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.

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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 $M$ ...
Ronald J. Zallman's user avatar
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51 views

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 ...
Mare's user avatar
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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 ...
Mare's user avatar
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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 ...
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6 votes
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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 ...
P. Grabowski's user avatar
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152 views

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. ...
LWW's user avatar
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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}\...
SoYu's user avatar
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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 $\...
SoYu's user avatar
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$\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^+...
James Cheung's user avatar
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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^+...
James Cheung's user avatar
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11 votes
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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 ...
Doelt_k's user avatar
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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 ...
Mikhail Bondarko's user avatar
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146 views

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 ...
Mare's user avatar
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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,...
Pavel's user avatar
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312 views

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 ...
Eugene Rabinovich's user avatar
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1 answer
194 views

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 ...
Najib Idrissi's user avatar
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1 answer
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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 $...
Ronald J. Zallman's user avatar
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1 answer
221 views

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 ...
Mare's user avatar
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10 votes
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306 views

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\...
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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,\...
Mare's user avatar
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2 votes
0 answers
72 views

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 ... \...
Mare's user avatar
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6 votes
0 answers
120 views

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 ...
Jim Stasheff's user avatar
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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 ...
Mare's user avatar
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2 votes
0 answers
56 views

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 $i>0$. ...
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2 votes
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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 ...
Ming Lu's user avatar
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16 votes
2 answers
688 views

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 ...
Tim Campion's user avatar
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3 votes
0 answers
232 views

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 $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
sdey's user avatar
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4 votes
2 answers
714 views

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 ...
Mare's user avatar
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4 votes
1 answer
2k views

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 ...
user267839's user avatar
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4 votes
0 answers
85 views

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 ...
Mare's user avatar
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4 votes
0 answers
142 views

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 ...
Cohen's user avatar
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5 votes
0 answers
140 views

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/...
Mare's user avatar
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3 votes
0 answers
47 views

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 $M=A \oplus D(A)$? ...
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4 votes
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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 ...
Mare's user avatar
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9 votes
0 answers
121 views

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 ...
Mare's user avatar
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5 votes
0 answers
89 views

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....
Mare's user avatar
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1 vote
0 answers
37 views

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 $...
Lukas Miaskiwskyi's user avatar
1 vote
0 answers
669 views

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 $$ ...
Federico Barbacovi's user avatar
5 votes
1 answer
168 views

(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^...
Mare's user avatar
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2 votes
1 answer
77 views

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。
Sun YongLiang's user avatar
2 votes
0 answers
59 views

$\Omega$-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 ...
Mare's user avatar
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2 votes
0 answers
63 views

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 ...
Mare's user avatar
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6 votes
1 answer
346 views

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 ...
Francesco Genovese's user avatar
4 votes
0 answers
85 views

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 $A$ ...
Mare's user avatar
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4 votes
1 answer
918 views

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 $\mathbb{R}^{\times}$....
user's user avatar
  • 323
6 votes
0 answers
144 views

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. ...
user11267981's user avatar
4 votes
0 answers
43 views

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/...
Mare's user avatar
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5 votes
0 answers
122 views

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 ...
Mare's user avatar
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7 votes
1 answer
528 views

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 ...
Tim Campion's user avatar
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