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16
votes
2answers
960 views

Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
13
votes
1answer
521 views

Motivation and potential applications of spectral algebraic geometry

Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry. Now I'm curious what future is there for spectral algebraic ...
4
votes
0answers
150 views

Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and ...
8
votes
0answers
146 views

List of known Fourier Mukai partners?

I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the Pfaffian-Grassmannian ...
4
votes
0answers
196 views

Reference request: category of sheaves of O-modules with coherent cohomology

Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of ...
18
votes
2answers
2k views

Derived algebraic geometry: how to reach research level math?

I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different. My goal is to study derived algebraic geometry, where derived ...
7
votes
0answers
160 views

$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
2
votes
0answers
77 views

Flat resolutions of DG-schemes

Recall that a DG-scheme is a pair $(X,\mathcal{O}_X)$, where $(X,\mathcal{O}^0_X)$ is a scheme, $\mathcal{O}_X$ is a sheaf of commutative DG-algebras over $(X,\mathcal{O}^0_X)$, and each ...
5
votes
0answers
238 views

Derived global functions on (derived) stacks $BG$ and $G/G$

In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we ...
3
votes
0answers
63 views

Derived Deformations of associative algebras

Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows: ...
6
votes
2answers
196 views

Direct proof that the model category of cdgas is left proper

Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
4
votes
0answers
227 views

On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
1
vote
0answers
183 views

Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms ...
2
votes
1answer
201 views

Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...
2
votes
3answers
354 views

Integral transform on noncommutative spaces

In their paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry" the authors show that for perfect stacks $X$ and $Y$ over $k$, and their $k$-linear $\infty$-categories of ...
1
vote
0answers
163 views

Classifying Spaces and Eilenberg-Maclane objects in the category of simplicial rings

[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, ...
6
votes
0answers
131 views

flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion. If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat. If $A$ is not noetherian, ...
22
votes
1answer
663 views

Which properties of a variety are detected by its derived category of coherent sheaves?

Context: I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about ...
3
votes
1answer
427 views

Hopf-algebras in associative ring spectra

I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative ...
4
votes
1answer
117 views

Proper Model Category

Let R be a commuative ring. Consider the category of simplicial R-modules with the projective model stucture. Can someone give me a precise reference which proves that this model category is proper? ...
16
votes
0answers
276 views

Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps $$ B G \longrightarrow B \mathrm{GL}_1(A) $$ for $A$ an $E_\infty$-ring carrying an oriented ...
2
votes
0answers
92 views

quotient a scheme by a stratified vector bundle

Let $k$ be a field. Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, ...
33
votes
2answers
1k views

What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
2
votes
0answers
101 views

How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?

First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$. We can see that $\mathcal{L}BG$ has the homotopy type of ...
10
votes
0answers
352 views

Thom Spectra and Hopf-Galois Extensions of Ring Spectra

So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...
13
votes
1answer
570 views

On a question motivating Lurie's treatment of formal moduli problems

Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is ...
11
votes
1answer
628 views

Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks. For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
12
votes
0answers
936 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 ...
7
votes
1answer
246 views

Which morphisms of ring spectra are of effective descent for modules?

There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...
4
votes
2answers
463 views

Small objects in categories

I would like to pick out small objects from a category. I would like to find such a notion which Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least ...
5
votes
1answer
245 views

Higher Descent Cohomology

Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on ...
4
votes
1answer
355 views

A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings

Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...
4
votes
1answer
310 views

Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology

Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this ...
6
votes
2answers
509 views

Higher commutators in E_n algebras and the Maurer--Cartan equation

Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The ...
4
votes
1answer
356 views

Why should we study deformations of perfect complexes

What are the advantanges of studying deformation of perfect complexes over the classical theory of deformation of coherent sheaves? Any references which elaborates on the applications on deformation ...
3
votes
0answers
354 views

Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...
3
votes
1answer
438 views

Pushout schemes/stacks

I would like to know in what generality do we have pushouts for schemes/stacks/derived stacks. More precisely, let $f : X \rightarrow Y$ be a proper flat surjective morphism of schemes of finite type ...
7
votes
2answers
520 views

What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?

I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...
8
votes
0answers
252 views

Derived (non-commutative) geometry, geometric constructions in explicit form

I'm interested in the following construction. Start with derived category of coherent sheaves witch equivalent to derived category of representations of some dg-algebra. Quasi-isomorphic dg-algebras ...
14
votes
1answer
694 views

When does the cotangent complex vanish?

The question is already in the title. Less succinctly, let's call a map $f:X \to Y$ of schemes $L$-trivial if its cotangent complex is quasi-isomorphic to $0$. Such maps have striking ...
23
votes
4answers
2k views

What is a simplicial commutative ring from the point of view of homotopy theory?

Let $k$ be a field. There are two natural categories to consider: The category of simplicial commutative $k$-algebras. The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of ...
6
votes
1answer
335 views

question about higher geometric stacks

I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Spec(A))= ...
7
votes
2answers
705 views

Defining ind-coherent sheaves and their singular support

Q1: My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of this paper, they are defined as being the ind-completion of the category ...
6
votes
0answers
549 views

The De Rham Stack and $\text{LocSys}$

Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$): $Maps(S, LocSys_G(X)) := ...
35
votes
1answer
2k views

Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general: $\bullet$ In the approach by ...
7
votes
1answer
370 views

If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence between objects of $\mathcal{D}(\mbox{Mod}_A)$ and $A$-module spectra?

In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the same thing as objects ...
0
votes
0answers
164 views

What means “extended concepts of symmetry”?

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting things like ...
6
votes
2answers
429 views

Somewhat general question that includes: “Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?”

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree? Example: Given a dg ...
8
votes
2answers
654 views

Blowing up a derived scheme

Is there a sensible notion of blowing up in any of the available frameworks for derived algebraic geometry? I am happy to remain in the affine setting, where I think the right question to ask is "what ...
16
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 ...