(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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6
votes
1answer
87 views

Injective dimension over enveloping algebra

Let $k$ be a field, and let $A$ be a commutative noetherian $k$-algebra. If a finitely generated $A$-module $M$ has finite injective dimension over $A$, does this imply that $M\otimes_k M$ has finite ...
2
votes
1answer
192 views

Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...
2
votes
1answer
196 views

Exact triple yields a distinguished triangle in derived category

In Methods of Homological Algebra before Proposition III.3.5 there is a short comment: "The next proposition shows that any exact triple [of complexes] can be competed to a distinguished triangle". ...
1
vote
0answers
126 views

Reference needed: Homology of the blow-up

Given an algebraic variety $X$ over the complex numbers. Let $V$ be a subvariety of $X$ and $\pi_V \colon X' \rightarrow X$ be the blow-up of $X$ at $V$. It is posible in general to compute the ...
3
votes
1answer
182 views

Is the space of smooth functions with compact support a DF space?

Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans ...
1
vote
0answers
152 views

Flat and injective quasi-coherent sheaves

Let $X$ be a quasi-compact semi-separated scheme and $$\varepsilon: 0\to A \to B \to C \to 0$$ be a short exact sequence of quasi-coherent sheaves. $\varepsilon$ is called a (categorical) pure exact ...
1
vote
0answers
426 views

Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written: Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...
4
votes
0answers
132 views

interpretation of homology of “non-commutative Koszul complex”

Let $A = Sym^*(V)$ be a polynomial ring. The Koszul complex $\cdots \to \wedge^2 V \otimes A \to V \otimes A \to A$ gives a resolution of the residue field $k$, so for any $A$-module $M$, the ...
15
votes
0answers
305 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. ...
6
votes
1answer
436 views

Spectral Sequences reference

What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep. I'm ...
1
vote
0answers
125 views

(Co)homology of classifying space of spin group $BSpin(n)$

In the answer for question: Homology of classifying space of spin group BSpin(n), it was shown that $H_i(BSpin(\infty),Z)$ is $0,0,0,Z$, for $i=1,2,3,4$. What is $H_i(BSpin(\infty),Z)$ or ...
0
votes
0answers
176 views

Second Quadrant Spectral Sequence

Let $\{E^{p,q}_r\}$ be a second quadrant spectral sequence (arising from a double complex), i.e. $E^{p,q}_r\neq 0$ only if $p\le 0$ and $q\ge 0$. In some papers I have seen such spectral sequences and ...
1
vote
0answers
68 views

Example H-unital algebra which is not unital

What is an example of an algebra which is H-unital (that is its Bar complex is acyclic) and yet it is not unital?
2
votes
1answer
66 views

dg-flat complexes and their characters

Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and ...
1
vote
1answer
93 views

Finite universal delta-functors

Let $F^\bullet : \mathcal{A} \to \mathcal{B}$ be a cohomological delta-functor which vanishes in degree strictly greater than $d$. Thus, $F^{d-\bullet}$ is a homological delta-functor. Now assume ...
1
vote
0answers
186 views

Soft Question: What does periodic cyclic theory measure?

Ex1) The cyclic homology of $\mathbb{C}[X,Y]$ and that of the algebra of functions on the sphere $S^2$ have the same periodic cyclic homology, clearly however these objects are topologically very ...
1
vote
1answer
110 views

Original sources for two theorems by Bass, Matlis and Papp

It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...
7
votes
1answer
224 views

Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?

Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...
1
vote
0answers
90 views

Chain homotopy of non-abelian category

How can one define the chain homotopy in non-abelian category? (The category I have in mind is the category of chain complexes of monoids.)
5
votes
3answers
682 views

is the tensor product of projective modules again projective?

