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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A structure on the groupoid of algebraic closures
Given a field $k$ let $\Omega(k)$ be the set of algebraic closures of $k$.
$\Omega(k)$ is obviously a groupoid. At each element $\bar{k}$ of $\Omega(k)$ we have its automorphism group over $k$, ...
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When is a monomial rational map on the projective space birational?
Let $k$ be an algebraically closed field of characteristic $0$.
For $\alpha :=(a_1,\dots,a_{n+1})\in \mathbb N^{n+1}_{\ge 0}$ , let $\bar x^{\alpha}:= x_1^{a_1} \dots x_{n+1}^{a_{n+1}} \in k[x_1,\...
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Does any derivation of commutative algebra preserve its nil-radical?
Given a commutative associative unital algebra over a field of characteristic zero.
Is it true that any derivation of it preseves its nil-radical?
More explicitly, let $D$ be a derivation of an ...
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When are the cotangent and tangent sheaves isomorphic?
Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
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Chain of closed irreducible sets on Zariski Riemann spaces
Let $A$ be a domain and $K=\mathrm{Frac}(A)$.
The Zariski Riemann space $\mathrm{ZR}(K,A)$ is the set of all valuation rings of $K$ containing $A$. It comes with a natural center map
\begin{align}...
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Krull dimension of a local ring and completion
Let $A$ be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal $\mathfrak{m}$ is of finite type.
Let $\hat{A}$ be its $\mathfrak{m}$-adic completion.
Do we have that $\...
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Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
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Detecting elements of nilpotent extensions via finitely generated ones
Let $K$ be a commutative unital ring field. Let $\pi:A \to K$ be a surjective homomorphism of commutative $K$-algebras with nilpotent kernel. (Recall that this means $\operatorname{Ker}(\pi)^n=0$ for ...
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Localization of symmetric monoidal categories and geometry
I have a series of vague questions, related to localization of symmetric monoidal categories.
Here is the context. Say we are working over a field of characteristic zero. Then the "one category ...
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Classification of Frobenius algebras of small dimensions
Despite (commutative) Frobenius algebras over a field $K$ being a very popular class of algebraic objects, it seems no attempt of classification (up to $K$-algebra isomorphism) for them has been ...
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(Krull) dimension of any associated graded ring of a ring R equals the dimension of R
I am not sure if this is appropriate for MO. If not, I shall be happy to take it to SE.
For a local ring $(R,m)$, given any proper ideal $I$, the (Krull) dimension (from here on dimension means Krull ...
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Module structure for $\mathbb{Z}$
I am interested to know which module structures we can define in the additive group of integers $\mathbb{Z}$.
It is easy to prove that $\mathbb{Z}$ does not admit a vector space structure. (For more ...
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Are monomorphisms between algebraic spaces representable?
The question in the title can be reformulated as follows. Let $f : Y \to X$ be a monomorphism of algebraic spaces where $X$ is a scheme. Is it true that $Y$ is a scheme?
If $f$ is locally of finite ...
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Isomorphism of real closed fields
Given two real closed fields $R_1$ and $R_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the ...
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Does ACC for principal ideal plus Krull dimension equal 0 imply DCC for principal ideals
Assume the ring is commutative and with 1.
We know that ACC + $\dim(R)=0$ imply DCC. However, if we only insist on the condition for principal ideals, can we conclude the same?
We know that having ...
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Splitting subspaces and finite fields
Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = \{\theta\...
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How did Hilbert prove the Nullstellensatz?
All of the many proofs of the Nullstellensatz I have seen use results from long after Hilbert’s time: Zariski’s lemma, Noether normalization, the Rabinowitch trick, model theory, etc. How did Hilbert’...
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Polynomiality of functions over residue rings
Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...
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A commutative variant of the exterior algebra
Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = ...
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Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$
Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
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On an application of the going-down theorem of Cohen-Seidenberg in Mumford
There is a following result in Mumford's red book of schemes (Chapter II Section 8). Here $R$ is a valuation ring with algebraically closed fraction field $k$.
Let $Z \subset \mathbb{P}^n_k$ be an ...
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Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford
In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
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Non-normal domain with algebraically closed fraction field
I am looking for an integral domain $A$ with the following properties:
$A$ is not integrally closed
$A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
There is an ...
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Intersection of primes in a regular local ring
Suppose $R$ is a regular local ring of dimension at least 3, and suppose $P_1, P_2$ are dimension 1 primes. Does there necessarily exist a dimension 2 prime $Q$ contained in both?
In other words, is ...
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Does "finitely presented" mean "always finitely presented", considered in general
I'm wondering about the question
"If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?"
I know this is true for groups and ...
