Questions tagged [homological-algebra]
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
2,611
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Free DGA given a map and cohomology groups
Why do Free DGAs on a morphism often give the same (co)homology as other (co)homologies?
Here is the example that comes to mind first:
Example: Let $R$ be a ring, let $A$ be an $R$-algebra, let $M$ ...
2
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51
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Selfinjective algebras with loops
Given a selfinjective finite dimensional algebra $A$ with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$.
Question:
Is A derived equivalent to an algebra with a loop in the quiver in ...
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53
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Strong no loop conjecture for uniserial modules
Let $A$ be a an Artin algebra. The strong no loop conjecture states that a simple $A$-module with $Ext_A^1(S,S) \neq 0$ has infinite projective dimension.
This conjecture was recently proved for ...
2
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44
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On monomial and $\Omega^d$-finite algebras
Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra.
It is well known that monomial algebras ...
6
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Why does every chain complex have a map into its cone?
In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...
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What does "standard Koszul morphism" mean?
I'm reading a paper 'D'Andrea, Carlos(RA-UBA), Dickenstein, Alicia(RA-UBA)Explicit formulas for the multivariate resultant. (English summary)
Effective methods in algebraic geometry (Bath, 2000).
J. ...
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198
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Equivalence of the category of covariant functors and the category of contravariant functors
Let $\mathcal{C}$ be a category. Then we have the category $\mathcal{C}^{\vee}$ of contravariant functors from $\mathcal{C}$ to $\mathcal{Sets}$ which is the category of sets. In the textbook "Sheaves ...
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Checking $\mathbb{K}_{U\times(a,b)}\ast\mathbb{K}_{[0,\infty)}\simeq \mathbb{K}_{U\times[a,\infty)}[-1]$ in derived category $D(X\times\mathbb{R})$
Let $D(X\times\mathbb{R})$ be the derived category of sheaves of $\mathbb{K}$-vector spaces on a smooth manifold $X\times\mathbb{R}$ where $\mathbb{K}$ is a ground field. Let $p_1:X\times\mathbb{R}\...
2
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372
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Computing injective resolution of some constant sheaves
I follow the notations on "Sheaves on manifolds" written by Kashiwara-Schapira. Let $\mathbb{K}$ be a ground field and $X$ be a smooth manifold. Let $D(X)$ be the derived category of sheaves of $\...
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55
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$\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^+...
4
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1
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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^+...
11
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Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
4
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Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?
Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...
4
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Testing whether a module generates $K_0(\mbox{mod-}A)$
Given a representation-finite (connected) quiver algebra $A$ and a module $M$.
Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K_0(\mbox{mod-}A)$?
Can ...
6
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1
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347
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Conditionally convergent spectral sequences with exiting and entering differentials
I have to deal with unbounded filtrations and want to use the conditional convergence of spectral sequences and the results from
[1]: J. Michael Boardman, Conditionally Convergent Spectral Sequences,...
6
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Homotopy transfer of cyclic L-infinity algebras
Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
5
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1
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Strøm model structure on nonnegatively graded chain complexes
Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes.
The ...
5
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143
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Simplicial Complex Induced by a Morphism
Let $(C, \otimes, I)$ be a symmetric monoidal abelian category. For an object $M$ in $C$ with a map $\rho : M \rightarrow I$, we can form the chain complex $M_*$ where $M_n = \otimes_{i = 1}^n M$ and $...
3
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1
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Ext-vanishing in abelian categories
Given an abelian category $A$ with enough projectives and enough injectives such that projectives do not coincide with injectives.
Can we have $Ext^i(I,P)=0$ for any $i>0$ and injective $I$ and ...
10
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306
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Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
2
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68
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Stable m-Calabi Yau property for Frobenius categories
Let $C$ be a Frobenius category. The stable category $\underline{C}$ is called $m$-Calabi Yau in case it is Hom-finite and there is a functorial duality
$D \underline{Hom}(X,Y)=\underline{Hom}(Y,\...
2
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72
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Equivalence from a tilting module
Let $A$ be a finite dimensional algebra. For a subcategory $C$ of $mod-A$ let $\overline{C}$ be the objects $X \in mod-A$ such that there exists an exact sequence $0 \rightarrow C_n \rightarrow ... \...