Let $R$ be a commutative ring and let $A_1$ and $A_2$ be (not necessarily commutative) $R$-algebras. Under which conditions on $A_1$ and $A_2$ is the following true: For every projective $A_1$-module ...
1
vote
0answers
91 views

Homology of the fixed points of the singular complex of a G-space

I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me. Suppose $X$ is a topological space and $G$ a ...
1
vote
0answers
89 views

Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature". The complex is referenced as "Calabi complex" in various citing ...
3
votes
1answer
374 views

Projectives and Injectives in Functor Categories

Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories? Here is a precise question. Let $C$ be a small category, whose ...
1
vote
1answer
137 views

Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ ...
10
votes
1answer
342 views

Homological criteria for finite generation and finite presentation of modules?

(I'm new here; if I'm doing something wrong please help me out.) In short, my question is: There are some results on the behavior of finite generation and finite presentation in exact sequences (of ...
6
votes
0answers
143 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 ...
7
votes
1answer
156 views

Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
2
votes
2answers
289 views

local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...
3
votes
1answer
232 views

Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
9
votes
1answer
422 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
2
votes
1answer
211 views

Showing a functorial isomorphism

I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory. The exercise in question is from chapter IV. So, let ...
3
votes
1answer
364 views

Hochschild homology of quiver algebras

Let $K$ be a field and $\Gamma$ a quiver (=multidigraph) and $K[\Gamma]$ its quiver algebra (free $K$-module on the set of all paths of length $\geq0$ where multiplication is concatenation if ...
3
votes
1answer
100 views

Are finite-dimensional representations of groups of type $\text{FP}_{\infty}$?

Let $G$ be a group (possibly infinite) and $k$ be a field. A module $M$ over $k[G]$ is said to be of type $\text{FP}_{\infty}(k)$ if it has a projective resolution each of whose terms is finitely ...
7
votes
1answer
290 views

In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
1
vote
0answers
111 views

Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects? What are some applications? For ...
6
votes
2answers
330 views

Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$? Is there any ...
0
votes
2answers
253 views

Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
3
votes
0answers
91 views

Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$? Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...
4
votes
2answers
767 views

Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has $$\otimes: D^{-}(X) \times D^{-}(X) \to ...
2
votes
2answers
243 views

Some questions on vanishing of Ext sheaves

Let $X$ be a complex manifold of dimension $3$ and $\mathcal{E}$ be a coherent sheaf such that $\dim supp(\mathcal{E})=1$. In this situation, I would like to know why we have the Ext sheaf ...
5
votes
1answer
137 views

Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras

Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where $$ C_n(A):=A^{\otimes n+1} ...
0
votes
0answers
96 views

Consistency of the u-invariant under field extension

A algebraic field extension L/k induces of homomorphism between the Wittrings. We get $\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...
3
votes
0answers
85 views

Lifting Lie algebra cohomology class to Hochschild cochain

Let $g$ be a Lie algebra, $h\subset g$ an ideal. The associative algebra $N=U(h)\subset U(g)$ can be viewed as a $g$-module. The Lie algebra cohomology ${\rm H}^*(g,U(g))$ is isomorphic to the ...
8
votes
3answers
262 views

Is 'the' homotopy colimit of a sequence of exact triangles an exact triangle?

Let $$ \begin{array}{rccccl} A_0&\to& B_0&\to& C_0&\to\\ \downarrow & &\downarrow&&\downarrow\\ A_1&\to& B_1&\to& C_1&\to\\ \downarrow & ...
2
votes
0answers
115 views

Sort of units for the Yoneda product (and/or in Hochschild cohomology)

In an abelian category $\mathcal A$ with enough projectives, we have the Yoneda pairing $$\operatorname{Ext}^p_{\mathcal A}(Y,Z)\otimes \operatorname{Ext}_{\mathcal A}^q(X,Y)\longrightarrow ...
3
votes
1answer
207 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
5
votes
0answers
286 views

Sign problem for the shift functor on DG modules

Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...
2
votes
1answer
132 views

Kernel of the induced map of the wedge product

Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of ...
3
votes
1answer
108 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
2
votes
1answer
474 views

A question about the universal coefficients theorems

This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (http://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem), so ...