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2
votes
0answers
147 views

Coordinate free Koszul-Tate resolution

Tate's original construction of the Koszul-Tate resolution involved choosing cocycles representing the cohomology to be killed. Where is it written in a coordinate free treatment, perhaps via a ...
1
vote
1answer
196 views

Cup-products and Transgression maps.

This question is related to Lyndon-Hochschild-Serre spectral sequence and cup products. I have the followin result by J.S Milne in his book Arithmetic duality theorems pg 105. Let $$0 \rightarrow C ...
1
vote
0answers
95 views

Explicit transgression maps for Group homology in LHS

This question is related to another question of mine (Here: Lyndon-Hochschild-Serre spectral sequence and cup products). I'm trying to figure out if some diagram commutes an one of the maps involved ...
6
votes
0answers
291 views

Lyndon-Hochschild-Serre spectral sequence and cup products

First here is my setup: Let $W$ be some group, and $C$ a normal subgroup of finite index, and let $W/C=G$. Now let $L$ be a a $G$-module on which $C$ acts trivially, so in particular we get on action ...
2
votes
1answer
139 views

Compatibility of connecting homomorphisms for Tor/Ext

This is a simple question about Tor and Ext functors. Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact sequences of $R$-modules. ...
0
votes
0answers
120 views

Split and pure exact sequence of sheaves

Let $X$ be a topological space and $$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$ be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if for each point $x\in ...
0
votes
2answers
326 views

Is this square a push-out square?

Consider the following diagram which lives in the category of $R$-modules. $$ \begin{array}{ccccccccc} 0 & \xrightarrow{i} & A & \xrightarrow{f} & B & \xrightarrow{q} & C ...
3
votes
1answer
221 views

Resolutions chain homotopic to projective ones

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...
4
votes
2answers
230 views

the first cohomology group of units of biquadratic number fields

Let $k=\mathbb{Q}(\sqrt{d_1}, \sqrt{-d_0})$, be an imaginary biquadratic number field and $H^1(G, U_k)$ the first cohomology group with $G=\mathrm{G}al(k/\mathbb{Q})$ and $U_k$ is the unit group of ...
1
vote
1answer
242 views

Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...
4
votes
4answers
195 views

Stratifications and Cohomology Computations

I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
3
votes
0answers
132 views

Repeated Homotopy Category of Chain Complexes

Consider an additive category $\mathcal{C}$. It is known that the category $Ch(\mathcal{C})$ of chain complexes in $\mathcal{C}$ is again an additive category and hence one can consider the category ...
8
votes
2answers
530 views

H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
0
votes
2answers
141 views

Flatness and tensor product of rings

Let $R_1$ and $R_2$ be two subrings of the ring $R$ which commute in $R$ so that we have a ring homomorphism $R_1\otimes_\mathbb{Z} R_2\rightarrow R$. Assume that $R$ is flat over $R_1$ and $R_2$. Is ...
9
votes
1answer
406 views

Examples of applications of the Freyd-Mitchell embedding theorem.

The Freyd-Mitchell embedding theorem states the following: Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor ...
1
vote
0answers
80 views

Cocompleteness of the category of small $A_\infty$ categories

To follow up on my previous question, is the category of small $A_\infty$ categories even cocomplete? Looking for reference.
0
votes
2answers
301 views

An exercise about Tor [closed]

Let $I$ and $J$ be two ideals in $A$. Show that $\operatorname{Tor}_{1} (A/I, A/J) =\frac {I \cap J} { IJ} $ and $Tor_{2} (A/I, A/J) =\ker(I \otimes_ {A}J \to IJ )$. The first Tor is not a ...
9
votes
0answers
237 views

Does there exist a Noetherian ring of finite injective dimension but higher Krull dimension?

Definition: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if (i) $R$ has finite left and right injective dimension (in which case it turns out ...
2
votes
0answers
149 views

About free resolutions of graded commutative algebras

Hi, I'm having troubles in adapting certain algebraic constructions to graded cases. We know that if $A$ is a commutative ring and $a_1,...,a_k$ are elements on $A$, there is a construction of the ...
13
votes
1answer
592 views

What is the purpose of section 3 of BBD?

I am not quite sure that this question is appropriate for Mathoverflow, yet I would be deeply grateful for any hint: what happens in section 3 of Beilinson A., Bernstein J., Deligne P., Faisceaux ...
3
votes
1answer
245 views

Commutativity of Tor

Let $A$ be a commutative ring with $1$ and $M,N$ be $A$-modules. Can you give a quick proof that $\textrm{Tor}_i(M,N) \cong \textrm{Tor}_i(N, M)$ using derived categories? In his Homological algebra ...
4
votes
1answer
249 views

Additive functors and Derived Categories

I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F ...
5
votes
2answers
370 views

Are subfunctors of left exact functors also left exact?

Consider the direct image functor $f_*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this ...
1
vote
1answer
310 views

Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$

Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that ...
2
votes
1answer
139 views

morphism of injective objects

Let $A,B$ be two bounded below complexes in module category, and $A \longrightarrow I$ (resp. $B \longrightarrow J$) a injective resolution. If $f: A \longrightarrow B$ is a morphism of complexes. My ...
13
votes
1answer
656 views

Homology in the $A_\infty$ World

This question is turning out to be a little long so let me start off with the headline. Given a differential graded algebra $A$, we can recover $A$ from its homology $HA$ if we know "the" ...
13
votes
1answer
533 views

How much of a variety can be reconstructed from codimension-zero data?

