All Questions
Tagged with rt.representation-theory homological-algebra
255 questions with no upvoted or accepted answers
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Ext in a selfinjective algebra
Given a finite dimensional selfinjective algebra $A$ and an indecomposable module $M$, which is not projective.
Let $v=DHom(-,A)$ be the Nakayama functor.
In case $Ext^{i}(v(M),M) \neq 0$ for some i, ...
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2
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0
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148
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Algebras where all indecomposable modules are rigid
Is there a classification of selfinjective algebras having $Ext^{1}(X,X)=0$ for every indecomposable module X?
Examples include trivial extensions of representation-finite hereditary algebras.
One ...
- 26.5k
2
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0
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70
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Short exact sequences in p-group algebras
Given a group algebra of a finite p-group over a field of characteristic p.
Tachikawa proved that in this case $Ext^{1}(M,M) \neq 0$ for any finite dimensional non-projective module $M$.
Can one give ...
- 26.5k
2
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0
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105
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Why $T'$ dosen't have projective direct summand?
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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2
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0
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145
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Question about Ext$^1$ in local commutative algebras
Given a local commutative (commutative only if needed...) selfinjective (non-semisimple) finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{...
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2
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203
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Could Partial Tiltings be studied as Almost Complete Tiltings?
The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end.
Here $k$ denotes an algebraically closed field,...
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2
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77
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Calculation of minimal right $\operatorname{add}(M)$-approximations
given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 \...
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2
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293
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Definition of 'Koszul Ring' (in BGS)
In the paper 'Koszul Duality Patterns in Representation Theory' by Beilinson et. al, they give the definition of a Koszul Ring:
A Koszul ring is a positively graded ring $A = \bigoplus_{j \geq 0} A_j$...
- 922
2
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107
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In which non-gorenstein algebras, are all maximal ideals Gorenstein injective modules?
In which non-gorenstein algebras, are all maximal ideals Gorenstein injective modules?
or Are the Gorenstein injective dimensions of all maximal ideals finite?
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1
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80
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Selforthogonal modules and finitistic dimension
Algebras $A$ are always finite dimensional over a field here.
A module $M$ is called selforthogonal if $\operatorname{Ext}_A^i(M,M)=0$ for all $i>0$.
Define the orthogonal finitistic dimension $\...
- 26.5k
1
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124
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Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
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1
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106
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
1
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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 ...
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1
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115
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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\...
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A question concerning extension groups between simple modules
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k with exactly m simple modules up to isomorphism. Let S be a simple left A-module. ...
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1
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108
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When is a Koszul algebra derived equivalent to its dual
Let $A$ be a finite dimensional Koszul algebra of finite global dimension.
Question: When is $A$ derived equivalent to its Koszul dual algebra?
I wonder whether there is an exact condition to ...
- 26.5k
1
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0
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37
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Coxeter period of representation-finite selfinjective algebras
Let $A$ be a representation-finite selfinjective (quiver) algebra, that we assume to be connected and non-semisimple. Define the Coxeter period $p_A$ of $A$ to be equal to the period of the Coxeter ...
- 26.5k
1
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49
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Invariance under derived equivalence of a Gorenstein projective bimodule
A module $M$ over an finite dimensional algebra $A$ is called Gorenstein projective in case there exists an exact complex $(P_i)$ of projective $A$-modules such that the complex stays exact after ...
- 26.5k
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Finding minimal copresentations of projectives in stable endomorphism rings
Let $A$ be a finite dimensional algebra (you can assume it is selfinjective in case this helps) and $M$ an $A$-module without projective summands.
Let $B=\underline{End_A(M)}$, the stable endomorphism ...
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1
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0
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92
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Symmetric stable categories
Let $A$ and $B$ be Frobenius algebras that are stable equivalent.
In case $A$ is symmetric, is $B$ also symmetric? (no, see the comment of Jeremy Rickard) Does it hold in case $A$ and $B$ are ...
- 26.5k
1
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30
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Right approximations for special modules in Frobenius algebras
Let $A$ be a commutative Frobenius algebra (we can assume $A$ is also local) given by quiver and relations.
Let $M_i=A/p_iA$ be a module where $p_i$ is a path in Q.
Let $N:=A \oplus \bigoplus\limits_{...
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1
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0
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55
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$\mathrm{Ext}^1$-ordering on ${}^IW^{\Sigma_\mu}$
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^+...
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1
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0
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57
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Modules with arbitrary large complexity
Let $A$ be a finite dimensional algebra with a module $M$ who has minimal injective coresolution $(I_i)$. Define the complexity $cx(M)$ as $cx(M):= \inf \{ t \geq 0 | \dim(I_i) \leq a i^{t-1}$ for all ...
- 26.5k
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141
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Question on vanishing Hochschild cohomology
Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$.
Question:
Is there a finite dimensional selfinjective ...
- 26.5k
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47
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Piecewise hereditary algebras of Dynkin type that are QF-3
Is there an easy classification of piecewise hereditary (which means derived equivalent to a hereditary algebra) algebras of Dynkin type
that are Quasi-Frobenius-3 (meaning that the injective envelope ...
- 26.5k
1
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0
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117
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Derived equivalences and the Coxeter polynomial
Let $A$ be a quiver algebra with an acyclic quiver and primitive idempotents $e_i$.
The Cartan matrix $C_A$ of $A$ is defined as the matrix with entries $dim(e_i A e_j)$ and the Coxeter matrix $\phi_A$...
- 26.5k
1
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0
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77
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n-Gorenstein algebras and tilting modules
Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$...
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1
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0
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77
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On $Ext_A^2(S,A)$ for algebras $A$
For Nakayama algebras $A$ with a linear quiver and at most 7 points (which are 196 algebras) the following was true:
$max \{ dim(Ext_A^2(S,A)) | S $ simple $\}= max \{ dim(Ext_A^2(X,A)) | X $ ...
- 26.5k
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0
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80
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When is a stable endomorphism ring selfinjective?
Let $A$ be a local symmetric finite dimensional algebra and $M$ an $A$-module with at least two non-isomorphic indecomposable non-projective summands.
In case $\Omega^1(M) \cong M$ in the stable ...
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1
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0
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64
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Questions on holonomic modules
An Auslander-Gorenstein ring is a noetherian ring R that has finite left and right selfinjective dimension and such that $fd(I_i) \leq i$ for all $i \geq 0$ for an injective coresolution of the ...
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1
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69
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Inequality for the global dimension of quiver algebras
Let $A$ be a finite dimensional algebra of finite global dimension given by connected quiver and relations.
Do we have $gldim(A) \geq \min \{ \text{injdim}(S)+\text{projdim}(S) | S$ simple $\} $ in ...
- 26.5k
1
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0
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82
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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
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0
answers
125
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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 ...
- 26.5k
1
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0
answers
142
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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 $pd ...
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1
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0
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60
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$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 $I=<V^2,Z^2,XY,VX+aXZ,VY+YZ,VX+Y^2,VY-X^2>.$
...
- 26.5k
1
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0
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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 ...
- 26.5k
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63
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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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1
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0
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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(...
- 26.5k
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0
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68
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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 ...
- 26.5k
1
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0
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46
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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
1
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0
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71
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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 $i \geq 1$) ...
- 26.5k
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361
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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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0
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27
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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 $N$ ...
- 26.5k
1
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0
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135
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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 ...
- 159
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57
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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
1
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0
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124
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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 $...
- 775
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0
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89
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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. $T^{\bullet}\in \mathcal{K}^b(A-proj)$ ...
- 775
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0
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120
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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
1
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0
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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 ...
- 741
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0
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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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