All Questions
Tagged with at.algebraic-topology homological-algebra
388 questions
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
7
votes
1
answer
310
views
Homotopy between posets
This is entirely a new area for me and I apologise in advance if the questions are silly.
In Quillen's paper "Homotopy properties of the posets of non-trivial $p$-subgroup of a group" (see ...
- 139
2
votes
1
answer
216
views
Compute the singular homology group modulo barycentric subdivision
Let $X$ be a topological space, and let $C(X)$ denote its singular chain complex with boundary operator $\partial$ and $n$-th chain group $C_n$. We know there exists a barycentric subdivision operator ...
- 807
3
votes
0
answers
181
views
Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
4
votes
1
answer
184
views
FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 275
4
votes
0
answers
112
views
Differentials on free algebras over operads
I am currently reading Cyclic Operads and Cyclic Homology by Getzler-Jones and have some confusions.
I am under the impression that given an (associative, say) algebra $A$ that an almost-free ...
- 195
2
votes
0
answers
84
views
Infinity-morphisms for operadic algebras
Is there an already studied notion of $\infty$-morphism between algebras over a quasi-free operad $P = (T(E), \partial)$?
If the operad $P$ is Koszul, or of the form $\Omega C$ for $C$ a cooperad, ...
- 215
3
votes
0
answers
79
views
Rational model for composition of linear isometries
There is a composition map on spaces of linear isometries (over $\mathbb{C}$ say)
$$
\mathcal{L}(\mathbb{C}^k, \mathbb{C}^\ell) \times \mathcal{L}(\mathbb{C}^\ell, \mathbb{C}^m) \longrightarrow \...
- 901
7
votes
0
answers
218
views
Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 171
1
vote
0
answers
58
views
Which sheaves are good for calculating extraordinary restriction?
Let $X$ be a sufficiently nice locally compact Hausdorff space and let $i:Y\subset X$ be the inclusion map of a sufficiently nice closed subspace. For example, one could take $X$ to be a locally ...
- 23.5k
8
votes
1
answer
232
views
Product structure in Milnor exact sequence
Let $h^*$ be a (multiplicative) generalized cohomology theory. Let $X$ be a CW complex which is a union of an increasing sequence $X_0 \subset X_1 \subset X_2 \subset \cdots$ of subcomplexes. Then ...
- 959
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
2
votes
0
answers
122
views
Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
- 215
2
votes
1
answer
199
views
Regular sequence in cohomology of Grassmannians
$\DeclareMathOperator\Gr{Gr}$Consider the polynomial ring $\mathbb{Z}[x_1,\dots,x_m, y_1,\dots,y_n]$, I want to prove that the sequence $$x_1 + y_1, x_2 + x_1y_1 + y_2, \dots, x_my_{n-1} + x_{m-1}y_n, ...
- 185
3
votes
1
answer
134
views
Are the two families of Johnson invariants of the Torelli groups related beyond the first one?
$\newcommand{\sp}{\operatorname{Sp}(H)}$
$\newcommand{\gr}{\operatorname{gr}}$
$\newcommand{\id}{\operatorname{id}}$
$\newcommand{\der}{\operatorname{Der}}$
Johnson has defined two families $\tau_k,\...
- 8,524
2
votes
0
answers
190
views
Connection on relative topological periodic cyclic homology
I have been looking Bhatt-Morrow-Scholze's paper:
https://arxiv.org/pdf/1802.03261.pdf
and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
- 3,758
3
votes
1
answer
144
views
Linearity of topological periodic cyclic homology
Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra ...
- 959
4
votes
1
answer
469
views
How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
- 959
5
votes
1
answer
471
views
Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...
- 339
5
votes
0
answers
187
views
Are there known minimal models for the cohomology of semisimple Lie algebras?
My student and I recently found a cute construction of a minimal model for the cohomology of a Lie algebra $\mathfrak{g}$. This is a "minimal model" in the sense that it is a minimal chain-...
- 1,335
1
vote
1
answer
610
views
The Krull dimension of the tensor product of rings
The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
- 35
4
votes
1
answer
230
views
Pontryagin product on the homology of cyclic groups
Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
- 103
3
votes
1
answer
233
views
On infinity-morphisms between algebras over algebraic operads
I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here.
Let $P$ be a Koszul operad.
In the book of Loday-Vallette "...
- 215
0
votes
0
answers
89
views
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, ...
