All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
6
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0
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506
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How can you see the minimal relations on a quiver from its bimodule resolution?
Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...
- 61
5
votes
2
answers
1k
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Describing the kernel of the exponential map as a homology group
I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I ...
- 5,637
5
votes
1
answer
261
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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
5
votes
2
answers
1k
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An example where finitistic dimension does not equal right global dimension?
The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
- 30.3k
5
votes
2
answers
479
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How to define cohomology of algebraic structures?
I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain
\begin{align*}
\cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
- 775
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....
- 26.5k
5
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1
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400
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Injective modules
Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。
- 169
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1
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145
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Commutator of finite global dimension algebras
Let $A=KQ/I$ be a finite dimensional quiver algebra of finite global dimension.
Is it true that the dimension of $A/[A,A]$ is equal to the number of simples of $A$?
Here $[A,A]$ is the vector space ...
- 26.5k
5
votes
1
answer
207
views
Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
- 26.5k
5
votes
1
answer
235
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Concept of an exact ideal of a module category
Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
- 696
5
votes
1
answer
226
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Frobenius algebras from symmetric polynomials
Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
- 26.5k
5
votes
1
answer
187
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Periodic objects in Frobenius categories
Let $A$ be a finite dimensional Gorenstein algebra and $C$ the stable module category of $A$.
Question: Does there always exist an indecomposable periodic object $X$ in C, that is an object with $\...
- 26.5k
5
votes
1
answer
268
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Global dimension 2 Nakayama algebras , 321-avoiding permutations and Fibonacci combinatorics
A $n$-LNakayama algebra (for $n \geq 2$) is a list $[c_0,c_1,...,c_{n-1}]$ with $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i -1 \leq c_{i+1}$ for all $i$.
They are in bijection with Dyck paths, ...
- 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?
...
- 775
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
5
votes
1
answer
573
views
"as close to being semisimple as it can possibly be."
I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.
In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns......
- 595
5
votes
1
answer
829
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Rigid monoidal and closed monoidal categories
I am trying to understand the relationship between rigid monoidal categories and closed monoidal
categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ ...
- 1,144
5
votes
1
answer
212
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On tilting and cotilting modules
Let A be an Artin algebra and assume all modules are basic, then a classical result says that tilting modules $T$ are in bijection with complete cotorsion pairs $(T^{\perp}, \check{ add(T)})$ (with ...
- 26.5k
5
votes
1
answer
151
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Derived equivalences of Artin algebras with finitistic dimension zero
Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero?
In case this is true, this might generalise the ...
- 26.5k
5
votes
1
answer
225
views
Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 26.5k
5
votes
1
answer
186
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(Stable) Auslander algebras in a specific example
Let $k$ be a field and $Q$ be the quiver with two vertices 1 and 2 and three arrows:
$a$ from 2 to 1, $b$ from 2 to 1 and $c$ from 2 to 2.
Let $I_1=\langle ab-c^2,ba\rangle$ and $I_2=\langle ab-c^2,c^...
- 26.5k
5
votes
1
answer
243
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Questions on group and Nakayama algebras from a book
Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series.
In the book "Classical artinian ...
- 26.5k
5
votes
1
answer
353
views
Existence of non-trivial reflexive modules
Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, ...
- 26.5k
5
votes
2
answers
226
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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
5
votes
1
answer
365
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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
5
votes
2
answers
281
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Isomorphism for Ext spaces for finite dimensional algebras
Let $A$ be an Artin algebra with enveloping algebra $A^e$.
Then we have $Hom_{A^e}(X,A^e) \cong Hom_A(D(A) \otimes_A X,A)$ for a bimodule $X$. (see for example in the article "A theorem of Green on ...
- 26.5k
5
votes
2
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360
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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
5
votes
0
answers
213
views
Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
- 26.5k
5
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0
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190
views
On the not so clear relationship between torsion theories and localization for a newcomer
Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...
- 83
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83
views
It there an algebra of the form $B_T$ with global dimension 3?
Let $A$ be the (symmetric Frobenius) algebra $A=K[x]/(x^3) \otimes_K K[x]/(x^3)$ over a field $K$, which is isomorphic to the group algebra of $C_3 \times C_3$, with $C_3$ cyclic of order 3, when $K$ ...
- 26.5k
5
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0
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142
views
A practical way to check whether a module is periodic
A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
- 26.5k
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0
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116
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An intelligent ant living on a symmetric quiver algebra - Does it have a way to find out whether it lives on a trivial extension?
For a given algebra $B$ over a field $K$ the trivial extension $T(B)$ of $B$ is defined as follows:
The underlying vectorspace is $T(B)=B \oplus D(B)$ where $D(B)=Hom_K(B,K)$ and the multiplication is ...
- 26.5k
5
votes
0
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76
views
Reference on two numbers associated to a module of finite homological dimension
Let $A$ be a finite dimensional algebra over a field $K$ with a module $M$ which has finite projective dimension and finite injective dimension.
Let $n \geq 1$.
Let $(P_i)$ be a minimal projective ...
- 26.5k
5
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0
answers
97
views
Periodics of Coxeter matrices for truncated Nakayama algebras
For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$).
So for example for $n=7$ and $r=3$ we ...
- 26.5k
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0
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132
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On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber
This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...
- 51
5
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0
answers
168
views
Higher analogue of the Auslander-Bridger transpose
Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$.
Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal ...
- 26.5k
5
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0
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113
views
On algebras where all indecomposables have no selfextensions
Let $A$ be a finite dimensional algebra (we can assume it is a connected quiver algebra).
Call $A$ extfree in case for every indecomposable $A$-module $M$ we have $\operatorname{Ext}_A^i(M,M)=0$ for ...
- 26.5k
5
votes
0
answers
114
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Extreme no loop conjecture for group algebras
Let $A=KG$ be a group algebra for a finite group $G$. Let $S$ be a simple $A$-module. The extreme no loop conjecture predicts that $Ext_A^1(S,S) \neq 0$ implies $Ext_A^i(S,S) \neq 0$ for infinitely ...
- 26.5k
5
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0
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140
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Open problems about Morita and derived invariants
Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
- 26.5k
5
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0
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91
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Bound on the sum of projective and injective dimension
Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category.
In proposition 1.2. of https://link.springer.com/article/10....
- 26.5k
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0
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125
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Stable equivalence and stable Auslander algebras
Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
- 26.5k
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105
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Derived invariant acyclic algebras
Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$.
For example local ...
- 26.5k
5
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0
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88
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Cluster-tilting object for a local non-selfinjective algebra
Let $A$ be a non-selfinjective (which is equivalent to non-Gorenstein) local finite dimensional algebra.
Is there a known example of such an $A$ having a cluster-tilting object?
Id be surprised to ...
- 26.5k
5
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0
answers
254
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Tannakian theory for Lie algebras
Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
5
votes
0
answers
380
views
A tensor product for dg-categories
For a finite group denote by $\mathbf{Ch}^G$ the dg-category of $G$-representations in chain complexes over a field.
Is there a tensor product $\otimes$ of dg-categories (similar to the Deligne ...
- 1,382
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
5
votes
0
answers
120
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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 ...
- 26.5k
5
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0
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303
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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
5
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0
answers
209
views
Ext^1 for a local finite dimensional selfinjective algebra
Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here:
http://arxiv.org/pdf/1609.00588.pdf.
There it is ...
- 26.5k
5
votes
0
answers
259
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divided power structure on Hocschild cohomology?
Does Hochschild cohomology of a cocommutative Hopf algebra over a field of positive characteristic have a natural divided power structure?
If not, perhaps a certain natural extra structure on the ...
- 1,526