All Questions
6,055 questions
10
votes
1
answer
852
views
Is it a valuation ring?
It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed.
Then, even if it is not Noetherian, would a one-dimensional local ring become a ...
- 5,146
2
votes
0
answers
169
views
The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
19
votes
2
answers
765
views
Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?
I asked this question on Mathematics Stackexchange (link), but got no answer.
Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.
Recall ...
- 3,738
4
votes
1
answer
280
views
Existence of module with periodic resolution
Let $R$ be a Gorenstein local ring. Does there always exist a module $M$ having eventually periodic minimal free resolution?
Any reference is also appreciated.
- 16.2k
8
votes
1
answer
333
views
Alterations and smooth complete intersections
Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d_k$ be a closed immersion into a projective space of high enough dimension.
Is there a ...
- 28.7k
2
votes
1
answer
191
views
Cohen-Macaulay fiber products
Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism.
Is the fiber product scheme $...
- 20.5k
4
votes
0
answers
179
views
Explicit construction for Cohen’s $p$-ring with imperfect residual field
Apologize if this is a below-research-level question. Asked in stack exchange but no response yet.
Let $p>0$ be a prime. Recall that a $p$-ring is a complete DVR $A$ with residual filed $k$ of ...
- 63.9k
2
votes
1
answer
250
views
Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
- 28.7k
8
votes
1
answer
1k
views
Software for computing multi-graded Hilbert series
The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-...
- 6,367
59
votes
4
answers
12k
views
Geometric meaning of Cohen-Macaulay schemes
What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various ...
- 42.9k
4
votes
0
answers
101
views
Theta functions in acyclic cluster algebras
Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $...
4
votes
0
answers
84
views
Dimension of a positively graded ring after a suitable localization
Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
2
votes
2
answers
369
views
Can a non-zero non-prime ideal become prime in a smaller ring?
All rings are assumed commutative and unital.
Some context (feel free to skip right to the questions below). I am trying to understand to what extent the property "being prime" for an ideal $...
- 42.9k
5
votes
0
answers
107
views
Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
- 187
1
vote
0
answers
83
views
Partial fraction decompositions for integral domains
I've recently been involved in a math conversation regarding partial fraction decompositions for rational numbers, but we seem to lack a formal definition and are unsure about whether there is some ...
- 2,858
4
votes
1
answer
268
views
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 $...
- 16.2k
0
votes
0
answers
63
views
A construction that sort of merges two semigroups to build a new one
Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
- 10.5k
3
votes
2
answers
379
views
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 ...
- 16.2k
5
votes
0
answers
181
views
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 ...
- 63.9k
3
votes
1
answer
238
views
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
6
votes
1
answer
443
views
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
1
vote
0
answers
116
views
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 ...
- 7,127
15
votes
1
answer
1k
views
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 ...
- 28.7k
0
votes
0
answers
122
views
Is there a name for this condition on a monoid?
Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
- 5,429
2
votes
1
answer
326
views
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 ...
- 20.5k
3
votes
1
answer
329
views
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
- 1
2
votes
1
answer
186
views
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). ...
- 2,368
10
votes
2
answers
449
views
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 ...
- 14.6k
4
votes
1
answer
151
views
Three preprints and one manuscript of Tamura on power semigroups
I'm reading Takayuki Tamura's article "On the recent results in the study of power semigroups", pp. 191-200 in Goberstein & Higgins' Semigroups and Their Applications, Kluwer, 1987 (the ...
- 188k
10
votes
3
answers
3k
views
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 ...
- 3,101
2
votes
0
answers
188
views
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
- 1
1
vote
1
answer
182
views
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^{...
- 4,600
2
votes
0
answers
164
views
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 ...
- 10.4k
2
votes
1
answer
340
views
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 ...
- 23k
2
votes
0
answers
68
views
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) ...
- 6,367
6
votes
1
answer
876
views
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 ...
- 404
3
votes
1
answer
531
views
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
226
votes
4
answers
16k
views
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 ...
- 4,713
4
votes
1
answer
288
views
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
0
votes
0
answers
156
views
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} ...
- 39
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 39
4
votes
1
answer
487
views
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
4
votes
1
answer
350
views
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}...
- 148k
2
votes
0
answers
114
views
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 ...
- 26.5k
1
vote
0
answers
103
views
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 ...
- 441
0
votes
0
answers
215
views
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 ...
- 12.9k
6
votes
1
answer
259
views
are endomorphisms "small" compared to the full transformations?
$\DeclareMathOperator\End{End}$Let $T_n$ be the full transformation semigroup/monoid of $[n]=\{1,\dots,n\}$. Let $\End(T_n)$ be the set of [endomorphisms][1] of $T_n$. Then, $\# T_n=n^n$ and
$$\# \End(...
- 63.9k
3
votes
0
answers
71
views
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\...
- 1,766
3
votes
0
answers
151
views
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 ...
- 10.5k
10
votes
0
answers
195
views
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