All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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
3
votes
0
answers
175
views
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
1
vote
1
answer
140
views
Is the Cartan permanent odd for finite global dimension?
Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix.
Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd ...
- 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
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 "...
- 26.5k
1
vote
1
answer
110
views
Question on strong cotilting modules
In https://www.sciencedirect.com/science/article/pii/0001870891900378 section 6 a cotilting module T over an algebra A is said to be strong in case $\hat{add(T)}$ coincides with the subcategory of ...
- 26.5k
5
votes
0
answers
92
views
Criteria for being representation-infinite for subcategories of quiver algebras
Let $A$ be a quiver algebra over a field $K$ (maybe we need algebraically closed?).
Then the following is two statements are well known:
In case $A$ is representation-infinite, every Auslander-Reiten ...
- 26.5k
2
votes
1
answer
157
views
A non-monoidal functor that respects fusion rules
Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and ...
- 159
3
votes
0
answers
427
views
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 ...
- 26.5k
10
votes
1
answer
920
views
Is the homotopy category of an abelian model category abelian?
A model structure on an abelian category $A$ is called an abelian model structure if the cofibrations are precisely the monomorphisms with cofibrant cokernel, and if the fibrations are precisely the ...
- 30.3k
6
votes
2
answers
332
views
Ext in symmetric algebras and group algebras
Let $A$ be a selfinjective algebra and for an indecomposable module $M$ define $\psi_M:= \inf \{ i \geq 1 | Ext_A^i(M,M) \neq 0 \}$.
Questions:
In case $A$ is symmetric, do we have $\psi_M \leq max \...
- 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
2
votes
0
answers
75
views
Double dual of the simple module in local algebras
Let $A$ be a local Artin algebra that is not selfinjective with simple module $S$.
Questions:
Can $S^{**}$ be indecomposable?
$S^{**}$ be somehow generally be described (for example as a ...
- 26.5k
1
vote
1
answer
174
views
Reference for a result of Auslander about the global dimension
One of Auslanders famous theorems is that he proved that the global dimension of a semiprimary ring is equal to the maximum of the projective dimensions of the simple modules of the ring. This result ...
- 26.5k
5
votes
2
answers
226
views
Algebras with all simples reflexive
Let $A$ be a ring. Recall that a module $M$ is called reflexive in case the canonical evaluation map $f_M : M \rightarrow M^{**}$ is an isomorphism. Here $(-)^{*}$ denotes the functor $Hom_A(-,A)$.
In ...
- 26.5k
2
votes
1
answer
98
views
Reflexive modules up to multiplicity
Call an indecomposable module $M$ over a ring $A$ (restrict to finite dimensional algebras if you like or if it helps) $n$-almost reflexive in case $M^{**} \cong nM$, when $(-)^{*}=Hom_A(-,A)$ and $nM$...
- 26.5k
3
votes
0
answers
169
views
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^{*...
- 26.5k
4
votes
1
answer
615
views
Characterisation of reflexive modules
Let $A$ be a semiperfect noetherian ring.
A module $M$ is called reflexive in case the canonical map $f_M: M^{**} \cong M$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. This is equivalent to say that ...
- 26.5k
4
votes
1
answer
189
views
Property of non-Gorenstein algebras
In the article http://www.sciencedirect.com/science/article/pii/S0021869301991306?via%3Dihub (see also the MO thread Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\...
- 26.5k
1
vote
0
answers
125
views
Global dimension of algebras under field change
Let $X$ be the collection of all fields (or if this is too large, the collection of all fields with cardinality at most the cardinality of $\mathbb{R}$).
Given a quiver algebra $A=FQ/I$ of finite ...
- 26.5k
1
vote
0
answers
142
views
Generalized strong no loop conjecture
Strong no loop conjecture: Let $A$ be an artin algebra and $S$ be a simple module in $mod A$, where $mod A$ denotes the right finitely generated module category. If $Ext_{A}^{1}(S,S)\neq 0$, then $pd ...
- 181
2
votes
0
answers
103
views
Representation dimension of Auslander algebras
Is the representation dimension of Auslander algebras known? Is there an example of such algebras with representation dimension larger than 4?
- 26.5k
5
votes
2
answers
360
views
Injective dimension of the Jacobson radical and global dimension
Given a finite dimensional algebra $A$ with Jacobson radical $J$.
Is the global dimension of $A$ equal to the injective dimension of $J$?
Together with Xiao-Wu Chen and Srikanth Iyengar we proved this ...
- 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
7
votes
0
answers
296
views
Secret exact sequence in path algebras of Dynkin type
Given a connected finite dimensional path algebra $A=KQ$ of Dynkin type with enveloping algebra $A^e= A^{op} \otimes_K A$.
I can prove that there is a canonical exact sequence connecting the ...
- 26.5k
0
votes
0
answers
76
views
Determination of the characteristic tilting module
Let $A$ be a finite dimensional selfinjective algebra and $M$ an indecomposable non-projective $A$-module such that the algebra $B:=End_A(A \oplus M)$ is standardly stratified.
Examples of such $B$ ...
- 26.5k
2
votes
0
answers
121
views
Ext of a Schur algebra
Let $A=A_n$ be the representation-finite block of a Schur algebra with $n$ simple modules for $n \geq 2$. Quiver and relations of $A$ can be found in 6.1. of https://arxiv.org/pdf/1607.05965.pdf . Let ...
- 26.5k
2
votes
0
answers
61
views
Question on outer Ext-products
For group algebras $A=KG$ over a field $K$ with finite group $G$ there exists an outer product on Ext:
$Ext_A^i(M,N) \otimes_K Ext_A^j(M',N') \rightarrow Ext_A^{i+j}(M \otimes_K M',N \otimes_K N')$.
...
- 26.5k
1
vote
0
answers
60
views
$Ext^i(D(R),R)$ for a certain commutative algebra
Let $k$ be a field which is not algebraic over a finite field and $a \in k$ an element of infinite multiplicative order.
Let $R=k[V,X,Y,Z]/I$ with $I=<V^2,Z^2,XY,VX+aXZ,VY+YZ,VX+Y^2,VY-X^2>.$
...
- 26.5k
1
vote
0
answers
230
views
Derived equivalence of algebras
Given two finite dimensional algebra $A$ and $B$ with generator-cogenerators $M$ and $N$. Define two algebras $X:=End_A(M)$ and $Y:=End_B(N)$.
Assume X and Y are derived equivalent. Are A and B ...
- 26.5k
2
votes
0
answers
63
views
QF-3 monoid algebras
A finite dimensional algebra $A$ is called QF-3 in case the injective envelope of the regular module is projective. For example all Frobenius algebras are QF-3.
Given a monoid algebra $kG$ of a finite ...
- 26.5k
1
vote
1
answer
149
views
Finding modules to check for finite global dimension
Given a finite dimensional algebra $A$ and a generator-cogenerator $M$ and let $B:=End_A(M)$. $B$ has finite global dimension iff every $A$-module has finite $add(M)$-resolution dimension, which is ...
- 26.5k
6
votes
0
answers
292
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 ...
- 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
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 ...
- 26.5k
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: $...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k