All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
4
votes
1
answer
241
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Questions in the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences"
I am reading the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences", the link is here:https://pub.uni-bielefeld.de/publication/1780235.
In the paper, $A$ ...
- 3,686
4
votes
0
answers
159
views
Finitistic dimension equal to the dominant dimension
Given a finite-dimensional self-injective algebra $A$ and an indecomposable non-projective module $N$, let $M:=A \oplus N$ and $B:=End(M)$.
Does $B$ always have dominant dimension equal to the ...
CommunityBot
- 1
0
votes
1
answer
337
views
Homological dimensions of tensor products of algebras
Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A ...
- 3,686
2
votes
0
answers
70
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Short exact sequences in p-group algebras
Given a group algebra of a finite p-group over a field of characteristic p.
Tachikawa proved that in this case $Ext^{1}(M,M) \neq 0$ for any finite dimensional non-projective module $M$.
Can one give ...
- 26.5k
6
votes
0
answers
266
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Tachikawa conjecture for commutative algebras
Let $A$ be a selfinjective algebra. A famous conjecture by Tachikawa says that $Ext^{i}(M,M) \neq 0$ for some $i>0$ and any nonprojective module $M$ (all algebras and modules are finite dimensional ...
- 26.5k
3
votes
1
answer
70
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2-periodic modules over p-group algebras
Given the group algebra of a p-group over a field of characteristic p. Can the 2-periodic indecomposable modules $M$ ($M$ with $\Omega^{2}(M)=M$) be classified? I am not experienced much with modular ...
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4
votes
0
answers
191
views
Finitistic dimension of Nakayama algebras
Given a connected (quiver) nonselfinjective Nakayama algebra with a circle as a quiver and at least two points.
Such an algebra is determined by the (Kupisch) sequence $[c_0,c_1,...,c_{n-1}]$, when ...
- 26.5k
4
votes
0
answers
237
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Derived equivalent algebras
Given a finite dimensional connected quiver algebra A, define $S_A$ as the set of quiver algebras derived equivalent to $A$ (up to isomorphism).
Questions:
Can one characterise algebras $A$,where $...
- 26.5k
1
vote
0
answers
27
views
Approximations of modules in a special setting
Given a local finite dimensional nonselfinjective algebra $A$ and $M:=A \oplus D(A)$. Can one find a general formula for the minimal right add(M)-approximation of a general indecomposable module $N$ ...
- 26.5k
3
votes
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
1
vote
1
answer
131
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Finite add(N)-resolution
Let $A$ be a local selfinjective algebra with indecomposable module $M$.
Let $N=A \oplus M$.
When there is an indecomposable module $U$ not in $add(N)$, having finite $add(N)$-resolution for some ...
CommunityBot
- 1
3
votes
0
answers
70
views
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
1
vote
0
answers
135
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Equivalence between blocks of BGG categories $\mathcal{O}$ which does not preserve highest weight structure
Let $\mathfrak{g}$ be a finite-dimensional (complex) semisimple Lie algebra. Then we denote the BGG category for $\mathfrak{g}$ by $\mathcal{O}$ as usual. It is well-known that $\mathcal{O}$ as well ...
CommunityBot
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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 ...
CommunityBot
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3
votes
0
answers
69
views
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
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 ...
- 26.5k
3
votes
1
answer
326
views
Whether Morita equivalence holds the following properties?
Let $A,B$ be two K-algebras over a field K.
$A$ and $B$ are said to be $Morita $ $equivalent$ if the category $Mod A$ and $Mod B$ are equivalent.
$A$ and $B$ are said to be $derived$ $equivalent$ ...
- 26.5k
1
vote
0
answers
57
views
Algebras with gorenstein dimension equal to the dominant dimension equal to one
Let algebras always be finite dimensional (and connected).
In https://arxiv.org/pdf/0809.4897v3.pdf , the algebras with global dimension equal to the dominant dimension equal to one are classified as ...
- 26.5k
0
votes
0
answers
72
views
Contravariant finiteness of a certain subcategory
Let $A$ be an algebra with finite dominant dimension $d \geq 1$ and $Dom_d$ the full subcategory of modules with dominant dimension at least $d$ and $Proj$ the full subcategory of modules of finite ...
- 26.5k
-1
votes
1
answer
116
views
How to show the following properties of $Coker(d^{-n-1})$?
