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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Endomorphisms of Artinian modules
The following claim is from a paper [On the moduli spaces of bundles on K3 surfaces, I, p. 358] of Mukai. Consider an artinian module $\mathrm{M}$ over a local ring, and let $\mathrm{M}_0$ be the ...
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What is the largest subcategory $C$ of a module category over an Artin algebra, such that $C$ is Krull-Schmidt (and abelian)? Does $C$ exist?
Let $A$ be an Artin algebra, $\text{Mod}\,A$ the category of $A$-modules and $\text{mod}\,A$ the category of finitely generated $A$-modules. It is well-known that $\text{mod}\,A$ is a Krull-Schmidt ...
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Understanding full set of sections as in Katz-Mazur
I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:
Being a "full set of sections" of $Z/S$ is something which is ...
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When every proper submodule of a free module is contained in a maximal submodule
Let $(R, m)$ be a commutative local ring (it is not Noetherian in general) and $F$ be a free $R$-module. Under what conditions every proper submodule of $F$ is contained in a maximal submodule.
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finite type k-subalgebras of k[[t]]
in the middle of working on some deformation related questions, I bumped into a question regarding finite type subalgebras of some completed algebras. I hope someone may know the answer, or may ...
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Gelfand ring in Bourbaki's exercises
In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that
The set $A^*$ ($=A^{-1}$) of invertibles is open.
The uniform structure ...
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How to compute the intersection of an ideal with the maximal order of a subfield?
I asked this earlier on math.stackexchange but I think this is a better place for this question.
Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be ...
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Does perfect fraction field imply perfect residue field?
Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect?
Thoughts:
If $A$ is ...
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Set of primes $p$ such that $\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset$
For which sets of primes $P$ is there a finite type $\mathbb{Z}$-algebra $A$ such that$$p\in P\iff\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset?$$Do all the finite $P$ arise this way?
$A=\mathbb{Z}/n$ works ...
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Why is this $ \mathbb{G}_{a} $ bundle trivial
Please tell me why the following example of a principal $ \mathbb{G}_{a} $-bundle over an affine ring is trivial. Let $ \{x_{1},x_{2},x_{3}\} $ a basis of $ \mathbf{V}^{\ast} $, $ c_{1}(t),c_{2}(t) $ ...
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Example request: seriously deficient homogeneous spaces
In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
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Existence of a translation-invariant basis of $\ell^2$
This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural).
I give four equivalent formulations of the same ...
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When are dual modules free?
Let $A$ be a commutative integral domain, with fraction field $K$. Let $T$ be a torsion-free finitely generated $A$ module, so $T \otimes_A K$ is a finite dimensional vector space $V$. Let $T^*$ be ...
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Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
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Are finitely presented algebras over VNRs projective?
Question: Let $A$ be a commutative von Neumann regular ring, and $B$ an $A$-algebra of finite presentation, i.e. $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Is $B$ a projective $A$-module?
In the ...
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Is every Dedekind domain the integral closure of some principal ideal domain?
I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ ...
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Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient
Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $$\sup\{k \mid R \text{ contains a direct sum of $k$ nonzero ideals }\}=\infty.$$ How can we ...
CommunityBot
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Completion of a local ring of a curve
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
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Question about a proof in Berthelot's crystalline book
Below is an excerpt from Berthelot's book on crystalline cohomology. I don't understand the last sentence, namely why it follows that $\sigma\circ \varepsilon$ is an isomorphism. For what it's worth, $...
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Ascent for projective modules
Let $A \subseteq B \subseteq C$ be commutative rings such that: $(1)$ $B$ is a projective $A$-module $(2)$ $C$ is a finitely presented $B$-module and $(3)$ $C$ is a flat $A$-module.
Does it follow ...
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Systems of regular parameters of a regular local ring
Let $R$ be a Noetherian regular local ring of dimension $n$ with maximal ideal $\mathfrak{m}$. Given two systems of regular parameters $\vec{u}=\left<u_1, \dots, u_n\right>$ and $\vec{v}=\left&...
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Sufficient conditions for $b\not\in I^2$ given that $b\in I$
Let $I$ be an $R$-ideal in a commutative algebra $B$ over a commutative ring $R.$ Given $b\in I$ I want to prove that $b\not \in I^2$.
