-2
votes
1answer
215 views

Can you overcome the 6th degree obstruction?

I read and am still thinking about a 3-year old paper from the Danish-Norwegian "Niels Abel Journal". Two authors, named Somethingson (not Jacobson) and another Somethingelseson (still not Jacobson), ...
21
votes
5answers
1k views

(Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting. The problem is to produce an example of the following ...
4
votes
0answers
151 views

What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact $ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...
0
votes
0answers
63 views

A $\mathbb{Z}$-basis of $K^*(U(n)/T)$

It is well-known that $K^*(U(n)/T)\cong \mathbb{Z}[x_1, \cdots, x_n]\left/\left.\left(s_i-\begin{pmatrix}n\\i\end{pmatrix}\right|1\leq i\leq n\right.\right)$, where ...
0
votes
2answers
219 views

What does a singular simplex with real coefficient mean [closed]

For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental ...
3
votes
2answers
501 views

Quotients in Sums of Rings

Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings $R\cong R_1\times R_2$. It is now straight forward, that any ideal ...
5
votes
2answers
981 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
6
votes
3answers
409 views

A lost lemma about periodicity in a grid of long exact sequences?

This is a question about finding references and hopefully a larger context for a lemma in homological algebra I proved recently. The motivation is to understand properties of characteristic classes of ...
15
votes
0answers
350 views

Algebraic closure as a fibrant replacement?

Emil Artin's construction of the algebraic closure of a field $K$ is as follows. Let $K_{0} = K$, and inductively let $\{x_f\}$ be a set of indeterminates indexed by the irreducible $f$ in one ...
17
votes
6answers
1k views

Pathological Examples of Dimension

I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
7
votes
4answers
607 views

Quotient rings of $C(X)$

Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
6
votes
1answer
201 views

Forgetting and tensoring up for very connective maps of $E_{\infty}$-rings

Does anything happen if I forget and tensor back up along a highly connective map of $E_{\infty}$-rings? Here's what I mean precisely: Let $f \colon A \to B$ be a $n$-connective map between ...
3
votes
2answers
588 views

Does the first singular cohomology of an ACM surface vanish?

Hi everybody, I am interested in the following: Let $I\subset S=\mathbb{C}[x_0,\ldots ,x_n]$ be a graded ideal such that $\operatorname{depth}(S/I)\geq 3$, and let $X^h$ denote the analytic space ...
8
votes
2answers
1k views

State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...
2
votes
4answers
683 views

Extension of Tate's result regarding Tor

In a 1957 paper (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255378502), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R ...
6
votes
1answer
562 views

Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R ...
8
votes
0answers
401 views

E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product

Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 ...
16
votes
5answers
1k views

A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
5
votes
1answer
437 views

Showing an Ext^2 element is zero

If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1_X(G,E)$, we need to show that this sequence splits. To produce a splitting ...
0
votes
2answers
248 views

Can all induced maps be described categorically.?. (or at least as generally as possible)

Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol. I am pretty confused about induced maps in different areas of algebraic topology; I do know how these induced maps are ...
11
votes
2answers
399 views

How do you compute the space of lifts of an E-infinity map?

Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps ...
5
votes
2answers
420 views

An algebraic proof of Mumford's smoothness criterion for surfaces?

(Disclaimer: I'm a beginner in this area, so welcome corrections.) Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
13
votes
3answers
2k views

Does homology detect chain homotopy equivalence?

Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.