Let $A$ be a k-algebra,where k is a fixed field. We denote by $\mathfrak{D}^b(A-mod)$ the bounded derived A-module category. A complex $Z^{\bullet}=(Z^i,d^i) \in \mathfrak{D}^b(A-mod)$ such that all $...
- 111
1
vote
0
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124
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Some questions in a paper of derived categoires and stable equivalence
I am reading the paper "Derived categories and stable equivalence", the link is here:http://www.sciencedirect.com/science/article/pii/0022404989900819.
At theorem 2.1, there is an equivalent functor $...
- 775
1
vote
0
answers
89
views
How to get the following functor of derived equivalent categories?
Let $A$ be a left coherent ring, that is, a ring for which the kernel of any homomorphisms between finitely generated projective modules are finitely generated. $T^{\bullet}\in \mathcal{K}^b(A-proj)$ ...
- 775
2
votes
0
answers
105
views
Why $T'$ dosen't have projective direct summand?
Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let ...
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0
votes
1
answer
77
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How to get $I_i \in add(\nu_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}_A(S,S)=0$?
Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let ...
- 26.5k
2
votes
0
answers
145
views
Question about Ext$^1$ in local commutative algebras
Given a local commutative (commutative only if needed...) selfinjective (non-semisimple) finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{...
- 26.5k
1
vote
1
answer
215
views
The projective modules of an algebra and the tilting module?
Let A be an algebra. We denote by by A-proj the full subcategory of A-mod consisting of projective modules. An A-module T is called a tilting module if $proj.dim(_{A}T)=n < \infty$, $Ext_{A} ^{j} (...
- 26.5k
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 ...
- 35.2k
2
votes
1
answer
450
views
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 ...
- 35.2k
2
votes
1
answer
277
views
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 ...
CommunityBot
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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 ...
- 26.5k
5
votes
0
answers
209
views
Ext^1 for a local finite dimensional selfinjective algebra
Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here:
http://arxiv.org/pdf/1609.00588.pdf.
There it is ...
- 26.5k
3
votes
1
answer
251
views
Some places I don't know of the paper "On the stable module category of a self-injective algebra"
Recently I am reading the paper "On the stable module category of a self-injective algebra", the link is here: http://www.ams.org/journals/tran/2000-352-05/S0002-9947-00-02232-7/S0002-9947-00-02232-7....
- 26.5k
13
votes
1
answer
669
views
Is a "smooth" finite-dimensional algebra separable modulo its radical?
Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
CommunityBot
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5
votes
0
answers
259
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divided power structure on Hocschild cohomology?
Does Hochschild cohomology of a cocommutative Hopf algebra over a field of positive characteristic have a natural divided power structure?
If not, perhaps a certain natural extra structure on the ...
- 1,526
11
votes
1
answer
812
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Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory
By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...
CommunityBot
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6
votes
0
answers
209
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Classification of representation-finite algebras up to stable equivalence of Morita type
Assume $K$ is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
- 8,529
6
votes
1
answer
775
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Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra
Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.
Tilting module theory play an important role in the ...
- 52.9k
0
votes
1
answer
324
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$Ext$ functor over a product of groups
Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 \...
- 790
5
votes
0
answers
246
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Partial formality of A-infinity structure implies formality
Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}...
- 2,450
14
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0
answers
891
views
Local proof of Grothendieck-Riemann-Roch theorem
There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of ...
CommunityBot
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2
votes
0
answers
203
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Could Partial Tiltings be studied as Almost Complete Tiltings?
The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end.
Here $k$ denotes an algebraically closed field,...
- 493
4
votes
0
answers
315
views
Compactly supported distributions as a projective G-module
For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
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4
votes
1
answer
273
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Homological characterisation of standardly stratified algebras using Ext
Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
- 3,686
2
votes
0
answers
77
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Calculation of minimal right $\operatorname{add}(M)$-approximations
given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 \...
- 398
4
votes
0
answers
175
views
Seeking an unpublished manuscript by Tetsuro Okuyama
Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...
- 30.3k
8
votes
2
answers
743
views
Confusion about Subcategories of Category $\mathcal{O}$
So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...
- 52.9k
2
votes
1
answer
395
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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 ...
CommunityBot
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4
votes
0
answers
85
views
Homological dimension of Joseph quotients
Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$.
Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique two-...
- 7,037
0
votes
1
answer
387
views
On a claim of Zagier on extending a map to cocycle
Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ \...
- 3,911
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 ...
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