Are there any sufficient conditions for showing that $b\not\in I^...
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Growth of dimension of a good filtration on finitely generated modules over polynomial rings
Let $S=K[x_1,\cdots,x_s]$ be a polynomial ring over a characteristic zero field $K$.
For a partition $\{1, \cdots, s\} = I_1 \sqcup \cdots \sqcup I_r$ of the variable index, let $\mathbb{x}_i^{\mathbb{...
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Smooth normalization and blow-up of the exceptional locus
Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\...
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Socle of a quotient of the ring of differential operators of a polynomial ring
I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...
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Künneth formula for de Rham cohomology with respect to an integrable connection
I am reading through https://stacks.math.columbia.edu/tag/0FM9 which proves that for $X,Y$ schemes over some base $S$ and $X \times _S Y \overset{p}{\rightarrow} X$ resp. $X \times _S Y \overset{q}{\...
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On the use of the fundamental exact sequence of K\"ahler differentials in a paper of Lyubeznik
Let $k$ be a field, $R := k[x_1, \cdots , x_n]$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading,...
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Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal
Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
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Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
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"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
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Flatness over a local noetherian ring
Let $(R,\mathfrak m)$ be a local noetherian ring, and $M$ an arbitrary $R$-module. Suppose that $\mathrm{Tor}_1(M,R/\mathfrak m)=0$. Does it follow that $M$ is flat?
The answer is positive when $M$ ...
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Graded-irreducible ideals are irreducible?
One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals?
Let $I$ be a graded ideal in a polynomial ring over a field....
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For a Cohen-Macaulay module $M$ of dimension $t$ over a local CM ring of dimension $n$, is $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay?
Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&...
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Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?
Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more ...
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Non-isomorphic smooth affine varieties dominating each other
Can non-isomorphic smooth affine varieties dominate each other?
In the projective case one can take isogenous abelian varieties.
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Can the notion of algebraic closedness be generalized to the rings with zero divisors?
Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero?
For instance, it seems that any polynomial of non-zero-divisor-...
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Are the rings $\mathbb{C}[X]/\langle X^2- c \mathrm{Tr}(X) X \rangle$ isomorphic when $c$ ranges over a neighborhood of 0?
Let $X=(X_{IJ})_{I,J=1,\ldots,N}$ be a family of $N^2$ indeterminates and consider the ring $$
R_{N,c}=\mathbb{C}[X] / J_c,\quad J_c=\langle X^2 -c \mathrm{Tr}(X) X \rangle .
$$
Here the notation ...
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Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $...
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Topologically finitely generated non-abelian isomorphic absolute Galois groups
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are topologically finitely generated, non-abelian and ...
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Non-abelian isomorphic absolute Galois groups of fields of different characteristic
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ ...
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Characteristic zero field with unique regular Noetherian dense unital subring
Let $F$ be a field. Call a unital subring $R\subset F$ dense if there is no subfield $K\subsetneq F$ such that $R\subset K$.
Is there a characteristic zero field $F$ such that the only regular ...
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Is there a notion of polynomial ring in "one half variable"?
Let $C$ be the category of commutative rings.
Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ?
(Here, we may assume those isomorphisms to be natural ...
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An invariance property of rational singularities
Let $X$ be a normal variety over a field of characteristic zero with rational singularities.
If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational ...
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Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?
The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.
Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
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"Formally unramified iff trivial Kähler differentials" using only universal properties?
For convenience we work with commutative rings instead of commutative algebras.
Fix a commutative ring $R$. Consider the functor $\mathsf{Mod}\longrightarrow \mathsf{CRing}$ defined by taking an $R$-...
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On the functors $\text{Hom}_R(k,-)$ and $k \otimes_R ( -)$ for Artinian local Gorenstein ring $R$
Let $(R, \mathfrak m,k)$ be an Artinian local Gorenstein ring, hence $\text{Hom}_R(k, R)\cong k$, and so
$\text{Hom}_R(k, R^{\oplus n})\cong k^{\oplus n} \cong k \otimes_R R^{\oplus n} , \forall n \ge ...
- 149k
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Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular
Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...
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2
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Projective modules over semi-local rings
Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.
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Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 12.9k
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1
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When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?
By a result of Klein-Nagata rings of the form $A_Q=K[x_1,...,x_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form.
Question 1: When is $A_Q$ a principal ...
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