Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Explicit Bézout cofactors
$\DeclareMathOperator\lcm{lcm}$This is a rather severe revision of a question I asked recently. We know over the integers that $\gcd(a^2,b^2)=\gcd(a,b)^2$. We might prove this via unique factorization....
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Do power sums determine the variables?
In my analysis research, I came across the following problem. Given $n$ positive real numbers $x_1,\dots,x_n$, consider the $n$-many power sums
$$ p_3 = x_1^3 + x_2^3 + \dots + x_n^3 , $$
$$ p_5 = ...
- 12.9k
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Finding generators and relations for special commutative algebras with a computer
Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$.
Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
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A criterion for whether an ideal is contained in a principal ideal
Let $R$ be a ring and $I$ an ideal. I am interested under which conditions the following holds:
Claim. Suppose that any two elements in $I$ have a non-trivial $\operatorname{gcd}$. Then $I$ is ...
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Ring structure of coinvariant of $W(U(4))$
I want to know the ring structure of the coinvariant of $W(U(4))$, where $W(G)$ is the Weyl group of G.
I know that the ring structure of the coinvariant of $W(U(3))$ is $\mathbb{Z}[x_1,x_2,x_3]$ with ...
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Idempotents in group rings of finite cyclic groups
For which fields $K$ and integers $n>1$ does the group ring $K(\mathbb{Z}/n\mathbb{Z})$ have idempotents distinct from $0$ and $1$?
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Finding étale slices
I'm trying to understand Luna's étale slice theorem by computing some examples. The theorem is usually phrased as an existence result. I wondered if there was a natural way to figure out the slice at ...
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Structure theorem for finitely generated $\Lambda$-modules - uniqueness part
In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}_p[[T]]$.
If $M$ is a finitely generated torsion ...
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Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
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What algebraic condition corresponds to injectivity of a morphism of varieties?
$\DeclareMathOperator{\Spec}{Spec}$
Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are ...
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Order 2 matrices with entries in the polynomial ring over a field are diagonalisable
This is a variant on the question posed here, in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}_n(A)$, where $A$ is a $k$-algebra for some field $k$ ...
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Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
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Is every matrix involution over a UFD diagonalisable?
Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}_n(A)$ diagonalisable?
This is of ...
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Localization of Bernstein center
Let $C$ be a $k$-linear category ($k$ an algebraically closed field) and $Z$ its Bernstein center (the ring of endomorphisms of the identity functor of $C$).
Are there natural assumptions that ...
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Split monomorphisms of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...
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Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals?
Excuse my naive question and please let me explain it:
In everyday life we experience 3 spatial "dimensions" + time etc.
Usually the 3 dimensions are represented by a coordinate system and ...
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Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?
Consider the hierarchy of relative geometric constructibility by
straightedge and compass. Namely, given a geometric figure $B$, a
set of points in the plane, we define that geometric figure $A$ is
...
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A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$
If $H = S_n$ then then the fundamental symmetric polynomials allow to write any $S_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\...
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Radical of an ideal in the polynomial ring with reducible generators
To find the radical of an ideal can be a very complicate task. Considering ideals in the polynomial ring, I am wondering if this task can be simplified in the case the generators of the ideal have the ...
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For any initial ideal $I$ of the ideal of maximal minors, is it true that $I^n = I^{(n)}$?
Let $X$ denote a generic $n \times m$ (with $n \leq m$) matrix and $R = k[X]$, where $k$ is any field. Let $J := I_n (X)$. It is well-known that $J^t = J^{(t)}$ for all $t$ (where $-^{(t)}$ denotes ...
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Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
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Symbolic powers of a prime ideal of height one
Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal ...
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Flatness of finitely presented algebras
Let $R$ be a commutative (noetherian, if needed) ring, let $f_1，\ldots，f_r\in R[x_1，\ldots，x_n]$ and $A=R[x_1，…，x_n]/(f_1，\ldots，f_r)$, when is $A$ flat over $R$?
I found a nice answer for the case $n=...
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Easy way to understand theta basis for X-cluster algebras of finite type?
For $\mathcal A$-cluster algebras of finite type, it is very easy to describe the theta-basis: it consists of the cluster monomials. Is there any similarly easy way to describe the theta-basis for $\...
