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2 votes
1 answer
141 views

Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$

Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring. There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
Justin Bloom's user avatar
7 votes
0 answers
225 views

Decomposing an endomorphism as a tensor product

$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
Pierre's user avatar
  • 2,287
3 votes
0 answers
137 views

Composition of Frobenius $n$-homomorphisms, characteristic-free?

This question is, as so often, a crossbreed of curiosity and laziness. The former has led me to an interesting, but somewhat unsatisfactory paper by Khudaverdian and Voronov (arXiv:2002.02395v2) and, ...
darij grinberg's user avatar
3 votes
0 answers
93 views

Commutant of irrep of $S_n$ (over local field)

Let $k$ be a field of characteristic zero and let $(V, \rho)$ be a finite-dimensional representation over $k$ of the symmetric group $S_n$. I would like to understand the commutant $\operatorname{End}...
bsbb4's user avatar
  • 363
6 votes
0 answers
178 views

Ext for commutative Gorenstein algebras

Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$. Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
Mare's user avatar
  • 26.5k
1 vote
0 answers
106 views

Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings

Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
uno's user avatar
  • 412
2 votes
1 answer
184 views

Gorenstein projective module over commutative local algebras

Let $A$ be a local commutative finite dimensional algebra over a field $K$. An $A$-module $M$ is called Gorenstein projective if $M$ is reflexive, $Ext_A^i(M,A)=0=Ext_A^i(M^{*},A)$ for all $i>0$ ...
Mare's user avatar
  • 26.5k
4 votes
0 answers
219 views

Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$

EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect. Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
Libli's user avatar
  • 7,300
8 votes
1 answer
356 views

Homological conjectures for finite dimensional commutative algebras

$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
Mare's user avatar
  • 26.5k
3 votes
2 answers
395 views

Cohen-Macaulay Representations

I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research. If yes, then what are some of ...
It'sMe's user avatar
  • 839
10 votes
0 answers
194 views

Singularity category of a hypersurface associated to $M_{11}$

For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
Dave Benson's user avatar
  • 16.2k
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 ...
Markuss Schmuckler's user avatar
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 $...
Iteraf's user avatar
  • 482
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 ...
Mare's user avatar
  • 26.5k
0 votes
1 answer
349 views

Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)

Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature? There are two sets of partition polynomials, not in the OEIS, that serve as the ...
Tom Copeland's user avatar
  • 10.5k
1 vote
0 answers
119 views

Germs of holomorphic functions and invariant functions

Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian. Now consider a ...
UVIR's user avatar
  • 803
3 votes
0 answers
249 views

Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)

In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as $$D\; X^n = F(\tfrac{...
Tom Copeland's user avatar
  • 10.5k
4 votes
1 answer
141 views

Kernels of actions on truncated polynomial algebra

Let $p$ be an odd prime, and let $k=\mathbb{F}_p$ be the field with $p$ elements. Let $G=\text{GL}_n(k)$. The group $G$ acts on the truncated polynomial algebra $A:=k[x_1,\ldots, x_n]/(x_1^p,\ldots, ...
Ehud Meir's user avatar
  • 5,039
3 votes
1 answer
173 views

$\Omega$ for noetherian semiperfect rings

Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$. Let $\Omega^n(mod A)$ be the category of $n$-...
Mare's user avatar
  • 26.5k
3 votes
1 answer
382 views

What is the name for algebras generated by elements, all of whose cubes vanish?

Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
Naysh's user avatar
  • 557
2 votes
0 answers
196 views

Commutative local rings which satisfy Krull-Remak-Schmidt

Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...
Mare's user avatar
  • 26.5k
4 votes
0 answers
114 views

Classification of 2-periodic triangulated categories

Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$. Question 1: Is there a ...
Mare's user avatar
  • 26.5k
0 votes
1 answer
265 views

Quiver representations over any commutative ring

I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image. Towards the end, he has this ...
It'sMe's user avatar
  • 839
6 votes
1 answer
394 views

On the Artin-Rees Lemma for non-commutative rings

Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
FPV's user avatar
  • 541
6 votes
1 answer
312 views