This is an improved version of this question. Maybe it should be an edit -- I'm not sure what the MO convention is. I'm curious, more or less, how much information one can get out of the derived ...
1
vote
1answer
296 views

How much of a variety can be reconstructed from codimension-zero data?

Suppose $X$ is a nice (finite type over $\mathbb{C}$, smooth and proper if necessary) variety. Suppose we're given the $\mathbb{C}$-points of $X$ as a set and for any finite subset $S\subset X$ we ...
1
vote
1answer
603 views

question on an exercise on homological algebra?

Suppose R has finite global dimension n, N is a f.g. module, F is a free module and Ext^n(N,F) is not equal to zero, then Ext^n(N, R) is not trival either. Note, HERE R is not Noetherian necessarily. ...
1
vote
2answers
350 views

Let $M$ be a $R$-Bimodule that happens to be projective, is its associated left $R \otimes R^{op}$-module projective too?

Let $k$ be a commutative ring, a $R$-Bimodule $M$ over a $k$-algebra $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being $R$ itself ...
1
vote
1answer
202 views

Defining Transfers Algebraically

I was trying to understand group (co)homology from a homological algebra point of view. Namely, given a group, $G$, one considers the category of (left) $\mathbb{Z}[G]$-modules, ...
4
votes
1answer
314 views

Mayer-Vietoris sequence in homology with local coefficients

Background. I'm trying to compute some homology groups using a Mayer-Vietoris argument, but I really need local coefficients. Question 1. What does the Mayer-Vietoris sequence look like when using ...
3
votes
0answers
227 views

Do exact functors commute with spectral sequences ?

Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let $$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...
1
vote
1answer
165 views

Compact generator of $D(\mathbb{P}^1)$

I suppose that Beilinson's compact generator (and, in fact, tilting object) $\mathcal{O} \oplus \mathcal{O}(1)$ in $D(\mathbb{P}^1)$ is the most well known example. I have the following simple ...
3
votes
2answers
323 views

Morphisms between $K_0$

I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map $$ f: \operatorname{K_0}(A) \to \operatorname{K_0}(B) $$ is it ...
0
votes
1answer
186 views

Spectral sequence for composition of global sections and tensor product of sheaves

Hi all, on the forum page http://www.groupsrv.com/science/about506648.html one can read the following (i cut out nonimportant parts): Question: Does anyone know any condition (non trivial) that ...
1
vote
1answer
319 views

Does the Čech cohomology always yield long exact sequences from short ones?

Does the Čech cohomology always give rise to a long exact sequences given a short exact sequence of sheaves? Clearly that cannot occur for sheaves on a paracomact (perhaps also Hausdorff, I'm not ...
3
votes
1answer
277 views

Restricting a Soft Sheaf to an Open is again Soft?

Hi everyone! Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :) Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are ...
2
votes
3answers
368 views

Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite. Let us assume we are given a group $G$ and a ...
1
vote
1answer
164 views

Question on resolutions for arbitrary chain complexes.

Consider a $\mathbb{Z}$-graded chain complex $A^{\bullet}$, I know that a bounded below complex is one such that $A^i = 0$ for $i$ sufficiently small, and a bounded above complex is one such that $A^i ...
7
votes
1answer
242 views

Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...
0
votes
1answer
277 views

Transfer map for group homology.

I'm trying to figure out what the transfer map looks like in a specific case. Here's the set up Let $G$ be a group and $H$ a subgroup of finite index, and let $h_{i}$ for $i=1,..,n$, be coset ...
8
votes
3answers
750 views

Serre Spectral Sequence of Representations

Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
7
votes
2answers
348 views

Recovering an abelian category out of its derived category

I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner. In Wikipedia it has been stated that since ...
0
votes
1answer
94 views

Explicit indecomposable monomorphism of finitely generated non-indecomposable Abelian groups.

This question is a follow-up of the question I asked here. Can you write down an explicit example of a monomorphism of finitely generated Abelian groups which is an indecomposable object in the ...
5
votes
0answers
232 views

Exactness is often an open condition. How often?

Let $\mathcal C$ denote an abelian category. Consider a formal $\mathbb Z$-graded object $V_\bullet$ in $\mathcal C$, by which I simply mean a $\mathbb Z$-indexed list $n\mapsto V_n$ of objects. A ...
7
votes
1answer
520 views

Universality of Ext functor using Yoneda extensions

Theses are simple and natural questions, but I could not find anything about it. If anyone has an answer or a reference this would be very much appreciated. Let $\mathcal{C}$ be an abelian category ...
7
votes
0answers
242 views

Do the Adeles Split?

I asked this question about a week ago here http://math.stackexchange.com/questions/288955/splitting-the-exact-sequence-of-the-idele-class-group, but got no answer so I thought I'd aske here and see ...
12
votes
2answers
363 views

Is the derived category of abelian groups a subcategory of the stable homotopy category?

An extension of the Dold-Kan equivalence gives an adjunction between the stable homotopy category and the (unbounded) derived category of abelian groups $SH \rightleftarrows D(Ab)$. Question 1: Is ...
4
votes
3answers
355 views

DG-projective vs. K-projective complexes

Hello! I'm a student learning the basics of working with the unbounded derived category $D(\mathcal{A})$. I arrived at the natural question, "is every K-projective complex formed out of projective ...