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73 views

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 $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$, ...
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1answer
256 views

Are chain complexes over a field always injective?

Question: Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and ...
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0answers
62 views

On a property of split short exact sequences [migrated]

Let $A_{\bullet}, B_\bullet$ and $C_\bullet$ be three short exact sequences of groups (not necessarily abelian) out of which $A_\bullet$ and $B_\bullet$ are split. Assume that there is again a short ...
8
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0answers
313 views

Pairing of cohomology and homology Künneth formulas

Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian ...
5
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2answers
311 views

Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory? By "interesting/useful", I mean "not merely ...
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0answers
77 views

$\text{Hom}(G,\mathbb{Z})$ [duplicate]

Fix a cardinal $\kappa$ and consider $\mathbb{Z}^\kappa$ with componentwise addition and the subgroup $$F_\kappa :=\{g:\kappa \to \mathbb{Z}: \{\alpha\in \kappa: g(\alpha)\neq 0\} \text{ is ...
3
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1answer
268 views

Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?

Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume ...
0
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1answer
101 views

Do levelwise quasi-isomorphisms of bicomplexes induce a quasi-isomorphism between the total complexes?

Let $C^{p,q}$ be a bicomplex with differentials $d_h:C^{p,q} \to C^{p+1,q}$ and $d_v:C^{p,q} \to C^{p,q+1}$ where $d_h \circ d_v = d_v \circ d_h$. Let $D^{p,q}$ be another bicomplex defined similarly. ...
11
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0answers
336 views

Lifting DG-categories to characteristic zero

The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...
4
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0answers
171 views

Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...
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0answers
41 views

Turning left modules into right modules over a homotopy Gerstenhaber algebra

For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$. In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course ...
6
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1answer
202 views

An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal

We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its ...
24
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2answers
599 views

A more natural proof of Dold-Kan?

The Dold-Kan correspondence gives an equivalence of categories between $SAb$, the category of simplicial abelian groups, and $Ch_{\geq 0}$, the category of non-negatively graded chain complexes of ...
5
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0answers
156 views

homology theory for affine and projective algebraic sets?

Given $f_1,\ldots,f_r\in K[x_1,\ldots,x_m]$, resp. homogeneous $f_1,\ldots,f_r\in K[x_0,\ldots,x_m]$, is there a chain complex built from these polynomials, such that any polynomial map $\varphi\!: ...
3
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2answers
283 views

What are cohomology of Lie algebra with coefficients geometrically?

I want to find analog of following two statements. Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi_1 (BG) =G$). Let ...
3
votes
1answer
124 views

Lyndon–Hochschild–Serre spectral sequence for not normal subgroup

Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup? What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$? What is the best technique ...
3
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1answer
85 views

Homogeneous polynomial vector fields tangent to the unit sphere

This question has something to do with that one. Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v_1,\ldots,v_n)$ whose components $v_j$ are homogeneous of ...
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0answers
87 views

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 ...
4
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0answers
73 views

Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...
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0answers
161 views

What is a Beilinson spectral sequence?

I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...
0
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0answers
92 views

A naturality question concerning the universal coefficient spectral sequence

I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence. Let X be a connected finite CW complex.Let $H$ be a ...
8
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0answers
253 views

Homological algebra is linearized homotopical algebra?

I have stumbled across statements like Homological algebra is linearized homotopical algebra. Chain complexes are linearizations of simplicial complexes. The Dold-Kan correspondence was ...
4
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2answers
221 views

Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes

It seems to be a well-known fact that homotopy (co)limits of (co)simplicial diagrams of nonnegatively graded (co)chain complexes in (Grothendieck) abelian categories can be computed by using the ...
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1answer
112 views

cohomology version of Cartan-Leray spectral sequence that deduces cup product

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology: If $X$ is a connected pace on which the group ...
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0answers
95 views

Exact Functors from Perverse Sheaves

Pretty sure this is a simple question, but there's something I'm missing here. For context, I'm reading 'An Elementary Construction of Perverse Sheaves' by MacPherson and Vilonen. A key aspect of the ...
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0answers
87 views

Can you always find a regular sequence consisting of monomials?

