All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
3
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417
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Finitistic dimension of an algebra
The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. (we look here only at finite dimensional modules).
It is ...
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3
votes
0
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111
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Strange modules over symmetric algebras
Let $A$ be a symmetric algebra (finite dimensional and connected) and define $\psi_M:=sup \{ i \geq 1 | Ext^{i}(M,M) \neq 0 \}$ (infinite if this Ext is nonzero infinitely often) for an indecomposable ...
- 26.5k
3
votes
0
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144
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Computation of nearby cycles, monodromy action and action of $sl_{2}$ on $\operatorname{gr}(Ψ)$ for Picard-Lefshetz family
Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{1}$ be a map that sends $(x,y)$ to $xy$. Let $U \hookrightarrow \mathbb{A}^{2}$ be the preimage $f^{-1}(\mathbb{A}^{2} \setminus \{0\})$ and $X:=f^{−1}(0)...
- 103
3
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0
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60
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Number of syzygy closed Nakayama algebras
Let $A$ be the algebra of $n \times n$ upper triangular matrices over a field $K$ and $J$ its Jacobson radical. Call an ideal $I$ of $A$ admissible in case $I \subseteq J^2$. Let $X$ be the set of ...
- 26.5k
3
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0
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197
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Questions on syzygies and Gorenstein algebras
Questions are about this paper: http://users.uoi.gr/abeligia/gorenstein.pdf and all algebras are finite dimensional
Question 1: In corollary 6.21 (2) there is a proof of the direction (e) implies (a):...
- 26.5k
3
votes
0
answers
205
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Finitistic dimension via tilting modules
is the following true (all algebras and modules are assumed to be finite dimensional):
The finitistic dimension of an algebra is equal to the supremum of projective dimensions of tilting modules?
It ...
- 26.5k
3
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0
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213
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Cohen-Macaulay Artin algebras
In http://link.springer.com/chapter/10.1007%2F978-3-0348-8658-1_8#page-1
Auslander and Reiten introduced Cohen-Macaulay Artin algebras as a generalisation of Gorenstein algebras. Let X be the full ...
- 26.5k
3
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0
answers
92
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Representation-finitness and Ext^1
Given a non-local selfinjective connected quiver algebra A with indecomposable module M with $Ext^{1}(M,M) \neq 0$. Can $B=End_A(A \oplus M)$ be representation-finite? The answer is no in case $A$ is ...
- 26.5k
3
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0
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70
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First Hochschild cohomology in a local selfinjective algebra
Given a non-semisimple local selfinjective finite dimensional algebra $A$ with enveloping algebra $A^e$. Can one have $Ext_{A^{e}}^{1}(A,A)=0$ (that is the first hochschild cohomology zero)? I can ...
- 26.5k
3
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0
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69
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Question on period of algebras
Given a finite dimensional selfinjective algebra $A$. By definition, the period of $A$ is the smallest integer $i >0$ such that $\Omega^{i}(A) \cong A$ as $A \otimes_K A^{op}$-modules. Is the ...
- 26.5k
3
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0
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324
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2 - Calabi Yau algebras and bimodule coherence
Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero.
In H. Minamoto "Ampleness of two-sided tilting complexes", ...
3
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0
answers
318
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When does Ext^2 vanish in a category of group representations.
Let $G$ be a linear algebraic group over field $k$ of characteristic zero. It is well known that the category of finite dimensional $k$--linear representations of $G$ is abelian, and that it is ...
- 4,015
2
votes
3
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299
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Can the 'linkages' between equivalent extensions of modules of an algebraic group be taken to have bounded length?
I have a conjecture concerning how "tightly" two equivalent $n$-fold extensions of modules over an algebraic group over a field might
be "linked". I suspect that the
question has been already ...
- 511
2
votes
1
answer
97
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When is $N^{*} \otimes_K M$ projective for a local Hopf algebra?
Given a finite dimensional local Hopf algebra $A$ over a field $K$ and two finite dimensional indecomposable modules $N$ and $M$. Is it known when the module $N^{*} \otimes_K M$ is projective? Can ...
- 26.5k
2
votes
2
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139
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Infinite radical ideal cubed equals zero for tame hereditary Artin algebras
Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...
