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.
1,097 questions with no upvoted or accepted answers
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Generalized edge map in spectral sequence of double complex
suppose we have a cohomologically indexed double complex $C^{\bullet,\bullet}$ with its spectral sequence
$$E_2^{p,q}=H^p_{vert}(H^q_{horiz}(C))\Rightarrow H^{p+q}(C)$$
and suppose that the horizontal ...
- 2,044
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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 ...
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A complex with homology $=R/p$
Given a Noetherian ring $R$ .
I am looking for a bounded complex $X$ of finitel geenerated projectives over $R$ whose homology is $R/p$. Infact I just need $X$ to have $\operatorname{Supp}(H(X)) = \...
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What happens if I take a doubly-free simplicial abelian group?
Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
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Is any deformation of an acyclic complex gauge equivalent to a trivial one?
This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...
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203
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Double complex of simplicial resolution
In his
lectures on condensed mathematics on page 30 Peter Scholze speaks of the double complex of a simplicial resolution. How is this defined?
In the next line, he writes that if $A_\bullet$ is a ...
user473423
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132
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Example of a periodic free resolution over a hypersurface
I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION,
WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud
I'm wondering what would be a nice example illustrating Theorem 6....
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What can be said about the derived functor of a composition between unbounded derived categories?
Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...
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pullback square in abelian category and derived categories
Let $\mathcal{A}$ be an Abelian category. Take objects $A,B,C$ and $D$ in $\mathcal{A}$, and morphisms $b:B\to A$, $b':B\to A$, $c:C\to A$, $e:D\to C$, $e':D\to C$ and $f:D\to B$ such that diagrams
$\...
- 519
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Distinguished triangles as generalizations of short exact sequences
If you'll have patience with me, I understand that this is not the first time that a variation of this question is being asked on MathOverflow, but alas, I am unable to truly make sense of those ...
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Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
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The coevaluation map for a projective module and its dual
$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
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Exactness of $I$-adic completion in a certain non-finitely generated case
I would like the functor
$$(-\otimes_{\mathbb Z} F)\hat{}: \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}_{\mathrm{f.g.}}\longrightarrow \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}$$
to be exact, where completion is w....
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Homotopy equivalent dg modules
$\newcommand\modl[1]{#1\text{-mod}}$Let $A:=(A,d_A)$ be a dg algebra. I would like to ask about isomorphisms in the homotopy category $H(\modl A)$ of the dg category $\modl A$ of dg modules over $A$.
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Another definition of singular homology
The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows:
Let $X$ be a topological space. A $n$-...
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Determinant of chain complexes
Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
user30211
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Connecting homomorphism and Baer sum in an abelian category
I would like to prove that the connecting homomorphism $\delta \colon \mathrm{Hom}_{\mathcal{A}}(N,M_3) \to \mathrm{Ext}_{\mathcal{A}}(N,M_1)$ from part (2) of Lemma 12.6.4 of the Stacks Project is ...
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Homology of a closed $3$-manifold with balls removed
This question has been posted on MSE with no answers.
Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
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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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Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$
I read that, using the fact that $\text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $\text{Ext}^* (A,B)$ (which I understand), we can give $\bigoplus_{i} \text{Ext}^i (A,...
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On the dimension of the cohomology of toric manifolds
Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...
- 1,227
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Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme
I'm seeking to understand the de Rham cohomology of a Hilbert scheme $K3^{[4]}$ of the K3 surface. By Beauville, this 8-dimensional compact manifold is Kaehler, irreducible, holomorphically symplectic ...
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248
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A possible generalization of "Group Cohomolgy"
The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor:
$$FIX: \mathcal{M_G} \to \mathcal{Ab}$$
where $FIX$ is the functor from the ...
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$Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]
Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space,
but the statement is easier if we also suppose $X$ ...
- 3,051
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147
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A question on the Bass-Papp theorem on injectives
Let $\{Q_{i}: i\in \mathcal{I}\}$ be a family of indecomposible injective modules over a commutative ring $R$ with identity. Is it true that the injective envelope of the direct sum of $Q_{i}$'s ...
- 471
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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 $B$ ...
