# 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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### Tensor product of an L-infinity algebra with the cochains on the 1-simplex

I would like to understand the $L_\infty$ structure on the tensor product of an $L_\infty$ algebra (over $\mathbb{R}$) $L$ with the normalized cochains on the one-simplex $N^*(\Delta^1)$. This latter ...

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125 views

### 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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**1**answer

158 views

### Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has this two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^...

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107 views

### Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...

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33 views

### Hermitian structure for complexes of vector bundles

Does it exist a different notion of Hermitian metric for complexes of vector bundles, besides the obvious data of a metric for each vector bundle?
Same question for connections. In particular is there ...

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**1**answer

105 views

### How to check that exceptional sequence of vector bundles on Fano variety is helix foundation

Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E_0,...,E_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$....

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329 views

### Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...

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155 views

### Derived invariance of the Cartan determinant

The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e_i$ is defined as the matrix having entries $c_{i,j}=\dim(e_i A e_j)$. The Cartan determinant is defined as the determinant of the ...

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103 views

### Injective resolution of the ring of entire functions

Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...

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135 views

### Gorenstein symmetric conjecture for arbitrary rings

The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...

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64 views

### $(A[1])^{\otimes n}\backsimeq (A^{\otimes n})[n]$? [migrated]

When $A$ is a $\mathbb{Z}$-graded module, $A[1]$ is the shift or suspension of $A$ (i.e $(A[1])^{i}=A^{i+1}$). May the $n$th power tensor of the shift be identified in this way?. Am I missing anything?...

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104 views

### homotopy MC element

A homotopy MC element is the Linfty analog of a Maurer-Cartan
element for a Lie algebra. Where is anything written about
homotopy MC elements as perturbations of strict MC elements?

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82 views

### Singular homology: Lifting simplices gives map in homology

Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$.
Then the ...

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**1**answer

151 views

### Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair

Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...

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**1**answer

140 views

### Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?

In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\...

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71 views

### Quartic link in a 5-sphere

In this post I would like to propose a quartic link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...

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156 views

### Generalizing the formula between Wu class and the Steenrod square

I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) .
\tag{eq.1}$$
...

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56 views

### Periodic modules in Frobenius algebras

Let $A$ be a finite dimensional Frobenius algebra and assume there exists an indecomposable periodic module $M$, that is $\Omega^n(M) \cong M$ for some $n$.
Question: Does this imply that there is ...

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51 views

### non zero differential in a spectral sequence

This is the situation:
Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...

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**1**answer

92 views

### Group cohomology with coefficients in a chain complex

Let us suppose that I'm in the following situation: I have a chain complex $(C,\partial)$ and say a finite group $G$ acting over $C$ up to homotopy, meaning that for each $g \in G$ I have a self ...

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109 views

### Injective Change of Rings

Sorry if this is too elementary, but when I was going to ask this question on math.stackexchange, I saw the same question with three up-votes and no answer. So I decided to post it here.
I am doing ...

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103 views

### Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$
\Omega_5^{...

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142 views

### Morphism of sites and abelian sheaf cohomology

Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi
$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$
By assumption, $f^{-1}$ is ...

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**1**answer

55 views

### Rings with only finitely many indecomposable reflexive modules

Let $R$ be a ring. Recall that a module $M$ is called reflexive in case the natural evaluation map $M \rightarrow M^{**}$ (with $M^{*}=Hom_R(M,R)$) is an isomorphism. A module is reflexive if and only ...

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201 views

### Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...

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**1**answer

346 views

### An inverse problem for Grothendieck rings of varieties

Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...

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105 views

### Is the existence of $A_{\infty}$-inverse a consequence of Homotopy Transfer Theorem?

Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map ...

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184 views

### If C is a cocomplete coalgebra, then $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism

I am reading the PhD thesis thesis of Kenji Lefèvre-Hasegawa and the corresponding errata by Bernhard Keller, my question is about the first error found in the thesis. Lemma 1.3.2.3 c states 'the ...

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96 views

### Homotopy coherent action on differential graded vector space

Suppose I have a group $G$ which acts on the homology $H_*(C)$ of a differential graded vector space $C$. Can I always lift this to a homotopy coherent $G$-action on $C$?
My first naive thought was ...

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90 views

### Second homology of finitely presented group with free abelianisation

It is known that for a presented group $G=F/N$ we have
$$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$
In general, the right side seems to be difficult to calculate. I am in the special ...

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130 views

### Deriving category of quadratic functors

Recall that a fuctor $T: \cal C \to \cal A$ from pointed small $\cal C$ with coproducts to additive and Karoubian — or, even better, abelian $\cal A$ is called quadratic if kernel of sum of obvious ...

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230 views

### mod p (odd) cohomology of dihedral groups

I've been trying to find the cohomology for the trivial module for $\operatorname{PSL}_2(r^n)$ over $\mathbb{F}_p$ for $2 \neq p \neq r$ and have managed to reduce this to the cohomology of a maximal ...

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78 views

### Cohomology of a chain complex over a polynomial ring

I asked this on SE but I did not get any answer; I got some progress but I hope here I can find some help to finish the problem out.
Let $R = F[x_1, \ldots, x_n]$ be a polynomial ring over a field $...

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52 views

### 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 ...

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25 views

### Complexity of the regular module

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 ...

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109 views

### Trivial action in the Hochschild-Serre spectral sequence

I probably don't understand something very basic about Hochschild-Serre spectral sequence.
Let $G$ be a group with normal subgroup $N$ and $M$ a $G$-module with trivial action. Then as far as I ...

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121 views

### $\omega$-categorical algebra

Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...

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95 views

### Proving that the exterior algebra is symmetric via the polynomial ring

Recall that a finite dimensional algebra $A$ over a field $K$ is called Frobenius in case $A \cong D(A)$ as right modules, and it is called symmetric in case $A \cong D(A)$ as bimodules (where $D=...

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134 views

### Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?

In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +_{obj W} W^{op})$, the free category on the ...

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107 views

### Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...

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418 views

### Does this algebra have finite global dimension ? (Human vs computer)

Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...

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114 views

### Algebras derived equivalent to quasi-hereditary algebras

Let an algebra always be finite dimensional over a field and connected.
It is well known that a quasi-hereditary algebra with $n$ simple modules has global dimension at most $2n-2$.
Questions:
1. ...

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**1**answer

264 views

### Can one always find a bigger global resolution

Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...

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64 views

### Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...

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**1**answer

73 views

### Alternating property of H_2(T, Z)

Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...

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63 views

### Finitistic dimension via a bimodule

Let $A$ be a connected finite dimensional basic algebra.
Question: Is there an indecomposable $A$-bimodule $W$ such that the finitistic dimension of $A$ is equal to the right projective dimension ...

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86 views

### Map from coproduct of products to product of coproducts

Consider obvious (imagine generators as $k \times l$ dot rectangle) epimorphism $$p: \Bbb Z^k * \dots (l \text{ times}) \dots \Bbb Z^k \to F(l)^{\times k}$$ where $F(l)$ are relatively free groups in ...

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81 views

### Could we characterize injective objects in the category of $G$-equivariant sheaves?

Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left....

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**1**answer

149 views

### Alternate property of H^2(T, Z) [closed]

Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...

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411 views

### Serre spectral sequence for de Rham cohomology

Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with
$$
E_2^{p,q} = H^p(M,\underline{H^...