Prove that $\overline{a}_{11}$ is a prime element in $R$

Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-...
It'sMe's user avatar
  • 839
3 votes
1 answer
111 views

Projectivity of some module

Let $k$ be a algebraically closed field and suppose that $A$ and $B$ are finite dimensional $k$-algebras. If we assume that $A$ is a symmetric $k$-algebra and $A\otimes_k I$ is a projective $A\...
Master Gang's user avatar
4 votes
0 answers
211 views

Diagonalization over valuation rings

Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
FPV's user avatar
  • 541
3 votes
0 answers
213 views

A question related to the polynomial ring $R[x]$ over some principal ideal domain $R$?

Let $R[x]$ be the polynomial ring over some principal ideal domain $R$. If $R[x]/I$ is free as a $R$-module for some ideal $I$, is $I$ a principal ideal which is generated by some monic polynomial in $...
Master Gang's user avatar
3 votes
1 answer
172 views

On the linearizability of the action of a finite group on a formal polydisc

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
FPV's user avatar
  • 541
2 votes
1 answer
257 views

Possible "algebraic" direction in hyperplane arrangements

I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the &...
It'sMe's user avatar
  • 839
7 votes
0 answers
275 views

Split epimorphism 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. Suppose further we have an ...
kevkev1695's user avatar
7 votes
1 answer
254 views

Group-like elements in quotients of group rings

$\DeclareMathOperator\Gr{Gr}$Let $R$ be a local ring, let $A$ be a finite abelian group, and let $I$ be a Hopf ideal of the ring $R[A]$. The quotient $R[A]\twoheadrightarrow R[A]/I$ induces a map on ...
Eric Ahlqvist's user avatar
2 votes
0 answers
78 views

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 ...
user125639's user avatar
3 votes
1 answer
240 views

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 ...
kevkev1695's user avatar
12 votes
2 answers
333 views

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 $\...
Hugh Thomas's user avatar
  • 6,292
8 votes
0 answers
291 views

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 $...
Will Sawin's user avatar
  • 148k
3 votes
1 answer
177 views

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 ...
Mare's user avatar
  • 26.5k
5 votes
1 answer
235 views

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 ...
kevkev1695's user avatar
4 votes
0 answers
196 views

Quillen–Suslin theorem in a more general context

Let $A$ be a finite dimensional local Frobenius algebra that is Koszul. Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...
Mare's user avatar
  • 26.5k
2 votes
0 answers
92 views

Expressing elements in Verlinde ideal in terms of generators

It is known that the level $k$ Verlinde ring of $SU(n)$ is $R(SU(n))/I_k$, where $I_k$ is the Verlinde ideal. A set of generators of $I_k$ is given by $\{V_{(k+i)L_1}:=\text{Sym}^{k+i}V_{\text{std}}| ...
No_way's user avatar
  • 383
14 votes
2 answers
1k views

Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings

$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....
stupid_question_bot's user avatar
0 votes
1 answer
128 views

About cluster variables obtained by (sequentially) mutating at exchangeable variables from an initial cluster

Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a ...
amator2357's user avatar
4 votes
0 answers
188 views

Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras

Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...
kevkev1695's user avatar
4 votes
0 answers
78 views

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]...
Libli's user avatar
  • 7,300
8 votes
1 answer
264 views

Class group of hypersurfaces of finite representation type

Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
Alessio's user avatar
  • 411
3 votes
0 answers
107 views

Do Frobenius algebras have a lattice basis and what lattices do appear?

Let $K$ be for simplicity be the field with two or three elements (or alternatively we could restrict to ideals containing only the field elements $-1$ or $1$ as coefficients). A (commutative) ...
Mare's user avatar
  • 26.5k
1 vote
0 answers
174 views

What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?

Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
Asvin's user avatar
  • 7,746
8 votes
1 answer
256 views

$\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$

Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves ...
Helen's user avatar
  • 83
5 votes
0 answers
132 views

On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory. Allow me to first give a minor introduction. Let $(...
user160167's user avatar
9 votes
2 answers
572 views

Decomposing a polynomial ring into Specht Modules

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$. I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
Karthik C's user avatar
  • 261