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Polynomial algebra and its special ideals
Consider a polynomial algebra $A=\mathbb{K}[x_1,\ldots,x_n]$ and its ideal $I$, such that $A/I=\mathbb{K}[y_1,\ldots, y_k]$. Is it true that there exist new polynomial generators $z_1,\ldots,z_n$ (in ...
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Is the restriction of a graded automorphism linearizable in characteristic zero?
This question follows up a previous one which was answered by Todd Leason. I want to impose two new requirements on the setup.
Let $k$ be a characteristic zero field. Let $A=k[x_1,\dots,x_n]$ be the ...
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Algebraic operation corresponding to "taking residues at roots of unity"
I'm not sure if this is more appropriate for MO or MSE, this is a question that came up in actual research, but it's of a somewhat elementary nature.
Let $R$ be the ring of rational functions with ...
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Tensor product of fields over integers
Inspired by this question we ask: Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$?
A field $K$ with the property that $K\otimes_{\...
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Stably triangular $\mathbb{G}_a$ actions
I try to solve the following problem:
For a polynomial algebra $\mathbb{K}^{[n]}=\mathbb{K}[x_1,\ldots , x_n]$ and its locally nilpotent derivation $D$, there exists an extension of $D$ to a ...
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Direct sum of injective modules over non-Noetherian rings
By the Bass-Papp theorem, if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists an injective module over $R$ non-Noetherian, that ...
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Is the perfection (perfect closure) presheaf a sheaf?
The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius. It does not commute with products, as was shown by YCor in ...
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The direct sum of injective modules need not be injective
The Bass-Papp Theorem asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective $R$-modules is injective. Thus every non-Noetherian ring carries a counterexample.
If
$$
I_1 ...
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Does perfection of rings commute with products?
The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius, so being a filtered colimit in Rings, the perfection ...
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Increasing the number of ideals in an exact sequence
In Broadmann and Sharp's book, Local Cohomology: An Algebraic Introduction with Geometric Applications, the exercise $3.2.4$ is about an exact sequence of the form $\DeclareMathOperator{\Hom}{Hom}$
$...
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When $I+J$ is a special indecomposable ideal of $R$
$\newcommand{\Ann}{\operatorname{Ann}}\newcommand{\Max}{\operatorname{Max}}$I am looking for an example of a commutatvive ring $R$ with $1$ having two ideals $I$ and $J$ such that $I\cap J\not=0$, $\...
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Henselization and completions of local rings & schemes
That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...
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Exactness of a certain sequence
Let $R$ be a commutative unitary ring and $I_1,..., I_n$ ideals in $R$. For each $p\in\{0,...,n-1\}$ consider the direct sums $\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}$ and define an $...
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Variants of Eisenstein irreducibility
In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
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Non-existence of projective covers
I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at:
http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf
In ...
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Partitioning a given set into root set of two polynomials or non zero set of two polynomials simultaneously
Let $p,q$ be two prime numbers and let $p<q.$ Also let, $\mathbb{Z}_p$ and $\mathbb{Z}_q$ denote the fields formed by integers modulo $p$ and modulo $q$ respectively (with respect to the modulo $p$ ...
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Galois group of a product of polynomials
How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
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Maximum cardinal of a set of linearly independent vectors in a module
A student asked me this, and I can't believe I never knew the answer to this.
Let $R$ be a commutative ring, and $M$ be an $R$-module.
If $M$ has a set of $n$ linearly independent vector for each $n\...
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The group of $k$-automorphisms of $k[x_1,\ldots,x_n,x_1^{-1}]$
Let $k$ be a field (of characteristic zero).
For $k[x_1,\dotsc,x_n]$ it is known that the affine and triangular automorphisms generate $G_n$, the group of automorphisms of $k[x_1,\dotsc,x_n]$,
see, ...
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Do there exist nonzero identically vanishing polynomials over infinite (or characteristic zero) reduced indecomposable commutative rings?
Let $R$ be an infinite, characteristic zero, commutative ring. I can furthermore suppose it is reduced and indecomposable (no nontrivial nilpotents or idempotents).
My question is whether there is a ...
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Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?
I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book
Greco, Silvio, ...
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Categorical view of Hilbert’s Nullstellensatz, and Zariski topology
Let k be algebraic closed field. then $\mathbb{A}_n(k)$ as $\operatorname{Hom}(k_n,k)$ and $V(\alpha)$ as $\operatorname{Hom}(k_n/\alpha,k)$ which is true by using noether normalization theorem. so ...
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Extensions of fraction field and residue field
Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$?
This question should be easy ...
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Low rank approximation
Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
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Compare degrees of a finite extension of domains and quotient domains
Let $A \subset B$ be a finite (finite type + integral) extension of integral domains and let $\mathfrak{p} \subset A, \mathfrak{q} \subset B$ be prime ideals such that $\mathfrak{q} \bigcap A =\...
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