(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

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21
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0answers
2k views

Sums of injective modules, products of projective modules?

Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension? Analogously, under what assumptions on R does ...
20
votes
0answers
286 views

Does the Tate construction (defined with direct sums) have a derived interpretation?

Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
17
votes
0answers
403 views

Other examples of computations using transfer of structure from the chains to the homology?

There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
16
votes
0answers
733 views

Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question. In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...
16
votes
0answers
867 views

Homotopy flat DG-modules

A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups ...
15
votes
0answers
324 views

Are dualizable modules finitely generated?

Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module. ...
14
votes
0answers
804 views

Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split

Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$ Actually this ...
14
votes
0answers
397 views

Refinement of concept of support of a module

My rings are commutative and noetherian. The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
13
votes
0answers
445 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 ...
12
votes
0answers
447 views

Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...
12
votes
0answers
1k views

Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$. In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology: The zeroth Cech ...
12
votes
0answers
515 views

References for a certain generalization of Hochschild cohomology?

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C ...
11
votes
0answers
365 views

Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
11
votes
0answers
371 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 ...
11
votes
0answers
444 views

Bar construction vs. twisted tensor product

One may study the cohomology of a space $E$ expressed as a homotopy pullback of $X$ and $Y$ over $Z$ using either the Eilenberg-Moore spectral sequence or the Serre spectral sequence for the fibration ...
10
votes
0answers
866 views

Complexes of representations with complementary central charges

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
9
votes
0answers
359 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 ...
9
votes
0answers
223 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. ...
9
votes
0answers
168 views

Is it possible to take “limits up to homotopy”?

Before I begin, an apology: it's been a while since I've done much analysis, so I might be misusing (or just missing) terminology. I have a chain complex $(V_\bullet,\partial)$ of topological vector ...
9
votes
0answers
183 views

The global dimension of fields

In the absence of the Axiom of Choice, it is not necessarily true that all vector spaces over a field have bases. What are the possible global dimensions of fields in a model of ZF in which AoC ...
9
votes
0answers
320 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 ...
9
votes
0answers
392 views

Are the supports of $Ext^i(M,N)$ eventually periodic?

Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules. Question: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic ...
9
votes
0answers
591 views

Who proved the exactness of Amitsur's complex ?

A foundational result in Grothendieck's descent theory and in his étale cohomology is the exactness of Amitsur's complex. More precisely, suppose we have an $A$-algebra $A\to B$; then there is a ...
8
votes
0answers
176 views

Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
8
votes
0answers
296 views

End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...
8
votes
0answers
173 views

Depth for non-commutative rings

The depth of a ring or module is one of the most basic invariants in commutative ring theory. Q1: Is there also a powerful notion of depth for non-commutative rings ? By a search in mathscinet, I ...
8
votes
0answers
190 views

Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...
8
votes
0answers
218 views

(Reduced) cyclic homology of a free product of unital algebras

Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...
7
votes
0answers
70 views

Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
7
votes
0answers
159 views

Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then it should hopefully be understood by most readers as saying ...
7
votes
0answers
150 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 ...
7
votes
0answers
259 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 ...
7
votes
0answers
318 views

Reference/ elementary proof of a result about projective dimension in group rings

Hello- I've had to use a result that sounds like it should be well-known, but I couldn't find any references, and my proof is rather unsatisfactory. I was hoping someone here could help! The problem ...
6
votes
0answers
179 views

Is there an obstruction which classifies “quasi-isomorphism but not chain equivalence”?

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...
6
votes
0answers
271 views

Transgression map spectral sequence of Ext

Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence ...
6
votes
0answers
54 views

A “lower-central” filtration of Steenrod algebra?

$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...
6
votes
0answers
475 views

How can you see the minimal relations on a quiver from its bimodule resolution?

Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...
6
votes
0answers
188 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 ...
6
votes
0answers
154 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 ...
6
votes
0answers
152 views

Split exact categories arising naturally

If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...
6
votes
0answers
349 views

Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
6
votes
0answers
501 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 ...
6
votes
0answers
310 views

homotopy pullbacks of tensor product chain complexes (towards Kunneth formula in diff cohomology)

I have editted this question from the previous version which did not obtain much attention. Suppose I have two diagrams of chain complexes: $A^* \rightarrow C^* \leftarrow B^*$ $\tilde{A}^* ...
6
votes
0answers
180 views

The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ...
6
votes
0answers
275 views

Fourier-Mukai bimodule

Let $X$ and $Y$ be two smooth varieties over some field, and let $E$ be a perfect complex on $X \times Y$. It looks like it is not possible to define a DG-functor $F_E : Perf(X) \to Perf(Y)$ such ...
6
votes
0answers
553 views

On the multiplicative structure in spectral sequences.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
6
votes
0answers
364 views

Quantum polynomial rings and singularities

Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
5
votes
0answers
142 views

Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors $$ ...
5
votes
0answers
84 views

Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$. Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
5
votes
0answers
219 views

Do differential objects form triangulated categories?

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to ...