All Questions
Tagged with rt.representation-theory homological-algebra
255 questions with no upvoted or accepted answers
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Exterior powers of the Cartan matrix and Dyck paths
(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
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3
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96
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Criterion for representation-finite algebras
Let $A=KQ/I$ a quiver algebra with acyclic $Q$.
Question: Is $A$ representation-finite if and only if $\tau^{-n}(A)=0$ for some $n \geq 1$?
Here $\tau$ is the Auslander-Reiten translate of $A$.
This ...
- 26.5k
3
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74
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Is any $n$-angulated category a $(n-2)$-cluster tilting subcategory of some triangulated category?
Geiss, Keller and Oppermann told us in "n-angulated categories" that some $(n-2)$-cluster tilting subcategory of a triangulated category is a $n$-angulated category.
$\require{wasysym}$
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3
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238
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What do the indecomposable objects of the homotopy category of chain complexes look like?
I am trying to understand indecomposable objects in the homotopy category of chain complexes. Let $\mathcal{A}$ be an abelian category. Denote by $C(\mathcal{A})$ the category whose objects are chain ...
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3
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112
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Finite global dimension via the Cartan determinant
Let $A=T(KQ)$ be the trivial extension algebra of a path algebra of Dynkin type $KQ$.
The indecomposable module of $A$ correspond to the roots of $Q$ (and not just the positive roots as for $KQ$).
Let ...
- 26.5k
3
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80
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Coxeter polynomials of graphs
Let $Q$ be a finite connected and directed graph with $n$ points.
Assume $Q$ is acyclic as a directed graph.
Let $C=C_Q$ be the Cartan matrix of $Q$, that is the matrix with entries $c_{i,j}$ being ...
- 26.5k
3
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102
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Frobenius algebras associated to posets and coalgebra structures
Let $P$ be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m).
Let k be a field, then the classical incidence algebra $kP$ has $k$-vector ...
- 26.5k
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48
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Questions on piecewise hereditary algebras
Let $A$ be a finite dimensional quiver algebra over a field $k$ that is quasi-tilted and representation-finite (this implies that $A$ is a tilted algebra). Assume that the Coxeter polynomial of $A$ is ...
- 26.5k
3
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99
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The union-closed sets conjecture for finite dimensional algebras
Say a finite dimensional algebra $A$ satisfies the right UC-condition if there exists an indecomposable projective module $P$ of $A$ such that $\operatorname{injdim}(\operatorname{top}(P))=1$ and $P$ ...
- 26.5k
3
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Quiver algebras of Dynkin type
Let $kQ$ be one of the Dynin path algebras of type $A_n , D_n $ or $E_i$ for $i=6,7,8$.
Question 1: How many (up to isomorphism) quiver algebras are there that are derived equivalent to $kQ$?
...
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3
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129
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Algebras with symmetric Cartan matrix
Let $A$ be a finite dimensional algebra with Cartan matrix $C_A$.$C_A$ being a symmetric matrix is equivalent to the Coxeter matrix being minus the identity matrix in case $A$ has finite global ...
- 26.5k
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82
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Derived equivalence for two modules
Let $A=K[x]/(x^n)$ and $M_1$ and $M_2$ two basic generator of mod-A and let $B_i=End_A(M_i)$.
$B_1$ and $B_2$ are derived equivalent in case $M_1 \cong \Omega^1(M_2)$ in the stable category.
Question:...
- 26.5k
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On grades of torsion modules in noetherian rings
Let $A$ be a (not necessarily commutative) two-sided noetherian ring with minimal injective coresolution $(I_i)$ of the regular module $A$ as a right module.
Say that $A$ has dominant dimension $n$ in ...
- 26.5k
3
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On a formula for the Auslander-Reiten translate
For an Artin algebra $A$ and an indecomposable non-projective module $M$ we should have that $\tau(M) \cong \nu \Omega^2(M)$ iff $Ext_A^i(M,A)=0$ for $i=1,2$. ($\nu$ being the Nakayama functor)
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Weakly symmetric rings and derived equivalences
A ring $R$ with Jacobson radical $J$ is called Frobenius in case $R/J \cong soc(R)$ as left and right $R$-modules and weakly symmetric in case we even have $R/J \cong soc(R)$ as $R$-bimodules.
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Classes of algebras where derived equivalence preserves the global dimension
Question: Are there known classes $X$ of finite dimensional algebras in the literature that have the property that in case $A, B \in X$ are derived equivalent, they share the same global dimension?
...
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Properties of sequences associated to Nakayama algebras
Assume Nakayama algebras are connected and given by quiver and relations. Note that such Nakayama algebras have global dimension at most $2n-2$ in case it is finite and the algebra has $n$ simples.
