Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,495 questions
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Are polynomial algebras over fields (that are not algebraically closed) tame?
Let $A$ be an algebra over a field $K$. Loosely speaking, an algebra is said to be tame if for each $d \in \mathbb{Z}_{>0}$ all but finitely-many of the indecomposable $A$-modules of $K$-dimension $...
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Zeros of higher Ext functors
I'm trying to understand module classes that are defined as the kernels of higher Ext functors (e.g., arising here; as this paper suggests, I'm coming at this problem outside of module theory). One ...
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The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?
In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...
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Flatness of certain subrings
The following question appears, more or less, here:
Let $k$ be an algebraically closed field of characteristic zero and let $S$ be a commutative $k$-algebra
(I do not mind to further assume that $S$ ...
- 28.7k
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Ring in which $x^n-x$ is central for every $x$
Let $R$ be a ring , $n \gt 1$, such that for all $x \in R$: $x^n-x \in Z(R)$, the center of $R$. Does it follow that $R$ is commutative?
For $n=2,3$ this is pretty straightforward to prove. But what ...
- 16.2k
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List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
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Is diagonalizability a local property?
Let $R$ be a commutative unital ring. Let $n\in\mathbb{N}_+$. Consider a $n\times n$-matrix $A=(a_{ij})_{i,j=1}^n$ with entries in $R$. A diagonal matrix is defined obviously. We can define several ...
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Krull dimension of the smooth locus
Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
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Finite subschemes of projective bundles
$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the ...
CommunityBot
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Derivations of differential operators
For a smooth affine variety $\operatorname{Spec} A$ over a ring $R$ we have an algebra of differential operators $\mathcal{D}_A$ (here I mean not the Grothendieck differential operators but PD-ones). ...
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What is the intersection of all ideals whose radicals are prime?
Let a fuzzy prime be an ideal of a commutative unital ring whose radical is prime (I'm not sure if this kind of ideal already has a name). Is the intersection of all fuzzy primes $\{0\}$?
Note this is ...
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Finite number of minimal ideals
What is the necessary condition on a ring that guarantees the number of minimal non-zero ideals to be finite? Neither Noetherian or Artinian condition seems sufficient, and the ring being semisimple ...
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Does going down property imply a corresponding map is open without "finiteness"?
Does the following proposition hold?
Proposition
Let f:A$\rightarrow$B be a ring homomorphism
If f has going down property then the corresponding map
$f^*$:Spec B$\rightarrow$Spec A is open map.
I ...
CommunityBot
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A question about surjective maps between quadratic algebras
Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...
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Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?
The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...
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flatness and exact sequences
Let $R$ be a commutative ring (with unit). Then if
$$0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$$
is an exact sequence of $R$-modules, with $M''$ $R$-flat, $M$ is flat if ...
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Sampling theorems for partition polynomials (associahedra, noncrossing partitions / parking functions)
Define the associahedra partition polynomial
$$
\begin{split}
A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\
& \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) ...
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Symmetric powers, localisation and Frobenius
I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
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Is the spectrum of this ring Noetherian?
Let $R = \mathbb{Z}/2\mathbb{Z}[X,X^{1/2},X^{1/4},\cdots,X^{1/{2^n}},\cdots]$
Is $\operatorname{Spec}R$ a Noetherian topological space?
Here is what I know.
$R$ is integral over $\mathbb{Z}/2\mathbb{...
- 63.9k
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A game on Noetherian rings
A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...
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The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings
I am looking for further proofs, preferably in the literature, of the following result:
Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in ...
- 7,127
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156
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Absolute integral closure of Noetherian local domain
Let $(R,\mathfrak{m})$ be a Noetherian local domain, let $K(R)$ be the total fraction field of $R$ and let $R^{+}$ be the absolute integral closure of $R$. Consider the $R^{+}$-module $K(R)\otimes_{R} ...
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Selforthogonal modules over Artinian Gorenstein rings
Let $R$ be a local artinian Gorenstein ring and $M$ a finitely generated $R$-module, then
$\mathrm{Ext}_R^1(M,M) = 0$ if only if $M$ is projective?
