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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 ...
GURI920826's user avatar
7 votes
1 answer
309 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 ...
GURI920826's user avatar
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 ...
Zhang Yuhan's user avatar
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 ...
groupoid's user avatar
  • 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 ...
Nicolas Guès's user avatar
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 ...
JD1874's user avatar
  • 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, ...
groupoid's user avatar
  • 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 \...
Niall Taggart's user avatar
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 ...
VadimKSt's user avatar
  • 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 ...
algori's user avatar
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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 ...
onefishtwofish's user avatar
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}...
Wilhelm's user avatar
  • 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_{...
asv's user avatar
  • 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 $...
groupoid's user avatar
  • 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, ...
atticusw's user avatar
  • 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,\...
Adrien's user avatar
  • 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 ...
Daniel Pomerleano's user avatar
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 ...
onefishtwofish's user avatar
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://...
onefishtwofish's user avatar
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 ...
Brian Shin's user avatar
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-...
user509184's user avatar
  • 1,335
1 vote
1 answer
609 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 ...
rr314's user avatar
  • 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 ...
Chase's user avatar
  • 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 "...
groupoid's user avatar
  • 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, ...
Inna's user avatar
  • 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 ...
Sergey Guminov's user avatar
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 ...
locally trivial's user avatar
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 ...
Richard Hepworth's user avatar
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 ...
Dmitrii Ivanov's user avatar
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 ...
Alexander's user avatar
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 ...
Matthieu Romagny's user avatar
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&...
xion3582's user avatar
  • 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 ...
Virgile Guemard's user avatar
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)) =...
Alexander's user avatar
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 ...
The Thin Whistler's user avatar
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 ...
Li Guanyu's user avatar
  • 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 ...
kelly maggs's user avatar
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 ...
Tessa van der Heiden's user avatar
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 ...
Daniel Shapero's user avatar
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 ...
Dora's user avatar
  • 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'...
UserIn's user avatar
  • 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 ...
R. van Dobben de Bruyn's user avatar
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 ...
algori's user avatar
  • 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 ...
Grisha Taroyan's user avatar
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 ...
Gabriel's user avatar
  • 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 ...
Praphulla Koushik's user avatar
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 ...
MathBug's user avatar
  • 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 ...
Eric's user avatar
  • 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 ...
Eduardo Longa's user avatar
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 $Z$, ...
ಠ_ಠ's user avatar
  • 6,025

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