All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
6
votes
0
answers
293
views
Representation-finiteness vs. $\tau$-tilting-finiteness
Setting: Throughout, $\Lambda$ is a finite dimensional associative algebra, $\operatorname{mod} \Lambda$ is the category of all finitely generated left $\Lambda$-modules, and all subcategories are ...
- 493
5
votes
1
answer
365
views
2TQFT and commutative Frobenius algebras
There is an equivalence between the category of commutative finite dimensional Frobenius algebras and 2 dimensional topological quantum field theories, see for example the book by Joachim Kock, which ...
- 3,830
4
votes
0
answers
273
views
Question on Han's conjecture
Let $A$ be a finite dimensional algebra with enveloping algebra $A^e$.
A conjecture of Han states that the Hochschild homology $Tor_{A^e}^n(A,A) \cong DExt_{A^e}^n(A,D(A))$ is nonzero infinitely often ...
- 26.5k
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 ...
2
votes
1
answer
207
views
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 ...
- 35.2k
2
votes
0
answers
135
views
Ext over a certain commutative algebra
Let $A$ be the algebra $K[x,y]/(x^2,y^2,xy,yx)$. Then $A$ is a 3-dimensional commutative algebra and $Ext^i(M,A) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$. Namely: $...
- 63.9k
1
vote
0
answers
63
views
Reference request for formula on global dimension
Given a finite dimensional algebra $A$ over an algebraically closed field $K$.
Let $A^e=A^{op} \otimes_K A$ be the enveloping algebra of $A$.
Who noted first that the global dimension of $A$ is equal ...
- 26.5k
2
votes
0
answers
33
views
Bounds on global and dominant dimension of certain algebras
Algebras are always finite dimensional over a field $K$.
Let $X_n$ be the set of representation-finite algebras over $K$ with $n$ simple modules.
Define $g(X_n):= sup \{ gldim(A) | A \in X_n $ and $A$...
- 26.5k
1
vote
0
answers
72
views
Question on Gorenstein projective modules
Call a finite dimensional algebra $A$ special in case the category of (finite dimensional) Gorenstein projective modules coincides with the category of finite dimensional modules $M$ such that $Ext^i(...
- 26.5k
2
votes
0
answers
48
views
Special modules over symmetric algebras
Let $A$ be a symmetric connected finite dimensional algebra over a field $k$.
Call a tuple of two modules $(X,M)$ (having no projective direct summands) cute in case $Ext^l(X,M) \neq 0$ for some $l \...
- 26.5k
3
votes
0
answers
81
views
Number of generalised tilting modules
This question is a modification of Number of tilting modules and was suggested by a comment of Dag Madsen. I wondered about this question too some time ago, but did not do much with it since my ...
- 26.5k
1
vote
1
answer
207
views
Tensor product of finite global dimension algebras
Given two finite dimensional connected algebras A and B over a field $K$ with finite global dimension.
Their tensor product is not necessarily of finite global dimension when the field is not ...
- 35.2k
3
votes
0
answers
417
views
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 ...
- 26.5k
2
votes
0
answers
81
views
Characterisation of Frobenius algebras via sequences
Given a commutative Frobenius algebra, finite dimensional over a field $k$.
We assume that the algebra is connected and in fact given by quiver and relations. Let $S$ be the unique simple modules of ...
- 26.5k
6
votes
1
answer
505
views
Global dimension of quiver algebra
Given a representation-finite (finite dimensional over a field) quiver algebra of finite global dimension. Is $eAe$ isomorphic to the field for at least one primitive idempotent $e$?
This is true for ...
- 4,713
2
votes
1
answer
77
views
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
15
votes
1
answer
961
views
Who conjectured the Cartan determinant conjecture
The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
- 26.5k
2
votes
0
answers
191
views
Global dimension of quiver algebras
Given a finite quiver $Q$, let $Y(Q)$ be the set of (isomorphism classes of) algebra $kQ/I$ (with admissible ideal $I$) that have finite global dimension.
For example in case $Q$ is acyclic, all ...
- 26.5k
0
votes
1
answer
153
views
Strange modules part II
Let $A$ be a finite dimensional symmetric algebra over a field (we can also assume that it is connected).
Call a non-projective indecomposable module $M$ strange in case $Ext^i(M,M)=0$ for all but ...
- 35.2k
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
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
5
votes
1
answer
577
views
When is the category of Gorenstein projective $R$-modules Frobenius?
Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a ...
- 1,690
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 ...
- 16.9k
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
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
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
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 ...
- 1,505
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
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 ...
- 5,039
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
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?
...
- 35.2k
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 $M$...
- 63.9k
9
votes
1
answer
736
views
Strange boundary-like map on tensor algebra: what is its kernel?
Let $k$ be a commutative ring and $L$ a $k$-module. The tensor algebra $\otimes L$ is $\mathbb{Z}$-graded and $\mathbb{Z}_2$-graded (an element of $L^{\otimes n}$ has degree $n$ and $\mathbb{Z}_2$-...
- 33.8k
7
votes
0
answers
433
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
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 ...
- 63.9k
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 ...
- 775
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 ...
- 4,679
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 $\textrm{add}\big(D(A)\big) \subseteq \textrm{add}(M)$. $M$ ...
- 1,690
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)),\ \...
- 55.9k
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-...
- 1,690
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:
...
- 876
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 ...
- 63.9k
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....
- 5,846
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