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Characterized maximal ideal [closed]
$\DeclareMathOperator\Alg{Alg}$Let $A$ be a commutative associative algebra with $1$ over $\mathbb{C}$. We define $\Alg(A,\mathbb{C}) $ to be the set of $\mathbb{C}$-algebra maps from $A$ to $\mathbb{...
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What does "linearly disjoint" mean for abstract field extensions?
All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
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Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules
Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
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Restricting maps between strict henselisations
$\require{AMScd}$I am currently thinking about (strict) henselisations but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict ...
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$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve
Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
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Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness ...
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Why doesn't local cohomology seem to be used as much in algebraic geometry?
In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations.
In algebraic geometry, we have local cohomology,...
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Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$
Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...
- 63.9k
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The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
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Alternative definitions of étale and formally unramified in Wraith
I have stumbled upon the following definitions in a paper by Gavin Wraith.
Definition 1. Say a ring morphism $A\overset \varphi \to B$ is formally unramified if every $b\in B$ admits:
$b_0\in B$ ...
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Image of multiplication map in tensor powers of finite-dimensional ring
Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$.
Then $R^{\otimes n}$ has a natural ring structure, together with an $...
- 149k
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Separable algebras and separably closed local rings (a.k.a strictly Henselian local rings)
Let $A$ be a local ring. Say a monic $f\in A[x]$ is unramifiable if it admits a simple root in its universal splitting algebra (equivalently, it admits a simple root in any ring over which it splits). ...
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Can any countably generated k-algebra occur as the ring of global sections of some variety?
In the answer to this question we saw that there exists a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.
I was talking about this previous ...
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Localization of the injective hull of a commutative non-Noetherian ring
Let $R$ be a commutative non-Noetherian ring and $m$ a maximal ideal. My question is whether the localization $E(R)_m$ of the injective hull $E(R)$ of $R$ is an injective $R_m$-module. This is true in ...
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Constant term extraction using combinatorial Nullstellensatz
$\DeclareMathOperator\CT{CT}$Given a Laurent polynomial $g$, let $\CT(g)$ denote its constant term.
Consider the specific Laurent polynomial
$$f_n(x_1,\dots,x_r)=\left(1+\prod_{j=1}^r(1+x_j)+\prod_{j=...
- 109k
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Existence of a finite resolution
I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance.
Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...
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Is being finitely generated module a local property?
There is this result on stack project, saying that let $S$ be a $R$-module and $f_1,...,f_n \in R$ that generates $R$, if $S_{f_i}$ is finitely generated $R_{f_i}$-module then $S$ is a finitely ...
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Quotients and associated graded
$\DeclareMathOperator\gr{gr}$Let $A = \cup_{i=0}^\infty F_i A$ be a filtered commutative ring, $I \subseteq A$ an ideal. Then we have a canonical surjection
$$ \gr(A)/\gr(I) \to \gr(A/I).$$
Under what ...
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Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?
Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
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When the annihilator of each nonzero submodule is prime
Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...
- 21
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Quasi-compact surjective morphism of smooth k-schemes is flat
I have precedently posted the same question on Math.Stackexchange (https://math.stackexchange.com/questions/4277856/quasi-compact-surjective-morphism-of-smooth-k-schemes-is-flat), but to no avail; I ...
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A "spectral theorem" to SVD reduction for every commutative *-ring
Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is ...
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Does irreducible polynomial remain reduced by pre-composition?
Let $f(x),g(x)$ be polynomials in $\mathbb{Q}[x]$. If $\mathrm{deg}(f)\geq2$ and $f$ irreducible, is the composition $f(g(x))$ always reduced (has no repeated irreducible factors)?
(If we do not ask $\...
- 109k
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Hensel's lemma, Bezout's identity, and the integers
Factorization in the ring $\mathbb{Z}[x]/(x^2+1)\mathbb{Z}[x]\cong \mathbb{Z}[i]$ is well known. For instance, $5$ and $13$ (and any prime $\equiv 1\pmod{4}$) are no longer prime.
The factorization ...
- 18.7k
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Concept of an exact ideal of a module category
Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
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Is a torsion-free abelian group finitely generated, if all of its localizations at primes $p$ are finitely generated over $\mathbb{Z}_p$?
Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}_\ell\otimes_{\mathbb{...
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