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Magnitude of ADR algebras
Let $A$ be a connected quiver algebra with $n$ simple modules and Jacobson radical $J$ and Loewy length $n+1$ (that is $J^{n+1}=0$ and $n$ is minimal with this property).
The ADR-algebra $B_A$ of $A$ ...
- 26.5k
3
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Quiver algebras with finite global dimension
Given a fixed connected quiver $Q$.
Are there only finitely many quiver algebra $KQ/I$ (I an admissible ideal) up to isomorphism which have finite representation type and finite global dimension?
- 26.5k
3
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92
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On NCR for finite dimensional algebras
Let $A$ be a finite dimensional algebra.
A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ ...
- 26.5k
3
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Inequality for the magnitude of quiver algebras
A conjecture on the global dimension of quiver algebras of finite global dimension states that the global dimension is bounded by the vector space dimension of the algebra.
The magnitude of a finite ...
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3
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48
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Endomorphism ring of a generator-cogenerator over acyclic algebras
Let $A$ be an acyclic quiver algebra, $M$ a generator-cogenerator and $B=End_A(M)$.
Questions:
Does $B$ have finite global dimension?
Does $B$ have finite global dimension in case $M=A \oplus D(A)$?
...
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3
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106
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Bounds for the finitistic dimension
The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension.
For finite dimensional algebras $A$ with radical cube ...
- 26.5k
3
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61
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Number of algebras stably equivalent to a given algebra
For $n \geq 2$ let $B_n$ be the algebra of upper triangular matrices over a field $K$.
Recall that two algebras are said to be stably equivalent in case their module categories modulo projectives are ...
- 26.5k
3
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96
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When is the category of complexes of finite type?
For a ring $R$ define the category of complexes of length $n \geq 2$ as the category $C_n(R)$ with objects the complexes of the form $0 \rightarrow X_n \rightarrow \cdots X_1 \rightarrow 0$ with the ...
- 26.5k
3
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Proving that the exterior algebra is symmetric via the polynomial ring
Recall that a finite dimensional algebra $A$ over a field $K$ is called Frobenius in case $A \cong D(A)$ as right modules, and it is called symmetric in case $A \cong D(A)$ as bimodules (where $D=...
- 26.5k
3
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Weak generators of the right-bounded derived category of a finite-dimensional algebra
The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
- 61
3
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Upper bound for embedding of submodules of projective modules
Assume we have a finite dimensional algebra $A$ with the following property:
Every indecomposable submodule of a projective module embedds into $A^n$ for a fixed $n$.
Is there a good method to ...
- 26.5k
3
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71
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$\Omega^2(S) \cong \tau(S)$ for simple modules
Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra?
Here $\tau$ denotes the Auslander-Reiten translate, which is ...
- 26.5k
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Can a acyclic quiver algebra be derived equivalent to a non-acyclic quiver algebra?
Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver?
(I moved this question from another thread Derived equivalences of Dyck paths , where the ...
- 26.5k
3
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Derived invariant algebras and cluster tilting objects
This question is perhaps a little stupid as I have barely any evidence for it, but another thread just reminded me about it so maybe someone has an idea or counterexample for this.
Let $A$ be a ...
- 26.5k
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Ext for simple modules in selfinjective algebras
Let $A$ be a finite dimensional symmetric algebra (given by a connected quiver and non-semisimple) and $S$ a simple $A$-module. (I am more interested in symmetric algebras, but selfinjective examples ...
- 26.5k
3
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208
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A new characterisation of hereditary algebras?
Let $A$ be a quiver algebra with global dimension at most two (or more generally finite global dimension) and $A^e=A^{op} \otimes_K A$ be its enveloping algebra.
Guess:Is $A$ hereditary if and only ...
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Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
- 26.5k
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When is the stable category abelian
For which Artin algebras $A$ is the stable module (that is the module category modulo projectives) category of finitely generated modules abelian?
If you like you may take rings that are not Artin ...
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3
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Characterisation of reflexive modules for general rings
A module $M$ over a general ring A is called reflexive in case the canonical evaluation map $e_M: M \rightarrow M^{**}$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. Is $M$ reflexive iff $M \cong M^{*...
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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 ...
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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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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 ...
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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)...
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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 ...
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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):...
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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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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 ...
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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
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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 ...
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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 ...
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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", ...
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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
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Is a triangulated category admitting a tilting object triangle equivalent to the unbounded derived category of the endomorphism ring of this object?
Let $\mathcal{T}$ be a triangulated category. We call an object $G$ tilting if
$G$ is compact, that is, $\mathrm{Hom}_{\mathcal{T}}(G, -)$ preserves all set-indexed coproducts;
$G$ is a generator, ...
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