All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
4
votes
0
answers
69
views
Auslander-Solberg algebras from non-rigid modules
Let $A$ be a Nakayama algebra and $M$ be the direct sum of all indecomposable $A$-modules $N$ with $Ext_A^1(N,N) \neq 0$.
The following is suggested by computer experiments with QPA:
Question: Is ...
- 26.5k
4
votes
1
answer
346
views
Verma module and vanishing of extension groups
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the associated Weyl group and let $\Phi$ be its root system.
We write $\Phi^+...
- 1,875
4
votes
0
answers
82
views
On strongly simply connected quiver algebras
Let $A$ be a representation-finite quiver algebra. In this case $A$ is simply connected if and only if its first Hochschild cohomology vanishes by a result of Buchweitz and Liu. $A$ is called strongly ...
- 26.5k
4
votes
0
answers
88
views
Minimal injective coresolution in the stable Auslander algebra
Let $A$ be a finite dimensional (connected) quiver algebra. Let $T(A)$ denote the full subcategory of coherent functors from $mod-A$ to $Ab$ that vanish on projective objects. $T(A)$ is equivalent to ...
- 26.5k
4
votes
0
answers
58
views
Interpretation of stable Hom in Nakayama algebras
Let $A$ be a Nakayama algebra with Kupisch series $[c_0,c_1,...,c_{n-1}]$ and Jacobson radical $J$ (given by quiver and relations). As is well known every indecomposable $A$-module is of the form $e_i ...
- 26.5k
4
votes
0
answers
85
views
Deciding whether two algebras are derived equivalent
Given two finite dimensional quiver algebras $A$ and $B$ (over a nice field in case that helps, for example a finite field).
Question: Can an there be a finite algorithm that decides whether $A$ ...
- 26.5k
4
votes
0
answers
43
views
Cartan determinants of minimal Auslander-Gorenstein algebras
Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/...
- 26.5k
4
votes
0
answers
71
views
Koszul and quadratic algebras with Gorenstein dimension 2
In proposition 2.19. of http://inmabb.criba.edu.ar/revuma/pdf/v48n2/v48n2a05.pdf it was mentioned that a finite dimensional algebra of global dimension 2 is quadratic if and only if it is Koszul.
...
- 26.5k
4
votes
0
answers
66
views
Periodic modules in Frobenius algebras
Let $A$ be a finite dimensional Frobenius algebra and assume there exists an indecomposable periodic module $M$, that is $\Omega^n(M) \cong M$ for some $n$.
Question: Does this imply that there is ...
- 26.5k
4
votes
0
answers
84
views
Finitistic dimension via a bimodule
Let $A$ be a connected finite dimensional basic algebra.
Question: Is there an indecomposable $A$-bimodule $W$ such that the finitistic dimension of $A$ is equal to the right projective dimension ...
- 26.5k
4
votes
0
answers
90
views
Number of hereditary modules of a hereditary algebra
Let $Q$ always denote a Dynkin quiver.
Given a connected path algebra $A=kQ$ and a module $M$, is there a useful criterion on $M$ when $End_A(M)$ is again a connected quiver algebra?
Call a module ...
- 26.5k
4
votes
0
answers
81
views
Sum of all projective dimensions of simple modules
Let $X_{n,t}$ be the set of all finite dimensional algebras (we can assume they are given by a connected quiver and admissible relations) that have global dimension equal to $n$ and $t$ simple modules....
- 26.5k
4
votes
0
answers
135
views
Question on syzygies
Given a finite dimensional algebra $A$ over a field $K$ with $\Omega^i(D(A)) \cong \Omega^{i+1}(D(A))$ for some $i \geq 1$, where $D(A)=\operatorname{Hom}_K(A,K)$.
Do we then also have $\Omega^{-i}(A)...
- 26.5k
4
votes
0
answers
210
views
Conjecture on tilting modules for an Auslander algebra
On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism
classes of modules, occurring as the $i$-th summand of ...
- 10.5k
4
votes
0
answers
61
views
Inequality for the global dimension
Let $A$ be a finite dimensional algebra with finite global dimension g and Loewy length l and dimension of the Jacobson radical being r.
Do we have $g \leq r-(l-2)$ ?
$g \leq r$ was proven in http://...
- 26.5k
4
votes
0
answers
228
views
Question on $n$-torsionless modules
Let $A$ be a finite dimensional algebra. Recall that a module $M$ is called $n$-torsionfree in case $Ext_A^i(D(A),\tau(M))=0$ for all $i=1,...,n$ when $\tau$ denotes the Auslander-Reiten translate. ...
- 26.5k
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
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
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
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
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
98
views
Bound for the global dimension of higher Auslander algebras
Let algebras be finite dimensional and connected.
Recall that an algebra $A$ is called a higher Auslander algebra in case it the dominant dimension coincides with the global dimension and both ...
- 26.5k
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 ...
- 10.4k
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
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
4
votes
0
answers
76
views
Minimal rank of a permutation resolution of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
- 14.2k
4
votes
0
answers
157
views
Endomorphismrings of maximal submodules.
The question I am interested in answering is the following:
Suppose that for a pair of $d$-dimensional modules $M$ and $N$ over a $k$-algebra ($k$ a field) $R$ we have that $\dim_k \rm{Hom}_R(X,M)\...