- 29
2
votes
1
answer
242
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
- 1,071
2
votes
1
answer
295
views
Projective objects in chain complexes of an abelian category: Further question
Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes
I am wondering why a level-wise projective chain complex $P$ which is split ...
- 511
11
votes
2
answers
856
views
Spectral sequences and short exact sequences
Suppose I take a short exact sequence of filtered chain complexes:
$$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$
We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
- 721
1
vote
0
answers
181
views
Non-trivial homotopy, but vanishing homology
I wonder if there are examples of 5-dimensional manifolds with vanishing integral second homology group, but non-vanishing second homotopy group? Or is it impossible by some Hurewicz theorem type of ...
- 129
5
votes
0
answers
154
views
Hochschild cohomology of path algebra as a cohomology of simplicial complex
M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link).
Is the opposite ...
- 91
17
votes
1
answer
1k
views
Who wrote `if only I could understand the equation $d^2=0$'?
I remember reading something like
if only I could understand the equation $d^2=0$
as an epigraph to a memoir on homological algebra. I think the author was Henri Cartan, and the epigraph may have ...
- 4,492
2
votes
0
answers
133
views
Formulation of cap product in group-equivariant sheaf cohomology + applications?
Originally asked on Math SE but it was suggested I move it here.
Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice&...
- 101
2
votes
0
answers
275
views
Homological algebra generalization of covering map
I would like to know if there exists an operation in homological algebra that generalizes the notion of covering maps for abstract chain complexes (over any field or ring, or maybe just certains where ...
4
votes
0
answers
317
views
What is the geometric interpretation of the first Hochschild homology group of path algebra constructed from a directed graph?
Let $\mathcal{G} = (V, E, s, t)$ is a directed graph, where $V$ - the set of its vertices, $E$ - the set of its edges, $s: E \rightarrow V, s((v_1, v_2)) = v_1$ and $t: E \rightarrow V, s((v_1, v_2)) =...
- 91
1
vote
0
answers
96
views
Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic
I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic.
I have come up with ...
- 1,805
1
vote
0
answers
197
views
topological functor of tor functor
The framework of Quillen's model categories gives us a very general way of defining things as derived functors. For instance, in this way one can realise the singular homology as Andre-Quillen ...
- 449
3
votes
1
answer
257
views
Why does this construction not give a functorial cone in the homotopy category of cochain complexes?
I have heard the expression recently that one should be careful when constructing cones in the homotopy category - namely, that this is not functorial. However, when working through some examples in ...
- 285
2
votes
1
answer
178
views
Can information theory characterise a (topological) space?
Consider an objective function f: $\mathbb{R}^n\rightarrow\mathbb{R}$, with a vector of variables $\theta$, i.e. $f(\theta)$, $\theta \in \mathbb{R}^n$. Depending on $f$, there can be interesting ...
1
vote
0
answers
86
views
Explicit form of boundary operators of topological cones
Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$.
For many problems in computational geometry, a key operation is to ...
- 840
12
votes
4
answers
2k
views
Applications of the Dold-Kan correspondence
The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...
- 129
1
vote
0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
6
votes
1
answer
478
views
Unbounded acyclic resolutions
Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
- 28.7k
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
3
votes
0
answers
148
views
An account of "Homologie nicht-additiver Funktoren. Anwendungen"'s results
Is there an account in English of results from "Homologie nicht-additiver Funktoren. Anwendungen" by Dold and Puppe? I am mostly interested in the spectral sequence of cross-effects which ...
- 1,374
4
votes
2
answers
228
views
Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?
Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...
- 711
0
votes
1
answer
239
views
Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book
Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the ...
- 6,023
1
vote
1
answer
310
views
Realize as homology a given polynomial ring
I am wondering if one can realize a polynomial ring as the homology of some chain complex in the same sense that the homology groups of a space with a cell complex structure is the homology of its ...
- 333
3
votes
0
answers
248
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
2
votes
0
answers
123
views
Every surface of sufficiently large genus separates
Let $M^3$ be a smooth closed orientable manifold.
Does there exist a non negative integer $g_0$ such that every closed orientable embedded surface $\Sigma \subset M$ of genus $g \geq g_0$ represents ...
- 2,095
7
votes
1
answer
567
views
Long exact sequences for parametrized cohomology
I'm reading Michael Shulman's articles on cohomology in HoTT here and here, as well as Floris van Doorn's thesis here.
Given $E: Z \to \mathsf{Spectrum}$ a family of spectra over a homotopy type $Z$, ...
- 6,025