Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Linear algebra. commutative algebra, a matrix having the maximal rank
I found a matrix that looks to have the maximal rank for my research.
I would really appreciate if some of you could give me any comments or suggestions.
Thanks in advance.
Sincerely, Yong-Su Shin
Let ...
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Injective hulls of quotient rings $R/p$
Let $R$ be integral domain and $p \neq 0$ a prime ideal.
It's well known that in category of $R/p$ modules the injective
hull of $R/p$ is $K=\operatorname{Frac}(R/p)$.
Is there a successful theory ...
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periodic cyclic homology and tilting in the sense of Scholze
Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
- 9,283
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Generators of $SL(n,\mathbb F_2)$? [closed]
Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...
- 37.4k
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module of differential and Weil restriction
Let $k$ a commutative ring. Let $A$ be an $k$-algebra of finite presentation $A = k[\underline{X}] / \langle \underline{P} \rangle$ and $K / k$ free algebra of rank $r$.
There is a $k$-algebra $A \...
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Characterization of projective modules in terms of Ext groups
This is from Hartshrone exercise 6.6 part (a).
Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following
$M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{...
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Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example?
Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example?
If a ring has an involution f, then f is an anti-automorphism;...
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Regular morphisms and formal power series
Let $A$ be a local noetherian ring. When (besides when $A$ is excellent) do we have that $\operatorname{Spec}(A[[t]])\rightarrow \operatorname{Spec}(A[t])$ is regular?
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Module of Kahler differentials of rings of integers of number fields
How does one prove that if $L/K$ is an extension of number fields with rings of integers $B/A$, then the module of Kahler differentials $\Omega^1_{B/A}$ can be generated by one element as a $B$-module?...
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Strongly graded rings
In Theorem 3.1 of Graded rings over arithmetical orders, the authors prove that for a strongly $\mathbb{Z}$-graded ring $R$, if $R_0$ is left and right Goldie and a maximal order in its (classical) ...
- 323
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Weil restriction over integers
If we have a finite (possibly ramified) map of Dedekind domains $f:D\to D'$ and a finite type affine $D'$-scheme $X'$ is there a functorial way to produce a finite type affine $D$-scheme $X$ that ...
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A question regarding base change of a smooth algebra via completion
Let $(R,m)$ be an excellent Noetherian local ring. Let $S$ be a smooth (i.e. $R \rightarrow S$ is flat and has geometrically regular fibers) Noetherian $R$-algebra. Let $T$ be the $m S$-adic ...
- 1,802
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Under what conditions is the polynomial of degree $6$ irreducible?
Let $k$ be a perfect field of characteristic $p \neq 2,3$ such that $\omega := \sqrt[3]{1} \in k$, where $\omega \neq 1$. Consider an absolutely irreducible (not necessarily homogenous) quadratic ...
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Automorphisms of the ring of Laurent polynomials
Is the group of automorphisms of the ring $\mathbb{F}[t,t^{-1}]$ of Laurent polynomials known? Here, $\mathbb{F}$ is an algebraically closed field of characteristic $0$ and I am considering not ...
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Asymptotics of Hilbert series for locally finite free graded-commutative algebras?
Let $A^\bullet$ be an $\mathbb N$-graded algebra over a field $k$, and let $d_A(n) = \dim A^n$ be the dimension of the $n$-th graded piece, so that $P^A(t) = \sum_n d_A(n) t^n$ is the Hilbert-Poincare ...
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Projective dimension of a sub-ideal
Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
- 30.5k
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When does a faithful module have an element with zero annihilator?
This is a follow up of the question Example of a finitely generated faithful torsion module over a commutative ring on MathSE.
Let $M$ be a finitely generated module over a commutative ring $R$ with ...
- 63.9k
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On the annihilator of a module
Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such
that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$?
Remark. The annihilator of a ...
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385
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Vector bundles on complete rings
Given a ring $A$ and an ideal $I$, consider the completion $\hat{A}$. What does usually mean by a vector bundle on $\hat{A}$? One way is to consider projective $\hat{A}$-modules. Another one is a ...
- 2,607
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Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind
Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension,
and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$.
Then $^\...
- 14.2k
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Computations of divisor class monoids
Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".
Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
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On resolution of singularities over an Artin ring
For a locally noetherian scheme $X$, Grothendieck conjectured that if $X$ is quasi-excellent then there is a proper birational map $Y \to X$ s.t. $Y$ is regular.
We now fix an Artin ring $R$ whose ...
- 66.3k
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Methods for multivariate polynomial equations over large finite fields
I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...
- 3,065
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560
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Filtration over tensor product
Let
$$ M \supset M_1 \supset \ldots \supset M_n \supset \ldots \text{ and } N \supset N_1 \supset \ldots \supset N_n \supset \ldots$$
be exhaustive decreasing filtrations of modules over a commutative ...
- 385
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336
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Intersections of strict transform and strict transform of intersections
Let $Z_1,Z_2$ and $Y$ be subvarieties of a locally complete intersection variety $X$ over $\mathbb C$.
