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1
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87 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, ...
1
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0answers
92 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 ...
8
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0answers
147 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
103 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
138 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
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1answer
102 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 ...
0
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0answers
52 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
129 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 ...
2
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1answer
104 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 ...
0
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0answers
66 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 ...
0
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0answers
36 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
76 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
votes
1answer
174 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 ...
2
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0answers
88 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
votes
0answers
179 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 ...
0
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0answers
41 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
110 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
323 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 ...
3
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1answer
286 views

Doing some homological algebra in triangulated categories

It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} ...
1
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0answers
80 views

Cohomological dimension of transcendental p-adic extensions

Let $q$ denote a quadratic form over a field $k$. The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$. Let $k = \mathbb{Q}_p$ for any ...
0
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0answers
54 views

Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
0
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0answers
40 views

question about Baer sum of extensions

Let $E_1$ and $E_2$ be extensions of $\mu_p$ by $\mathbb{Z}/{p\mathbb{Z}}$. Assume that $E$ contains $E_1$ and $E_2$ both, and $E_1 \cap E_2 = \mathbb{Z}/{p\mathbb{Z}}$. Then, does $E$ contain their ...
1
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0answers
59 views

Homology and Exterior Square

Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto ...
0
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1answer
217 views

A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
5
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3answers
290 views

Poincare duality for (co)homology of Lie algebras?

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200 it ...
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1answer
92 views

terminology: “complex” and “sequence” in homological algebra

It appears that the terms "complex" and "sequence" are used synonymously in homological algebra. But there seem to be collocations (in the linguistic sense) that prefer one of those words. For ...
2
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0answers
57 views

relations between derived categories of ind-A and A

Let $A$ be an abelian category and $indA$ be its ind category. I want to know the relations between $D^b(A)$ and $D^b(indA)$. For example, I find in another question that if $A$ is thick in $indA$, ...
3
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2answers
518 views

Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold). Let $f_*\colon ...
0
votes
0answers
49 views

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 $Rep_G$, ...
14
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2answers
733 views

Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra? Is it easier if we impose any of the three conditions: characteristic zero; ...
2
votes
2answers
160 views

Standard homology result on double complexes

Suppose you have got a double complex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows ...
0
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0answers
41 views

Corestriction map for the relative homology (cohomology) group

Let $G$ be a group and $N$ be its normal subgroup. Is there any concept of corestriction map for the relative homology (cohomology) group $H_n(G,N,-)$ ($H^n(G,N,-)$) such that when $N=G$ it is the ...
1
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0answers
44 views

a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...
6
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1answer
223 views

Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?

My question is: Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by ...
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0answers
234 views

Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
2
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1answer
239 views

How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...
0
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0answers
96 views

Golod Shafarevich Inequality and Inequalities among higher Cohomology groups

As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...
2
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0answers
117 views

Perf($\mathscr{A}$) and perfect chain complexes

Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ ...
1
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0answers
71 views

When localizing a category at a multiplicative system, is there a lemma utilizing Ore Condition/Cancellation once for all roof-independence proofs?

For example, given an additive category $\mathcal{C}$ with a (both sided) multiplicative system S, the localization $S^{-1}\mathcal{C}$ is also additive. Yet to prove that the addition of morphisms in ...
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2answers
581 views

(co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
6
votes
1answer
244 views

Idea and intuition behind Penrose transform

I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application. My knowledge of differential ...
2
votes
0answers
99 views

the algebraic theory of obstruction of a homology theory

In general homology group of a complex is $Ker d/ Im d$(regardless of grading). However, in many case the square $d\circ d$ is not zero. For example, the study of Floer thoery needs A infinity ...
12
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1answer
292 views

What is the geometric fixed points of an (equivariant) Eilenberg Maclane Spectrum?

The following was posted to math.stackexchange to no avail: http://math.stackexchange.com/questions/908756/an-exercise-in-homology-computation-what-is-the-geometric-fixed-points-of-an-e The question ...
1
vote
1answer
147 views

Bounds for Betti numbers

Why the graded Betti numbers of ideal $I \subset k[x_1 , \cdots , x_n]$, are bounded by the graded Betti numbers of $\mathrm{gin}(I)$? (Where $\mathrm{gin}(I)$, is the generic initial ideal of $I$ ...
2
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1answer
115 views

Is a inverse limit of indecomposable again indecomposable?

In truth, I do not need in the general case. Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$. If $\mathbb{N}$ is the ...
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0answers
189 views

What is the abutment filtration of the second spectral sequence of hypercohomology?

I have been recently learning about spectral sequences, following mainly Illusie's notes and EGA, and I am about to write some expository notes, but there are still some points that I was not able to ...
4
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1answer
362 views

How to prove a Proposition of Rouquier?

Proposition 7.15 in Rouquier's paper (see Publication paper) "Dimension of triangulated categories J. K-theory, 1 (2008) 193-256" as follows, for details please see Rouquier's paper (or arXiv): ``Let ...
5
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1answer
227 views

Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be $$ \text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!, $$ where $ P_{M} $ denotes the Hilbert ...
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0answers
107 views

Definition of 'Koszul Ring' (in BGS)

In the paper 'Koszul Duality Patterns in Representation Theory' by Beilinson et. al, they give the definition of a Koszul Ring: A Koszul ring is a positively graded ring $A = \bigoplus_{j \geq 0} ...
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0answers
54 views

Change of relative base

If $k$ is a commutative ring, $A$ a $k$-algebra and $\phi: k \rightarrow k'$ is a morphism of rings then how (/under what conditions) can the relative homology functors $Ext_{A/k}(-,-)$ and ...