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4
votes
1answer
142 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 ...
1
vote
0answers
186 views

Learning roadmap in Algebra [on hold]

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
votes
1answer
101 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
votes
0answers
88 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
votes
0answers
94 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
vote
0answers
55 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 ...
10
votes
2answers
478 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
224 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
96 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
votes
1answer
254 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 ...
0
votes
1answer
135 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
votes
1answer
108 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 ...
1
vote
0answers
154 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
votes
1answer
352 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 ...
4
votes
1answer
164 views

$ mult(R/I) = d_1 \cdots d_r \quad \Rightarrow \quad f_1,\dots,f_r \quad \text{is a $R$-regular sequence?}$

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
1
vote
0answers
102 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} ...
0
votes
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 ...
5
votes
0answers
148 views

Can we “complete” model categories to compute derived functors in the usual way?

Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the ...
4
votes
1answer
146 views

AdicCompletion$\dashv$Torsion adjunction on spectra?

It seems to me that in slight paraphrase the central result of the article Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386) (theorems 6.11 and ...
10
votes
4answers
308 views

The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...
1
vote
0answers
37 views

Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones

This question is strictly related to this other one. Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let be a diagram in $Z^0(\mathcal A)$, where the rows are ...
7
votes
1answer
258 views

Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H_i(G, \mathbb{Q})$, or equivalently $H_i(BG, \mathbb{Q})$ (topological ...
2
votes
0answers
101 views

Unbounded derived category that is not left-complete

Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...
3
votes
2answers
238 views

Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
2
votes
0answers
107 views

Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...
1
vote
2answers
205 views

If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if ...
1
vote
0answers
157 views

On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces, $$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
2
votes
1answer
142 views

Interpretation of Hochschild Homology groups

In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc.... In ...
2
votes
1answer
196 views

Castelnuovo Mumford Regularity

Let $$ ‎‏0 ‎‎\rightarrow‎‎‏ ‎E_{n-1} ‎‎\rightarrow ‎... ‎‎‎\rightarrow ‎E_1 ‎‎‎\rightarrow ‎E_0 ‎‎‎\rightarrow I \rightarrow 0‎ ‎,‎ $$ be a minimal free resolution of ideal $I$, where $$ E_p = ...
9
votes
1answer
239 views

Motivation behind the definition of hochschild cohomology

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
1
vote
0answers
80 views

generalisation of the universal coefficient spectral sequence

Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective ...
0
votes
2answers
132 views

Smooth Affine algebras are Calabi-Yau

Are all smooth affine algebras over a field Calabi-Yau? I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
0
votes
1answer
83 views

Formally smooth map from a regular ring

Let $A,B$ be two commutative noetherian rings. Let $f:A\to B$ be a formally smooth homomorphism. If $A$ is a regular ring (in the sense that all its localizations are regular local rings), does this ...
3
votes
0answers
126 views

Pursuing an abelian categorical proof of the Zassenhaus Lemma

Fix an abelian category and suppose $M',M,N',N$ are sub-objects of a given object such that $M'\subset M$ and $N'\subset N$. Then there exists a canonical isomorphism $\frac{M'+(M\bigcap ...
2
votes
0answers
194 views

Tensor product of pullbacks of abelian categories

Does the tensor product of abelian categories commute with pullbacks? In more detail: Let $k$ be a field. We consider $k$-linear small abelian categories ...
3
votes
0answers
82 views

When does a commutative DGA have a finitely generated quasi-free resolution?

Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question ...
6
votes
0answers
131 views

Coherent sheaves and Mitchell's embedding theorem

Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...
5
votes
0answers
109 views

Vanishing of Andre-Quillen homology and injective dimension

Let $(A,m,k)$ be a commutative local ring. Assume that for all $n\ge 3$, the Andre-Quillen homology modules $H_n(A,k,k)$ vanish. Does this imply that $A$ has finite injective dimension over itself? ...
2
votes
1answer
91 views

Does semi-free behave well under totalization

Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this ...
8
votes
1answer
255 views

Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that $K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$. A important part in the proof of the Bloch-Kato conjecture is to ...
0
votes
0answers
95 views

How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture: If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...
5
votes
1answer
160 views

Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question. Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states: "If $\varphi$ is any anisotropic class in ...
4
votes
1answer
132 views

Interpretations of differentials in hypercohomology spectral sequences as Yoneda products

I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups. More ...
0
votes
1answer
71 views

Bounded algebras of finite global dimension

Let $k$ be a field, $Q$ an acyclic finite quiver and $I$ an admissible ideal of $kQ$. I am looking for a reference for the fact that the bounded algebra $kQ/I$ has finite global dimension.
1
vote
0answers
77 views

A factorization system on ${\rm Ch}(R)$

This is a sort of follow up to this MO question. Let $R$ be a ring (eventually with good properties) and $\mathrm{Ch}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
1
vote
0answers
207 views

Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative). Setup Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying: If $u$ is a unit ...
5
votes
2answers
337 views

Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question. Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
7
votes
1answer
321 views

DG enhancements of $\ell$-adic derived categories

This question is similar in flavor to Existence of dg realization for 6 functors Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...
5
votes
1answer
144 views

Is a Gorenstein ring a quotient of a local complete intersection

The title says it all - Suppose you are given a noetherian Gorenstein local ring $(A,m,k)$ of finite Krull dimension. Does there exist a local complete intersection ring $B$ such that $A$ is a ...
2
votes
0answers
82 views

How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?

First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$. We can see that $\mathcal{L}BG$ has the homotopy type of ...