All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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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 ...
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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 $pd ...
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$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 $I=<V^2,Z^2,XY,VX+aXZ,VY+YZ,VX+Y^2,VY-X^2>.$
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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 ...
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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 ...
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72
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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(...
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Ext in Gorenstein algebras
My computer suggests that the following is true for Nakayama algebra (it also found not counterexample for arbitrary algebras):
Let $A$ be an algebra of finite Gorenstein dimension $g \geq 1$, then ...
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Ext in selfinjective algebras
Is there a selfinjective finite dimensional algebra $A$ with a non-projective module $M$ such that $Ext^{i}(M,M)=0$ for all $i=1,2,...,2n-1$ when $n$ denotes the number of simple module of that ...
- 26.5k
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Non-Gorenstein projective maximal Cohen-Macaulay module
In http://s.web.umkc.edu/segal/papers/refl.pdf the authors gave the first example of a maximal Cohen-Macaulay module $M$ (that is a module over an algebra A with $Ext^{i}(M,A)=0$ for all $i \geq 1$) ...
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Property of the syzygy functor of $\operatorname{\underline{mod}} A$
Let $A$ be an artin algebra. We denote by $\operatorname{mod}A$ the category of finitely generated left $A$-modules. We denote by $[A]$ the ideal of morphisms between $A$-modules which factor through ...
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Approximations of modules in a special setting
Given a local finite dimensional nonselfinjective algebra $A$ and $M:=A \oplus D(A)$. Can one find a general formula for the minimal right add(M)-approximation of a general indecomposable module $N$ ...
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Equivalence between blocks of BGG categories $\mathcal{O}$ which does not preserve highest weight structure
Let $\mathfrak{g}$ be a finite-dimensional (complex) semisimple Lie algebra. Then we denote the BGG category for $\mathfrak{g}$ by $\mathcal{O}$ as usual. It is well-known that $\mathcal{O}$ as well ...
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Algebras with gorenstein dimension equal to the dominant dimension equal to one
Let algebras always be finite dimensional (and connected).
In https://arxiv.org/pdf/0809.4897v3.pdf , the algebras with global dimension equal to the dominant dimension equal to one are classified as ...
- 26.5k
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Some questions in a paper of derived categoires and stable equivalence
I am reading the paper "Derived categories and stable equivalence", the link is here:http://www.sciencedirect.com/science/article/pii/0022404989900819.
At theorem 2.1, there is an equivalent functor $...
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How to get the following functor of derived equivalent categories?
Let $A$ be a left coherent ring, that is, a ring for which the kernel of any homomorphisms between finitely generated projective modules are finitely generated. $T^{\bullet}\in \mathcal{K}^b(A-proj)$ ...
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Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$
Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
- 257
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336
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generators for derived category
Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...
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238
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Is this a pure monomorphism?
Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
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337
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Homological dimensions of tensor products of algebras
Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A ...
- 26.5k
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On a claim of Zagier on extending a map to cocycle
Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ \...
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Algebra with all modules non-rigid 2
Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective indecomposable module $M$, with the condition that at least one such module exists....
- 26.5k
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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 ...
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223
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Representation dimension of a special algebra
Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...
- 1,755
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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
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76
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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
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112
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Some places I can't understand in the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra"
I am reading the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra", the link is here: https://arxiv.org/pdf/1608.04212.pdf
There are some places I can't ...
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116
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A question on the paper "The classification of algebras by dominant dimension"
I'm reading the paper "the classification of algebras by dominant dimension" by Bruno J.Mueller, the link is here http://cms.math.ca/10.4153/CJM-1968-037-9.
In the proof of lemma 3 on page 402, there ...
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72
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Contravariant finiteness of a certain subcategory
Let $A$ be an algebra with finite dominant dimension $d \geq 1$ and $Dom_d$ the full subcategory of modules with dominant dimension at least $d$ and $Proj$ the full subcategory of modules of finite ...
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77
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How to get $I_i \in add(\nu_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}_A(S,S)=0$?
Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let ...
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324
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$Ext$ functor over a product of groups
Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 \...
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How to show the following properties of $Coker(d^{-n-1})$?
Let $A$ be a k-algebra,where k is a fixed field. We denote by $\mathfrak{D}^b(A-mod)$ the bounded derived A-module category. A complex $Z^{\bullet}=(Z^i,d^i) \in \mathfrak{D}^b(A-mod)$ such that all $...
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