers
      90 views

      Standard reference/name for “initial ideals $\Leftrightarrow$ associated graded rings”

      Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...
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      votes
      1answer
      98 views

      Symmetric polynomials in two sets of variables

      Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...

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