All Questions
Tagged with reference-request ac.commutative-algebra
402 questions
6
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Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?
A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...
- 6,229
6
votes
1
answer
587
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Study of convex polytopes via commutative algebra
Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...
- 211
6
votes
1
answer
437
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"Archimedeanising" an ordered field
If $K$ is an ordered field, let $B$ be the subring comprising the $x \in K$ such that $|x| \le n$ for some $n \in N$, and let $I$ be the ideal of $B$ comprising the infinitesimal elements (i.e. the $x ...
- 879
6
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2
answers
850
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Decomposition of finite algebras over finite fields
Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
- 32.6k
6
votes
1
answer
422
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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=...
- 43.2k
6
votes
1
answer
670
views
Base change of trace for Gorenstein or Cohen-Macaulay morphisms
This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'...
6
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1
answer
641
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The Jacobian ideal generates the socle of a complete intersection
This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
- 805
6
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1
answer
245
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Reference request for results that involve the transcendence degree
Currently I'm reading "On the Decidability of the Real Exponential Field" by Macintyre and Wilkie and the Proof of Theorem 1.1 (page 462-464) uses two algebraic results that involve the ...
- 123
6
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0
answers
68
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Vector algebra in a Tarski space
By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...
- 41.8k
6
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0
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194
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"Cluster algebra" structure for finite distributive lattices
Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets).
For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...
- 24.2k
6
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0
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171
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Name for property of mixed characteristic DVR: admits regular local homomorphism from DVR with finite residue field
Does anybody happen to know if there is already a name in the literature for the following property of a mixed characteristic DVR: that there exists a local homomorphism that is regular into the ...
- 4,111
6
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0
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224
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Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite ...
- 10.5k
6
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0
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671
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Flat + locally of finite presentation + monomorphism = open immersion
It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$:
Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then $...
- 15.9k
6
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0
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243
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I. Kaplansky, Going up in polynomial rings, unpublished manuscript, 1972
Anyone got a copy of this article?
- 1,388
6
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0
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881
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.5k
5
votes
3
answers
677
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Spectrum and scheme of the commutative group-algebra of an abelian group.
The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...
- 263
5
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3
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1k
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adjoint of multiplication operator in a commutative algebra
Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
- 6,614
5
votes
1
answer
308
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Reference request for the group of units of a power series ring in one variable
Let $\mathbb F_p$ denote the finite prime field of $p$ elements. What reference can be recommended for an analysis of the structure of the group of units of the power series ring $\mathbb F_p[[x]]$? ...
- 1,638
5
votes
2
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964
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Krull-Schmidt Analogue for Complete / Graded Rings
Over the ring $\mathbb{Z}$, all finitely-generated modules decompose uniquely as a direct sum of indecomposable submodules; that's the Krull-Schmidt theorem.
I'm given to understand that if a (...
- 146
5
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2
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754
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A version of Hilbert's Nullstellensatz for real zeros
$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
- 128k
5
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4
answers
660
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Relative version of Hilbert syzygy theorem
I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a ...
- 15.6k
5
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1
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299
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The universal multiset for a finite scheme - reference request
If $X$ is a finite set of size $n$, then by listing the elements of $X$ we get a canonical element of the symmetric power $X^n/\Sigma_n$, which we can call the universal multiset for $X$.
Now let $X$ ...
- 56.9k
5
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1
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223
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Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
- 6,214
5
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1
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2k
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Generalizing Dedekind's Factorization Theorem
A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
$...
- 14.6k
5
votes
2
answers
760
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Jordan-Holder vs Harder-Narasimhan
Let $M$ be a module over an algebra or a group. I am interested the following decreasing filtration:
$F^0M=M$;
$F^iM$ is the smallest sub-module of $F^{i-1}M$ such that the quotient is semi-simple....
- 2,384
5
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1
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151
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Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 3,230
5
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1
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959
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Rings of $S$-integers are finitely generated as rings
Let $K$ be a global field (number field or algebraic function field over a finite field), $\mathcal{V}$ the set of $\mathbb{Z}$-valuations on $K$, $S \subseteq \mathcal{V}$ a finite set. The ring of $...
- 277
5
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1
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339
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Can completely multiplicative functions be extended to $\overline{\mathbb{Q}}$ or further?
I'm looking for a subject of study that handles the following question. I'm not the most familiar with algebra; I have a strong working knowledge and that's about it, but I've been considering ...
user78249
5
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1
answer
2k
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Length of a module over different rings
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...
- 1,391
5
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1
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448
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A question about the Buchsbaum-Eisenbud-Horrocks Conjecture
It's known that Mark E. Walker proved the "weaker" version of Buchsbaum-Eisenbud-Horrocks' Conjecture (BEH). Although the claim was stated to hold in arbitrary field $k$, Walker's proof does not seem ...
- 43.2k
5
votes
2
answers
369
views
Links between tight closure and deformation theory
I am looking for links between tight closure and deformation theory. As a sample question:
Question 1. Are there geometric interpretations in terms of deformation theory of
Frobenius rationality?
...
- 32.1k
5
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1
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126
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Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
- 2,356
5
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2
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601
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Reference book for understanding Hilbert Series/functions
For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".
https://en.wikipedia.org/wiki/...
user129588
5
votes
1
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383
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Euler characteristic and rational Poincaré series
$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-...
- 125
5
votes
1
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225
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Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 26.5k
5
votes
2
answers
732
views
The Weyl algebra modules which are also rings
Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...
- 389
5
votes
1
answer
208
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Zariski openness of Newton non-degenerate polynomials
Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ ...
- 53
5
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1
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674
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Resolution of a module as an $A_\infty$ module over resolution of an algebra
The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...
- 1,545
5
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0
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107
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Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
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0
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160
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Cohen-Macaulayness of rings of polynomials vanishing at points
Let $V$ be a finite dimensional vector space, let $L_1$, $L_2$, ..., $L_r$ be subspaces and let $w_1$, $w_2$, ..., $w_r$ be positive rational numbers. Define a graded ring $R$ where $R_d$ is those ...
- 156k
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0
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137
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A particular family of symmetric functions (sums of powers of sums of subsets)
For any $m,k$ define
$$ f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k. $$
Do these symmetric polynomials have a name and any theory?
- 37.7k
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87
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Reference request: Étale base change of differential-graded algebras
I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...
- 171
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0
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255
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Is the category of topologically free $k[[h]]$-modules locally presentable?
$\newcommand{\colim}{\operatorname{colim}}$
Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is
$$
\hat M:=\lim M/h^nM,
$$
...
- 8,524
5
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0
answers
788
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Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
5
votes
0
answers
132
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Asymptotics of Hilbert series for locally finite free graded-commutative algebras?
Let $A^\bullet$ be an $\mathbb N$-graded algebra over a field $k$, and let $d_A(n) = \dim A^n$ be the dimension of the $n$-th graded piece, so that $P^A(t) = \sum_n d_A(n) t^n$ is the Hilbert-Poincare ...
- 63.9k
5
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0
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324
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Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
- 10.5k
5
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0
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132
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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 $(...
- 51
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0
answers
122
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Are affinoid algebras over nontrivially valued fields Jacobson?
It is well-known that for any field $k$ with valuation the Tate algebra $k\{T_1,\dots,T_n\}$ is Jacobson (see Bosch-Güntzer-Remmert for nontrivial valuations; for trivial valuations those are just ...
- 28.2k
5
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0
answers
166
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When do the spectra of overrings glue to a proper morphism?
This question is motivated by the construction of blowups.
Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between.
Let $X ...
- 1,338
5
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0
answers
92
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Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
- 1,755