6
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0
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120
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Cyclic version of Lie algebra cohomology
Lie algebra cochains have a differential $d$ where $d^2 =0$ because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. $L_\infty$ algebra cochains have a ...
4
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82
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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 ...
2
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56
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Characterisation of representation-directed algebras
A representation-directed algebra $A$ over a field $K$ has the property that for every indecomposable module $M$ we have $\mathrm{End}_A(M) \cong K$ and $\mathrm{Ext}_A^i(M,M)=0$ for all $i>0$.
...
2
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1
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304
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Simple object of $k[X,Y]/(Y^2)$
Let $k$ be a field. Let $A=k[X,Y]/(Y^2)$ be the quotient of polynomial ring $k[X,Y]$. Let $\mathcal{C}$ be the category of finite-dimensional $A$-modules $M$ with the action of $X$ nilpotent, and of ...
16
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2
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688
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How complicated can a finite double complex over a field be?
A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is ...
3
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Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes
Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
4
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2
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714
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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 ...
4
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1
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Definition of dualizing complex
Sorry for a not research level question asking for a definition but unfortunately I nowhere found a source which explains the construction presented below in a satisfactory way.
This question refers ...
4
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0
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85
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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 ...
4
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0
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142
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Extensions of rings
Let $K$ be a commutative but not necessarily unital ring. Let $C$ be a unital commutative ring.
An extension is the data of a unital commutative ring $R$, a homomorphism $\phi:K\rightarrow R$, a ...
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/...
3
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Endomorphism ring of a generator-cogenerator over acyclic algebras
Let $A$ be an acyclic quiver algebra, $M$ a generator-cogenerator and $B=End_A(M)$.
Questions:
Does $B$ have finite global dimension?
Does $B$ have finite global dimension in case $M=A \oplus D(A)$?
...
4
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0
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58
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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 ...
9
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0
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121
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Cartan determinant of stable categories
Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$.
Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...
5
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0
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89
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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....
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An exact complex of tensor families
Given a field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, some natural numbers $q,n,m$ with $q \leq n$ and $m < n$ and a polynomial algebra $S := \mathbb{K}[\eta_1, \dots, \eta_n]$. We can consider $...
1
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0
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Total complex of complexes
When we have a double complex of vector spaces $V^{p,q}$, we can produce a complex either taking direct sums or products along the anti-diagonals. Then, the differential in this new complex will be
$$ ...
5
votes
1
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168
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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^...
2
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1
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77
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Whether a partial tilting complex has a complement
I want to know whether a partial tilting complex has a complement。if the answer is obvious?or to what kind of algebra this is true。
2
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0
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59
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$\Omega$-periodic modules in selfinjective algebras
Given a representation infinite (connected) selfinjective algebra $A$ with an indecomposable $\Omega$-periodic module $M$.
Does $A$ then have infinitely many indecomposable $\Omega$-periodic ...
2
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0
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63
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Constructing stable equivalences for finite dimensional algebras
Given a finite dimensional (non-selfinjective) algebra $A$.
Is there a method (for example using QPA) to construct algebras stable equivalent to $A$?
Such a thing is easily possible for derived ...
6
votes
1
answer
346
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The category of complexes over a dg-algebra is Grothendieck (it has a generator)
Let $A$ be a dg-algebra over some commutative ring $k$. We have an abelian category $\mathrm{C}(A)$ of (right) $A$-dg-modules. I've read in a few sources that $\mathrm{C}(A)$ is a Grothendieck abelian ...
4
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85
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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$ ...
4
votes
1
answer
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First homology group of the general linear group
The abelianization of the general linear group $GL(n,\mathbb{R})$, defined by $$GL(n,\mathbb{R})^{ab} := GL(n,\mathbb{R})/[GL(n,\mathbb{R}), GL(n,\mathbb{R})],$$ is isomorphic to $\mathbb{R}^{\times}$....
6
votes
0
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144
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Hochschild cohomology of the $A_\infty$-category of paths
I would like to describe the Hochschild cohomology (in the sense of $A_\infty$-categories) of the following $A_\infty$-category associated to a topological space $X$:
It has points of $X$ as objects.
...
4
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0
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43
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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/...
5
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0
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122
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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 ...
7
votes
1
answer
528
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Example of a ring where every module of finite projective dimension is free?
I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective.
Note that self-injectivity says ...