- 236
4
votes
2
answers
771
views
Finitistic dimension conjecture for quadratic algebras
The finitistic dimension of a finite dimensional algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. The finitistic dimension conjecture says ...
- 26.5k
3
votes
2
answers
1k
views
Dual of a projective module
Let $R$ be a noncommutative ring with unit, let $P$ be a projective left $R$-module, and denote $^{\vee}\!P := \,_R\mathrm{Hom}(P,R)$. One often sees it written that projectivity implies an ...
3
votes
1
answer
244
views
Left module which cannot be made into a bimodule?
Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...
3
votes
1
answer
339
views
If the Hom-space of finite length modules is generated by single elements, must the elements be conjugate?
Let $A$ be an Artin $k$-algebra for a commutative artinian ring $k$ (e.g. $A$ is a finite dimensional algebra over a field $k$). Let $X,Y$ be finite length left $A$-modules. If $\text{Hom}_A(X,Y)$ is ...
- 696
3
votes
1
answer
240
views
Split monomorphisms of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...
- 696
3
votes
2
answers
214
views
History of an open problem on partial tilting modules
The following is an open problem:
Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
- 26.5k
3
votes
1
answer
301
views
If a bimodule is "generated" by single elements, must the elements be conjugate?
Let $A$ and $B$ be Artin $k$-algebras for a commutative artinian ring $k$ (e.g. $A$ and $B$ are finite dimensional $k$-algebras for a field $k$). Let $M$ be an $A$-$B$-bimodule of finite length over $...
- 696
3
votes
1
answer
129
views
Extensions for simple modules over group algebras
Let $G$ be a finite group and $K$ a field with field extension $L$ ($K$ perfect and $L$ finite field extension first for simplicity),
Let $S$ be a simple $KG$ module.
Viewed as a $LG$-module $S$ ...
- 26.5k
3
votes
1
answer
565
views
Finiteness of cohomology group
Suppose $G$ is a finite Galois group, and $M$ is an infinite $G$-module. When can I say that $H^1(G, M)$ is finite?
I know this not true in general. Is it true under certain assumptions on $M$?
To be ...
- 143
3
votes
1
answer
189
views
Question on $\operatorname{Ext}$ in a local Frobenius algebra
Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional).
Question:
Is there an example 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 ...
- 7,789
3
votes
1
answer
118
views
Weakly symmetric Frobenius algebras
Let $A$ be a finite dimensional Frobenius algebra and $e$ and idempotent of $A$.
It is well known that the algebra $eAe$ does not have to be a Frobenius algebra. But if $A$ is additionally symmetric, ...
- 26.5k
3
votes
1
answer
446
views
A set of objects classically generates the full subcategory of compact objects iff it generates the whole category
Sorry in advance if my question doesn't have the level of this community.
I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
- 224
3
votes
1
answer
252
views
Higher Extension Group Question
Suppose we have an associative unital ring $R$, and we have an $R$-module $M$ with a length 3 socle filtration, i.e. write
$$soc(M) \text{ for the socle of } M,$$
$$soc^2(M) \text{ for the preimage ...
- 1,077
3
votes
1
answer
258
views
Invertible bimodules which are isomorphic in the stable module category
I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...
- 15.2k
3
votes
1
answer
106
views
The kernel of the morphism from the Picard group to the stable Picard group of a self-injective algebra
Let $\Lambda$ be a finite-dimensional self-injective algebra (over an algebraically closed field, if necessary). Let $Pic(\Lambda)$ be the group of natural isomorphism classes of self-equivalences $...
- 15.2k
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 $derived$ $equivalent$ ...
- 775
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
3
votes
1
answer
98
views
Finding automorphisms and cyclic modules via QPA
Given a symmetric finite dimensional algebra $A$ over a finite field with enveloping algebra $A$.
Assume we know that $\Omega_{A^e}^i(A) \cong A_{f}$, where $f$ is some automorphism of the algebra $A$....
- 26.5k
3
votes
1
answer
163
views
Identity for $Ext^1$ for special algebras
Let $A$ be a finite dimensional algebra and assume all modules are also finite dimensional. A module $M$ is said to have dominant dimension at least $n$ in case the term $I_i$ for $i=0,1,...,n-1$ are ...
- 26.5k
3
votes
1
answer
237
views
Finding all selforthogonal indecomposable modules
Given a finite dimensional algebra $A$ with finite global dimension such that there are only finitely many basic tilting modules. Then every selforthogonal indecomposable module $M$ (that is a module ...
- 26.5k
3
votes
1
answer
126
views
Strong cotilting module for radical square zero algebras
Given a connected Artin algebra $A$ (a quiver algebra $A=kQ/I$ if it helps) with radical square zero. Can the basic strong cotilting right $A$-module $T$ be explicitly written down?
A cotilting ...
- 26.5k
3
votes
1
answer
146
views
Bijection on tilting modules
Given a finite dimensional hereditary algebra A and let $X_A$ denote the set of tilting $A$-modules.
Questions:
1.Is there a "canonical" bijection from $X_A$ to $X_A$ that sends $A$ to $D(A)$?
...
- 26.5k