- 63.9k
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Is a complete local ring determined by its values in local fields?
Let $A$ be a complete, Noetherian, local ring with finite residue field of characteristic $p$. If $F$ is a non-Archimedean local field, then we will denote the ring of integers of $F$ by $\mathcal{O}...
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How many minimal relations are needed to obtain a Frobenius algebra?
Let $A_n:=K \langle x_1,x_2,...,x_n \rangle$ be the non-commutative polynomial ring in $n$-variables over the field $K$ and let $J=\langle x_1,...,x_n \rangle$ be the ideal spanned by the $x_i$.
An ...
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Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?
In B. Bhatt's lecture notes[1], Remark 4.2.5 says
... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete.
which amounts to the following pure algebraic question.
Statement ...
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215
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On linear schemes
Let $X$ be a smooth projective curve and $T$ is any smooth variety. Let $E$ be a family of vector bundles over $X\times T$ which is flat over $T$. Then there exists a scheme $Y$ over $T$ such that for ...
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Automorphisms of matrix algebras and Picard group
This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
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Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
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Local cohomology and residues of rational functions at 0 and $\infty$
Let $a_1,\dots,a_s$ and $b_1,\dots,b_t$ be positive integers, where
$s,t>0$. Choose $c\in\mathbb{Z}$. Let $M_c$ be the real vector
space spanned by all monomials $x^\alpha y^\beta=x_1^{\alpha_1}\...
- 50.8k
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Motivation for definitions of donor and receptor in Salamander Lemma?
$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes:
$$\require{AMScd}\begin{...
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
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Example of a certain type of Cohen-Macaulay ring
Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.
$R$ is of finite type over $k$ and is a domain;
for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $...
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Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
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Normal forms of ADE singularities
Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...
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Product/intersection of two ideals
Let $I_{1}$ and $I_{2}$ be two ideals in polynomial ring $C[u,v,t]$, defined as $I_1 = \langle t-u^3, (v-u^5)^2\rangle$ and $I_2 = \langle u^{11} + v^{11} + t^{11} + 3\rangle$. Is there a method for ...
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Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
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Comparison of depth of two monomial ideals
Let $I$ and $J$ be two monomial ideals of $R=K[x_1,\dotsc,x_n]$ where $K$ is a field.
Could we say $\mid\operatorname{depth} (R/I)- \operatorname{depth}(R/(I:J))\mid\leq 1$, where $(I:J)$ is colon ...
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On the analogy between $p$-derivations and derivations
$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such ...
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Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
- 39.3k
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Examples of stretched artinian local ring
In Sally's Paper stretched artinian local ring is defined as :
Let $(R, \mathfrak{m})$ be an Artin local ring of length $\lambda.$ If $\nu$ is the embedding dimension of $R$, that is, $\nu$ is the ...
- 16.2k
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Is there a Hopf algebra-style description of chain complexes?
An old chestnut is that filtered objects are the same as sheaves over $\mathbb A^1 / \mathbb G_m$.
Question: Is there a similar description of chain complexes?
More precisely, if $\mathcal C$ is a ...
- 5,760
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Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
- 12.9k
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116
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Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 10.5k
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5
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Additive integer-valued functions on the module category
This is inspired by the theorem mentioned in Why is this theorem attributed to Serre?. But I'm not sure if it's research level. If not, please feel free to vote for closing.
Let $R$ be a ring and ...
- 1,446
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Cataland: Facets and partition polynomials of cluster complexes
Figure 25 on pg. 101 of "Cataland: Why the Fuss?" by Christian Stump, Hugh Thomas, and Nathan Williams depicts cluster complexes (CCs) associated with generalized $(m)$-Narayana / ...
- 2,175
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Is there a good notion of higher-rank archimedean norm?
Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
- 64k
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Intersection of two modules (and sub-modules) under tensors
I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
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Effective bound on "Jacobian rank" for (regular) planar algebraic curves
Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...
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Hilbert polynomials of graded algebras evaluated at negative numbers
Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
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