All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
3
votes
0
answers
111
views
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
2
votes
0
answers
80
views
Coxeter polynomials of Nakayama algebras
Are two Nakayama algebras with a linear quiver derived equivalent if and only if they have the same coxeter polynomial?
Derived categories of Nakayama algebras appear in interesting contexts (see for ...
- 26.5k
2
votes
0
answers
86
views
Testing the Cartan determinant conjecture via Gorenstein algebras
Let $A$ be a Gorenstein algebra (of infinite global dimension) with finitely many indecomposable Gorenstein projective modules and $X$ the basic direct sum of all indecomposable Gorenstein projective ...
- 26.5k
1
vote
0
answers
68
views
Ext in Gorenstein algebras
My computer suggests that the following is true for Nakayama algebra (it also found not counterexample for arbitrary algebras):
Let $A$ be an algebra of finite Gorenstein dimension $g \geq 1$, then ...
- 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
3
votes
0
answers
144
views
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
1
vote
0
answers
46
views
Ext in selfinjective algebras
Is there a selfinjective finite dimensional algebra $A$ with a non-projective module $M$ such that $Ext^{i}(M,M)=0$ for all $i=1,2,...,2n-1$ when $n$ denotes the number of simple module of that ...
- 26.5k
3
votes
0
answers
60
views
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
votes
0
answers
197
views
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
1
answer
354
views
Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?
Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...
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1
vote
0
answers
71
views
Non-Gorenstein projective maximal Cohen-Macaulay module
In http://s.web.umkc.edu/segal/papers/refl.pdf the authors gave the first example of a maximal Cohen-Macaulay module $M$ (that is a module over an algebra A with $Ext^{i}(M,A)=0$ for all $i \geq 1$) ...
- 26.5k
0
votes
1
answer
98
views
Algebra with all modules non-rigid 2
Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective indecomposable module $M$, with the condition that at least one such module exists....
- 26.5k
1
vote
1
answer
133
views
Algebra with all modules non-rigid
Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective module $M$.
Is $A$ selfinjective?
Is $A$ local?
- 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
5
votes
2
answers
224
views
Properties of right rejective subcategories
I am reading this paper finiteness of representation dimension, on page 1012 there is a place I don't understand:
Why $\mathcal{C}(, \mathbb{F}(X)) \rightarrow [\mathcal{C'}](,X)$ is an isomorphism?
...
- 775
6
votes
1
answer
448
views
Hochschild cohomology of certain local algebras
Let $K$ be a field and $A$ the algebra $K\langle x_1,...,x_n\rangle/J^m$ for $n \geq 1$ and $m \geq 2$, where $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring in $n$ variables over ...
- 26.5k
4
votes
0
answers
127
views
Injective dimension is infinite?
Let $A$ be a non-selfinjective finite dimensional algebra and $M$ a nonprojective module with $Ext^{i}(M,A)=0$ for all $i \geq 1$. It is easy to see that $M$ has infinite projective dimension. Does $M$...
- 26.5k
7
votes
0
answers
432
views
What is the endomorphism cooperad?
In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
- 2,074
24
votes
1
answer
1k
views
About the abelian category of endofunctors of $\mathsf{Vect}$
Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $...
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1
vote
0
answers
361
views
Property of the syzygy functor of $\operatorname{\underline{mod}} A$
Let $A$ be an artin algebra. We denote by $\operatorname{mod}A$ the category of finitely generated left $A$-modules. We denote by $[A]$ the ideal of morphisms between $A$-modules which factor through ...
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5
votes
1
answer
261
views
Some questions on division algebras
Given a field $K$, is there a finite dimensional quiver algebra, such that any finite dimensional division algebra is isomorphic to End(M)/rad(End(M)) for some indecomposable finite dimensional module ...
- 26.5k
2
votes
0
answers
102
views
Ext in a selfinjective algebra
Given a finite dimensional selfinjective algebra $A$ and an indecomposable module $M$, which is not projective.
Let $v=DHom(-,A)$ be the Nakayama functor.
In case $Ext^{i}(v(M),M) \neq 0$ for some i, ...
- 26.5k
4
votes
1
answer
423
views
Alternating sum of symmetric and exterior powers vanishes
Let $V$ be a vector space with some extra structure (maybe to be general, an object of an abelian tensor category?), where we can form tensor products and exterior powers $\Lambda^i V$ and symmetric ...
- 3,139
11
votes
0
answers
818
views
How to compute Ext-groups for categories without enough injectives/projectives?
I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
- 507
0
votes
0
answers
112
views
Some places I can't understand in the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra"
I am reading the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra", the link is here: https://arxiv.org/pdf/1608.04212.pdf
There are some places I can't ...
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5
votes
0
answers
120
views
Ext$^1(D(A),A)$ for hereditary algebras
Let $A$ be a hereditary (non-semisimple) finite-dimensional algebra over a field $K$. Let $M:={\rm Ext}^{1}_A(D(A),A)$ ($ \cong D(\tau((D(A)))$ as left modules) and let $A^e$ be the enveloping 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 $\textrm{add}\big(D(A)\big) \subseteq \textrm{add}(M)$. $M$ ...
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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
1
vote
1
answer
158
views
How to get that one module is tilting iff the other one is?
Let $A$ be an algebra over a field k. $D$ is the standard duality functor. A module $_AM$ is called a generator if $add(A) \subseteq add(M)$, a cogenerator if $add(D(A)) \subseteq add(M)$. $M$ is n-...
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5
votes
0
answers
303
views
Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
- 2,450
1
vote
2
answers
157
views
On some modules with bounded syzygies
Let A be a finite dimensional algebra. Call an indecomposable module M cool in case $\Omega^{i}(M)$ is nonzero and indecomposable for every $i \geq 1$ and $dim(\Omega^{i}(M))$ is bounded.
Questions:
...
- 26.5k
5
votes
1
answer
374
views
Tachikawa conjecture for commutative algebras proven?
The Tachikawa conjecture states that $Ext^i(M,M) \neq 0$ for some $i \geq 1$ for every non-projective module $M$ over a selfinjective finite dimensional algebra.
In theorem 4.6. of http://maths.nju....
- 26.5k
2
votes
0
answers
148
views
Algebras where all indecomposable modules are rigid
Is there a classification of selfinjective algebras having $Ext^{1}(X,X)=0$ for every indecomposable module X?
Examples include trivial extensions of representation-finite hereditary algebras.
One ...
- 26.5k
3
votes
1
answer
348
views
Finitistic dimension equal to cofinistic dimension for QF-3?
Given a finite dimensional algebra A such that the regular module embeds into a projective-injective module (such algebras are called QF-3 algebras and generalise Frobenius algebras).
Define the ...
- 26.5k
7
votes
0
answers
266
views
Closed formula for some dimension
This question has a background from representation theory/homological algebra, but I state everything in elementary terms here:
Call an n-CNak an n-tupel $[c_0,c_1,...,c_{n-1}]$ where $c_0=c_1=...=c_{...
- 26.5k
3
votes
0
answers
205
views
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
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 ...
- 26.5k
3
votes
0
answers
213
views
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
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 ...
- 26.5k
2
votes
0
answers
70
views
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
views
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
4
votes
0
answers
237
views
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
28
votes
4
answers
3k
views
Yoga of six functors for group representations?
I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can ...
- 7,789
14
votes
1
answer
835
views
Special configurations on a circle from a homological algebra problem
Here is the short version of the combinatorial problem:
Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
- 26.5k
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
3
votes
0
answers
92
views
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
4
votes
1
answer
241
views
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, $A$ ...
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3
votes
1
answer
70
views
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 ...
- 26.5k
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