3
votes
0answers
126 views

Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?

Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...
9
votes
2answers
506 views

Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base? In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
9
votes
0answers
487 views

Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...
20
votes
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 ...
3
votes
1answer
149 views

Higher homotopy groups of a Zariski closed subset of $\mathbb C^n$

Suppose $V\subset \mathbb C^n$ is a Zariski closed subset. Is it true that the higher homotopy groups $\pi_i(\mathbb C^n-V)$ vanish for $0<i<$ some number depending on the codimension of $V$ in ...
6
votes
2answers
257 views

Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper? Since weak ...
10
votes
1answer
324 views

Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types

I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might ...
4
votes
0answers
313 views

Interaction petit topos - gros topos

I know that there are already several discussions on this topic (and they already helped me a lot, so far), but I couldn't find one completely solving my problem. Question 1. Fix a scheme $X$. I know ...
3
votes
1answer
218 views

$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?

I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through ...
8
votes
1answer
435 views

Etale cohomology of and forms of algebraic groups

Let $k$ be a field, and $K$ its separable closure. Consider two different $k$-schemes, $X$ and $Y$, which become isomorphic upon extension of scalars to $K$: $X_K \cong Y_K$. Then the etale ...
1
vote
3answers
380 views

prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...
15
votes
3answers
1k views

Is there a scheme corresponding to the unit interval?

Can someone complete the following table? $\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & ...
6
votes
2answers
381 views

Algebraic K-theory and Homotopy Sheaves

Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory ...
0
votes
1answer
190 views

Second homotopy of a torus complement in the 4-sphere

Let $T$ be the boundary of a solid torus in $S^4$. Are there any theorems or methods which would help one to compute $\pi_2(S^4 -T)$? Or to say if, e.g., it had finite rank and no torsion? More ...
22
votes
3answers
1k views

What is the algebraic geometry version of the spheres?

In topology the spheres $S^n$ are the "simplest" closed manifolds, and they are like "Dirac's delta at $n$" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact ...
14
votes
3answers
2k views

Conjectures in Grothendieck's “Pursuing stacks”

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this ...
17
votes
2answers
532 views

Explicit computations of the étale homotopy type?

Hi, I'm currently trying to learn about etale homotopy for schemes as introduced by Artin-Mazur. I know that by the Artin-Mazur comparision theorem, it is possible to compute the etale homotopy type ...
3
votes
0answers
140 views

axiomatizing the abelian part of the topological fundamental groupoid functor on algebraic varieties

let Vv be the category of complex algebraic varieties defined over $\bar Q\subset C$, and let $\pi_1^{top}:Vv\longrightarrow Groupoids$ sending a variety V into its (strict) fundamental groupoid ...
8
votes
4answers
1k views

Lurie's “Virtual fundamental classes” and “Geometric derived stacks”

In his thesis, Jacob Lurie mentioned two work in preparation (by him), namely "Virtual fundamental classes and the motivic sphere" and "Geometric derived stacks". Now that much is written in the DAG ...
7
votes
3answers
584 views

Can we define homotopy groups using Tannakian categories

This is another vague question. Hope you guys don't mind. Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi_1(T,F)$, to ...
3
votes
2answers
212 views

When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?

Let $A$ be a simplicial commutative ring over a field $k$ of characteristic zero (or a cdga in non-positive degrees with differential of degree -1). Let $M$ be a perfect $A$ module. If necessary, ...
6
votes
0answers
219 views

Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...
12
votes
1answer
1k views

can a common mortal understand why the affine line is not smooth in brave new algebraic geometry?

In the introduction to HAGII Toen and Vezzosi write that in brave new algebraic geometry (that is, algebraic geometry over the category of symmetric spectra) Z[T] is not smooth over Z. I am told that ...
3
votes
0answers
167 views

Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor

Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
13
votes
2answers
1k views

obstruction theories in algebraic geometry

I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction ...
12
votes
2answers
912 views

Spaces that are both homotopically and cohomologically finite

Is it true that every connected space with 1) just finitely many nontrivial homotopy groups, all finite, and 2) just finitely many nontrivial rational cohomology groups, all finite rank, is ...
4
votes
2answers
332 views

Which $H$--groups satisfy the rigidity property of abelian varieties?

Let us call a group object $G$ in a category $\mathcal C$ rigid, if it has the following property: For every group object $X$ in $\mathcal C$, every morphism $G\to X$ in $\mathcal C$ respecting the ...
13
votes
2answers
747 views

Homotopy type of Hilbert schemes of points of $\mathbb C^2$

Let $X=\mathbb C^2$, let $X^{[n]}$ be the Hilbert scheme of length $n$ 0-cycles in $X$, and let $X^{[n]}_0$ be the closed subscheme formed by the 0-cycles supported at 0. As far as I know $X^{[n]}_0$ ...
5
votes
1answer
549 views

Flat Principal Connections and Homotopy Groups?

I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...
15
votes
1answer
761 views

Does derived algebraic geometry allow us to take quotients with reckless abandon?

So one of the major problems with the categories of schemes and algebraic spaces is that the "correct quotients" are oftentimes not schemes or algebraic spaces. The way I've seen this sort of thing ...
3
votes
0answers
515 views

A Question about a theorem in Toën's notes “Lectures on dg-categories”

So I am trying to learn a bit about dg categories from Toën's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand ...
24
votes
1answer
808 views

Identifying the stacks in Devinatz-Hopkins-Smith

I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what ...
4
votes
1answer
213 views

Model category with formally smooth morphisms as fibrations?

Let's view the category of algebraic spaces as a full subcategory of the category of "spaces" over the opposite category of commutative rings, that is, the category of sheaves on $CRing^{op}$ in the ...
8
votes
1answer
673 views

Interdependence between A^1 homotopy theory and algebraic cobordism

I would like to learn something about $\mathbb{A}^1$-homotopy theory. I know about standard references on the subject, but before dwelling into studying them I have a doubt which some expert could ...
25
votes
2answers
1k views

What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale ...
9
votes
1answer
665 views

Formalism of homotopy theory of schemes

I have some vague knowledge about the philosophy that schemes should be thought of as similar to topologic spaces, and we should divide everything by homotopy, and that the space should be actually ...
19
votes
3answers
2k views

Homotopy theory of schemes examples

Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?
5
votes
2answers
687 views

Higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks ...
10
votes
2answers
1k views

Finiteness conditions on simplicial sheaves/presheaves

Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
3
votes
4answers
361 views

E_\infty spectrum corresponding to Z_p

First of the questions about derived algebraic geometry from a noobie. The way I understand it, every discrete ring R corresponds to some ring spectrum whose ...
18
votes
4answers
1k views

Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...