- 696
2
votes
2
answers
140
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Finding exceptional regular representations of $\tilde{D}_4$ efficiently
Let $A$ be the path algebra of the quiver $\tilde{D}_4$. I would like to find its exceptional regular representations with as little computation as possible.
Of course, we can compute the whole ...
- 1,071
2
votes
1
answer
100
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In search of disconnected indecomposable self-injective finite-dimensional algebras
I wanted to know if it is possible to construct an indecomposable self-injective finite-dimensional algebra $\Lambda$ whose Auslander-Reiten quiver $\Gamma_\Lambda$ is not connected. I'd love to see ...
- 15.2k
2
votes
1
answer
207
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Question on Ext for finite dimensional algebras
Given a finite dimensional non-Gorenstein algebra $A$, do we have $Ext^i(D(A),A) \neq 0$ for infinitely many $i$? (We can assume A is local or commutative if that helps).
All I can show is that such ...
- 26.5k
2
votes
1
answer
450
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The definitions of a generator module?
Recently I have seen two definition of a generator module:
1) A generator for a category $C$ is an object $G$ such that for any two parallel morphisms $f,g:X \rightarrow Y$ with $f \neq g$, then ...
- 775
2
votes
1
answer
307
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Gaps in the projective dimensions of simple modules
Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $n$ simple modules.
Let $d_1<d_2<...<d_r$ be the sequence of projective dimension of simple $A$-modules in ...
- 26.5k
2
votes
1
answer
200
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Projective dimensions of simple modules in acyclic quiver algebras
Given a quiver algebra $A=KQ/I$ with acyclic $Q$. Then $A$ has finite global dimension, lets say $g$.
Question: Is there for any $0 \leq i \leq g$ a simple module with projective dimension equal to ...
- 26.5k
2
votes
1
answer
959
views
homology under exact functors
Let $\mathcal{A}$ be an Abelian category, $X$ be a complex, $F$ be a contravariant exact functor. I am wondering whether F preserves the homology of X, that means whether $H^{i}(FX)=F(H^{-i}(X)),\ \...
- 327
2
votes
1
answer
395
views
Projectivity of torsion-free modules over integral group rings
Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...
- 2,998
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 $i>0$ ...
- 26.5k
2
votes
1
answer
165
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Rep infinite, but $\tau$-tilting finite
Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting ...
- 839
2
votes
1
answer
169
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Combinatorial problem on periodic dyck paths from homological algebra
edit: I added conjecture 2 that looks much more accessible.
Here is the elementary combinatorial translation of the problem (read below for the homological background):
Let $n \geq 2$.
A Nakayama ...
- 26.5k
2
votes
1
answer
203
views
$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$
Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
- 26.5k
2
votes
1
answer
98
views
Reflexive modules up to multiplicity
Call an indecomposable module $M$ over a ring $A$ (restrict to finite dimensional algebras if you like or if it helps) $n$-almost reflexive in case $M^{**} \cong nM$, when $(-)^{*}=Hom_A(-,A)$ and $nM$...
- 26.5k
2
votes
1
answer
77
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Complexity one modules that are not periodic
Questions:
Is there a module of complexity one that is not periodic over a selfinjective algebra over a finite field?
Is there a module of complexity one that is not periodic over a symmetric algebra ...
- 26.5k
2
votes
1
answer
119
views
How to get that $\Omega^2_{\Lambda}(N) \cong \textrm{Hom}_A(M,Y)$?
Let $A$ be an algebra over a field k. A module $_AM$ is called a generator if $\textrm{add}(A) \subseteq \textrm{add}(M)$, a cogenerator if $\textrm{add}\big(D(A)\big) \subseteq \textrm{add}(M)$. $M$ ...
- 775
2
votes
1
answer
213
views
How to get $Hom_A(M,N) \cong Hom_{B^{op}}(Hom_A(N,T),Hom_A(M,T))$?
I am reading the paper"Dominant dimensions, derived quivalences and tilting modules", the link is here:http://link.springer.com/article/10.1007/s11856-016-1327-4.
On page 22,Lemma 4.2 says that let M ...
- 775
2
votes
1
answer
203
views
Proving indecomposability of special modules
I'm reading the following paper: http://math0.bnu.edu.cn/~huwei/paper/Holm-Hu-1.pdf
On page 795 and 796 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon $A_n:=k[x,...
