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108 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 ...
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0answers
142 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 ...
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0answers
50 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
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1answer
58 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 ...
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1answer
87 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 ...
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0answers
168 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 ...
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1answer
99 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
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1answer
207 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 ...
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0answers
84 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.)
3
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3answers
438 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
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0answers
82 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 ...
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0answers
84 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
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1answer
274 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 ...
0
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1answer
109 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
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1answer
266 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
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0answers
128 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
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1answer
150 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
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2answers
256 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
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1answer
195 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 ...
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1answer
402 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
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1answer
178 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
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1answer
292 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
96 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 ...
6
votes
1answer
215 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 ...
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0answers
97 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
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2answers
280 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
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2answers
204 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
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0answers
90 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 ...
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2answers
529 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
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2answers
199 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
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1answer
104 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} ...
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93 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
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0answers
75 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 ...
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3answers
210 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 & ...
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0answers
97 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
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1answer
201 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
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0answers
256 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
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1answer
90 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
96 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
454 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 ...
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0answers
377 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 ...
7
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2answers
286 views

Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex

Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence $$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...
2
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0answers
110 views

semi-orthogonal decompositions and embeddings

This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition $$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$ Suppose ...
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0answers
99 views

syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...
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0answers
249 views

generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...
3
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0answers
234 views

2 - Calabi Yau algebras and bimodule coherence

Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero. In H. Minamoto "Ampleness of two-sided tilting complexes", ...
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4answers
1k views

origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange). I have been "brought up" as an algebraic ...
1
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1answer
253 views

Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...
7
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2answers
275 views

Why the term “monad” in homological algebra?

Which is the origin and the reason for the choice of the term "monad" in the sense of homological algebra? Does this concept have any relation whatsoever to the "monads" from category theory?
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0answers
167 views

Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...