All Questions
Tagged with homological-algebra quivers
19 questions
0
votes
0
answers
103
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Matrix of the minimal projective presentation of a $\tau$-rigid module
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
- 839
9
votes
1
answer
236
views
Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\...
- 5,407
2
votes
0
answers
86
views
Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an ...
- 839
2
votes
1
answer
165
views
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
1
vote
0
answers
52
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Is the Schofield semi invariant defined at $V/IV$?
Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
- 839
1
vote
0
answers
115
views
Prove that $B$ is a directing module
Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
- 839
1
vote
1
answer
218
views
A result of Schofield in the case of quivers with relations
Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\...
- 839
2
votes
2
answers
140
views
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 ...
- 6,292
2
votes
0
answers
135
views
How to compute the derived functor of bounded derived categories of hereditary algebra?
Let $\Lambda$ be
a finite dimensional algebra given by the quiver
$$\cdot\leftarrow\cdot\leftarrow\cdot\rightarrow\cdot.$$
It can be view as an triangulated matrix algebra.
$$\Lambda={A\ \ M\choose0\ ...
- 63.9k
12
votes
1
answer
577
views
Embedding of a derived category into another derived category
I am considering the following two cases:
Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...
- 1,161
11
votes
0
answers
202
views
Quiver and relations for blocks of category $\mathcal{O}$
In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ .
...
- 26.5k
8
votes
3
answers
1k
views
Intuition behind the canonical projective resolution of a quiver representation
Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...
- 63.9k
6
votes
2
answers
768
views
Auslander-Reiten theory for Gorenstein algebras
In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
- 1,244
6
votes
2
answers
329
views
Path algebras are formally smooth
In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2).
I have two questions. First, how to show this claim and ...
- 1,505
1
vote
0
answers
82
views
Endomorphism ring of a cotilting module
Given a basic tilting or cotilting right module $T$ over an algebra $A$ given by quiver and relations, is there a "linear-algebra method" to decide whether $\operatorname{End}_A(T) \cong A?$
Here "...
- 1,161
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
6
votes
0
answers
209
views
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
1
vote
1
answer
229
views
Bounded algebras of finite global dimension
Let $k$ be a field, $Q$ an acyclic finite quiver and $I$ an admissible ideal of $kQ$.
I am looking for a reference for the fact that the bounded algebra $kQ/I$ has finite global dimension.
- 3,686
1
vote
1
answer
277
views
finitely presented representations
Let Q=(V,E) be a direct graph where V is the set of all its vertices and E denotes the set of all its arrows. $X$ is called a representation of Q by modules if it is a functor from Q to R-Mod. i.e. $X(...
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