- 1,755
2
votes
1
answer
173
views
Using the mapping cone to show that a chain map defines a stable equivalence between two symmetric algebras
This question is about an argument in the proof of Theorem 9.8.8 in Linckelmanns Block Theory of Finite Group Algebras. I need to understand the argument in order to do something similar in my ...
- 75
2
votes
1
answer
127
views
Ext between a module and its higher Auslander-Reiten translate
Let $A$ be a representation-finite algebra and $M$ an indecomposable module with finite projective dimension $g >0$.
Question 1: Do we have $dim(Ext_A^g(M, \tau_g(M)))=1$? Here $\tau_g(M)=\tau ( \...
- 26.5k
2
votes
1
answer
200
views
Characterisation of minimal projective resolutions via the Euler characteristic
Let $A$ be a finite dimensional $K$-algebra (where $K$ is a field) and $M$ a finitely generated $A$-module.
Let $\psi: 0 \rightarrow P_r \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$
...
- 26.5k
2
votes
1
answer
250
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Example of a projective bimodule with isomorphic left and right duals
What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying
i) $M$ is projective as both a left and right $R$-module
ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual ...
2
votes
1
answer
98
views
A weaker version of strongly graded algebras
Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$...
2
votes
1
answer
206
views
Extensions of lattices
Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\...
- 23
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1
answer
118
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Example to periodic symmetric algebras
In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.
Is there an easy counterexample for the following:
In case $A$ is additionally a periodic ...
- 26.5k
2
votes
1
answer
389
views
Calabi-Yau algebra for finite dimensional algebras
I read the article "Defomrations of algebras in noncommutative geometry" by Schedler.
In Definition 3.7.9. he gives the definition of Calabi-Yau algebra of dimensi on d as algebras that are ...
- 26.5k
2
votes
1
answer
257
views
First Hochschild cohomology of $A=K[x]/(x^n)$
Given the algebra $A=K[x]/(x^n)$ for some field $K$ and natural number $n \geq 2$ with enveloping algebra $A^e=A \otimes_K A$.
It is easy to see that the 1. Hochschild cohomology of $A$ is nonzero ...
- 26.5k
2
votes
1
answer
130
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Questions about dominant dimension
Let $A$ be a finite dimensional algebra over a field K. Let $M$ be an $A$-module and $0 \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots$ be a minimal injective resolution of $M$. The dominant ...
- 775
2
votes
1
answer
369
views
Nomenclature question: a morita-invariant way to say finite-dimensional?
Say $\mathcal{C}$ is the Abelian category of finitely-generated modules over some $k$-algebra $A$. Then an object $M\in \mathcal{C}$ is finite-dimensional over $k$ if and only if $\text{Hom}(P, M)$ is ...
- 7,952
2
votes
1
answer
233
views
The projective and injective modules of $End_A(V)$?
Let A be a finite-dimensional k-algebra,where k is a fixed field. All modules of A are finitely generated left modules. Suppose X is an A-module. We denote by add(X) the full subcategory of A-modules ...
- 775
2
votes
1
answer
372
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Ext groups in the equivariant derived category
I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange.
I am starting to learn about perverse sheaves, the ...
2
votes
1
answer
244
views
Computing (relative) cohomology classes on quotient (vector) space via Hodge theorem
I am working on a graded vector space $V = \bigoplus_{i\in \mathbb{N}}V_i$ (which is a parabolic Verma module in the sense of [1], but let's ignore such specifics) with a positive definite inner ...
- 121
2
votes
1
answer
183
views
Almost split sequences coming from bimodules
Let $A$ be a finite dimensional algebra with enveloping algebra $A^e$.
Auslander and Reiten proved in "On a theorem of E. Green on the dual of the transpose" that
$Hom_A(Tr_{A^e}(A),M) \cong ...
- 26.5k
2
votes
1
answer
212
views
Injective dimension of $A/AeA$
Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $e$ the idempotent such that $eA$ is the direct sum of all indecomposable projective-injective $A$-modules. Do we have $g=...
- 26.5k
2
votes
1
answer
157
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A non-monoidal functor that respects fusion rules
Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and ...
- 159
2
votes
1
answer
277
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Criteria for a finite-dimensional $k$-Algebra to be basic and elementary
I have the following question:
Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$.
I'm ...
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