Consider the strict transforms of $Z_1$ and $Z_2$ in the blowup $Bl_YX$, the question is: when ...
- 31
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on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
- 30.5k
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In a commutative Koszul algebra, does every ideal generated by a subset of variables have linear resolution?
Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset ...
- 553
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The Kronecker--Hurwitz property for rings of integers in global function fields
In Ireland and Rosen's book on number theory they give a proof of the finiteness of the class group of a number field which they attribute to Hurwitz, but which is essentially due to Kronecker (as I ...
- 3,823
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Extending finitely many primitive elements of $\mathbb{Z}^{n+1}$ to bases with non-trivial intersection
Given a finite number of primitive elements $v_1,\dots,v_k\in\mathbb{Z}^{n+1}$ (i.e. the gcd of the entries of each $v_i$ is $\pm1$), is it always possible to find an element $v\in\mathbb{Z}^{n+1}$ ...
- 291
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Normal Macaulayfications
Given any reduced excellent scheme $X$, there exists a Macaulayfication $\pi : Y \to X$. In other words, there exists a proper birational map $\pi$ from a Cohen-Macaulay scheme $Y$ to $X$. These ...
- 3,065
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Ring isomorphism of multivariate polynomials/functions
It's well-known that over an infinite integral domain $R$, the ring of univariate polynomials $R\left[X_{1}\right]$ is isomorphic to a ring of one-argument "polynomial functions" (see, for ...
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(Infinite) free resolution of $R/(x-z, y-w)$ for $R=\mathbb C[x,y,z,w]/(xy-zw)$
For a Noetherian local ring $R$, Koszul complex is a useful tool to construct finite free resolution of any quotient of $R$ by regular sequences $R/(x_1,\dots, x_r)$.
Let $R=\mathbb C[x,y,z,w]/(xy-zw)$...
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Euler characteristic and rational Poincaré series
$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-...
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Two conjectures by Gabber on Brauer and Picard groups
In a paper I need to make reference to two conjectures by Gabber, from
Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37
...
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Wild ramification in composite fields
Let $ F $, $ F_i $ for $ i = 1, 2 $ and $ E $ be function fields over a finite field with characteristic $ p $ such that $ F \subseteq F_i \subseteq E $ and $ E = F_1 \cdot F_2 $ is the composite ...
- 3,065
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Algebraically closed ring extension
Suppose that $B \rightarrow A$ is an extension of rings where $A$ and $B$ are integral $k$-algebras ($\mathrm{char}\,k = 0$) finitely generated over $k$.
It is well known that if $B \rightarrow A$ is ...
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How general are Gröbner degenerations?
While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...
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Do commutative rings with "interesting" Jacobson radicals turn up "in nature"?
Let $R$ be a commutative ring. Let's say that the Jacobson radical $J(R)$ of $R$ is uninteresting if
$J(R)$ coincides with the nilradical, or
$J(R)$ is the intersection of a finite number of maximal ...
CommunityBot
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Lci local rings with isolated singularity are irreducible?
Let $R$ be a noetherian local ring; I say it has isolated singularity if its spectrum is regular outside the closed point. Such rings certainly don't need to be irreducible, for example the ...
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To show equivalence and full faithfulness of a functor PRESERVED under an action of a finite flat algebra
I have explained the two questions and then showed my effort on question $(1)$ as follows (Please at least check my effort below and suggest to make it perfect):
Let $R, S,T$ be three commutative ...
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Existence of a global completion functor
Is there any such thing as a ``global completion functor'' for commutative Noetherian rings?
More specifically, let $R$ be a commutative Noetherian ring. For each maximal ideal $m$ of $R$, let $W_m :=...
- 1,802
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670
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Derivation of formal power series
The basic idea of this question is to see if there is any other derivations than 'formal derivations'.
Let $\mathbb{K}$ be a field. Given a commutative $\mathbb{K}$-algebra $A$, a derivation of $A$ is ...
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How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?
I am describing the question details, though the main question is short as below.
Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $...
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When are classes with prescribed reducts "pseudo"-elementary?
Let $\mathsf{Set}$ be the class of all sets and let $\mathcal{L}$ be a first-order language. Let $M \subseteq \mathsf{Set}$ be a set of $\mathcal{L}$-structures and let $$\mathfrak{Th}_{\in}(M) = \\ \{...
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An example of two elements without a greatest common divisor
Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.
"Easy" means that I can explain it to my ...
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Ascend and descend properties for arithmetically Cohen-Macaulay/Gorenstein varieties
I had few questions regarding varieties admitting embeddings that make them arithmetically Cohen-Macaulay or Gorenstein varieties. A projective variety is called arithmatically Cohen-Macaulay/...
- 5,901
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1
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85
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Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$
The answer to this MO question says the following:
Lemma 1. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ ...
- 19.6k
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Is a tower of locally-free modules locally a tower of free modules?
Suppose we have a (commutative, unital) ring $R$ and a (commutative, unital) $R$-algebra $A$ such that $A$ is projective of constant rank $n$ as an $R$-module. This condition is equivalent to there ...
- 23.4k
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Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
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Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...
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