Let $\mathbb{k}$ be a field, and let $S=\mathbb{k}[x_1,x_2,\ldots,x_n]$. Let $M$ be an $S$-module. A sequence $$f_1,f_2,\ldots,f_r$$ of polynomials in the maximal ideal $\langle x_1,\ldots,x_n\rangle$ ...
0
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1answer
80 views

Hochschild cohomology of the skew group ring D(X)#G in the complex analytic case

Let $X$ be a n-dimensional complex compact manifold and let $G$ be a finite subgroup of $Aut(X)$ acting by biholomorphic maps on $X$. I would like to compute the Hochschild cohomology group ...
3
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0answers
61 views

Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite? (Assuming that $A\neq Z(A)$).
3
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1answer
85 views

Existence of small projective dimensioned modules

Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$. Then do either of the ...
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0answers
51 views

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 ...
3
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1answer
175 views

Classical deformation of algebras

Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$. ...
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1answer
138 views

Ext functor for more than two modules? [closed]

The question is natural. Let's just work in the category of modules over a ring. Pick three modules $M_1, M_2, M_3$. Consider consecutive extensions of these modules, i.e., consider M, such that we ...
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0answers
109 views

Strange invocation of Shapiro's lemma

I'm having trouble understanding a claim in a paper I'm reading. To avoid having to explain a lot of notation, I'll abstract the claim a bit. Assume that $G$ is a group with a subgroup $H$. Also, ...
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0answers
108 views

Ext and cup products and subvarieties

I am trying to understand Remark 11.3 in Huybrechts's amazing book on derived categories (FM transforms in AG). He starts with smooth projective varieties $j\colon Y \subset X$ and aims to describe ...
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0answers
193 views

Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
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0answers
115 views

Question about a theorem of Goodwillie on periodic cyclic homology

In his paper Cyclic homology, Derivations and the Free Loopsace, Goodwillie defines periodic cyclic homology for differential graded algebras (A,d) concentrated in non-negative degree. Why does he ...
7
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1answer
147 views

Is there a ring which is not Hermite but is coherent?

Call a commutative unital ring $R$ Hermite if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from ...
1
vote
1answer
141 views

Regular rings and formally smooth algebras

Let $A\rightarrow B$ be a commutative $A$-algebra. If $A$ is a field and $B$ Noetherian and formally smooth over $A$, then it is known that $B$ must be a regular ring. Is there a partial converse of ...
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0answers
63 views

Lie bialgebras cohomology

I am wondering if there exist a universal construction of (co)-chain complex associated to a given algebra to study deformations as Hochschild homology for associative algebras, or Chevalley-Eilenberg ...
4
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0answers
141 views

spectral sequence of a bicomplex equipped with a group action

Let $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ be complexes of $\mathbb{C}$-vector spaces. We suppose that $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ are equipped with ...
3
votes
1answer
113 views

On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned: Is the map $g_3 ...
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0answers
80 views

Criterion for global dimension of subring

All rings are assumed to be associative and unital. If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
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0answers
41 views

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]. ...
1
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1answer
80 views

Whitehead's second Lemma and invariants of exterior square

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
5
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1answer
182 views

What does an endomorphism in a triangulated category give rise to?

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...
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0answers
91 views

Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$

Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
2
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0answers
231 views

What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...
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0answers
45 views

Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
6
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0answers
121 views

Stable, tame A-infinity isomorphism?

Linear duality provides a correspondence between differentials on free tensor algebras and A-infinity algebras. Under this correspondence, what is the A-infinity counterpart of "stable, tame ...
17
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1answer
364 views

Is such a map null-homotopic?

Suppose I have (semi-infinite) chain complexes $$ \cdots \rightarrow A_i \rightarrow A_{i+1}\rightarrow \cdots$$ $$ \cdots \rightarrow B_i \rightarrow B_{i+1}\rightarrow \cdots$$ over an additive ...