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Group cohomology with coefficients in a permutation module (follow-up)
This is a follow-up question to Group cohomology with coefficients in a permutation module (which was for the case of a normal subgroup)
Let $A$ be an Abelian group, and let $G$ be a finite group ...
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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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Generalising the Mayer-Vietoris principle
My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \...
user81684
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Two questions on homotopy category
I am learning Homotopy category of chain complexes from wikipedia. There are two places I do not understand：
1.Let $K(A)$ (respectively, $D(A)$) be a homotopy category (respectively, derived category)...
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A question on the paper "The classification of algebras by dominant dimension"
I'm reading the paper "the classification of algebras by dominant dimension" by Bruno J.Mueller, the link is here http://cms.math.ca/10.4153/CJM-1968-037-9.
In the proof of lemma 3 on page 402, there ...
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Contravariant finiteness of a certain subcategory
Let $A$ be an algebra with finite dominant dimension $d \geq 1$ and $Dom_d$ the full subcategory of modules with dominant dimension at least $d$ and $Proj$ the full subcategory of modules of finite ...
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Is the pull back of a compact generator under field extension again a compact generator?
Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums. A compact ...
- 9,019
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Right split for homomorphism onto $S_\infty$
Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...
- 2,882
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Conditions for splitting of short exact sequence?
Assume $K$ is a number field and $E$ is an elliptic curve defined over $K$.
Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\...
- 1,805
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139
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Relating overlapping simplicial complexes
Let $X$ be a simplicial complex and let $A,B\subset X$ be
subcomplexes such that $C=A\cap B$ is a non-empty simplicial
complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$,
$C_{\cdot}...
- 547
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Computations of derivations, $d^2$, of a (Grothendieck) spectral sequence
The maps $d^1:E_1\rightarrow E_1$ have a nice description. Is there any text providing us with a description of the higher derivations $d^2:E_2\rightarrow E_2$ arising from a Grothendieck spectral ...
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Shift functor and the origin of the linear decalage isomorphism
Let $(\mathbf{Vec}_\mathbb{Z}(\mathbb{K}),\otimes,\tau)$ be the symmetric monoidal category of $\mathbb{Z}$-graded $\mathbb{K}$-vector spaces, where $\otimes$ is the tensor product of $\mathbb{Z}$-...
- 2,074
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Analogue of Baer's injectivity criterion for comodule algebras
Let $H$ be a Hopf algebra and $A$ an $H$-comodule algebra; denote by $M^H_A$ the category of right $(H,A)$-Hopf modules [i.e. $A$-module, $H$-comodules, everything is compatible with everything else]. ...
- 651
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84
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Yoneda extension in the category of representations
Assume $G$ is a group scheme over a field $k$ and consider the categories $Rep_G$ of finite dimensional representations of G and $REP_G$ of all representations of G. For two objects $A,B$ in $Rep_G$, ...
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223
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Trivial extensions by torsion-free groups
Let $A$ be an abelian group. Recall that $A$ is
($\bullet$) a Whitehead group if $\text{Ext}(A,\mathbb Z)=0$,
($\bullet$) a free abelian group if $\text{Ext}(A,D)=0$ for every abelian group $D$.
...
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Consistency of the u-invariant under field extension
A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...
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algebras of infinite injective dimension
Are there any connected graded Noetherian algebra of infinite injective dimension but has a balanced dualizing complex?
Thanks a lot.
- 141
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421
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Split and pure exact sequence of sheaves
Let $X$ be a topological space and
$$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$
be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if
for each point $x\in X$,...
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90
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Quick question on chain maps and maps induced by truncations.
Let $A^\bullet$ be the complex:
$\cdots \rightarrow A^{n-2} \xrightarrow{d^{n-2}} A^{n-1} \xrightarrow{d^{n-1}} A^{n} \xrightarrow{d^{n}} A^{n+1} \xrightarrow{d^{n+1}} A^{n+2} \xrightarrow{d^{n+2}} \...
- 456
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349
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cokernel for $L_\infty$-algebra morphisms
As I have asked a wrong question previously, I edited a bit.
It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal ...
- 1,271
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301
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Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
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