All Questions
Tagged with ag.algebraic-geometry derived-algebraic-geometry
159 questions
4
votes
0
answers
254
views
Homotopy colimit description of stacks
Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...
- 3,574
2
votes
0
answers
132
views
Compact generation of quasicoherent sheaves on mapping stack
Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
- 3,019
23
votes
3
answers
2k
views
Where does one go to learn about DG-algebras?
The theory of differential graded algebras (in char 0) and their modules has numerous applications in rational homotopy theory as well as algebraic geometry.
I'm looking for a reasonably complete ...
- 610
8
votes
1
answer
979
views
Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
- 1,761
8
votes
1
answer
2k
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Elementary (English) reference for the cotangent complex?
I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.
I understand that the standard ...
- 928
4
votes
0
answers
246
views
Tannaka duality for $DG$ indschemes
In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism
$$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$
where $X$ and ...
- 3,019
6
votes
0
answers
196
views
Specific Example of a Morphism of Schemes for which the Push-Pull Morphism is not an Isomorphism
Consider a Cartesian diagram of schemes as follows:
$\require{AMScd}
\begin{CD}
X \times_Z Y @>{\tilde{\pi}}>> Y\\
@VV{\tilde{\phi}}V @VV{\phi}V\\
X @>{\pi}>> Z
\end{CD}$
From the ...
- 422
12
votes
0
answers
324
views
Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack
I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
...
- 595
2
votes
0
answers
181
views
Why does the following construction describe the Serre functor?
In the book "Spectral Agebraic Geometry" that Jacob Lurie is currently writing, he gives a construction (11.1.5.1), which describes the Serre functor: it has already been shown that any proper $A$-...
- 21
16
votes
1
answer
1k
views
$\infty$-operads and $E_\infty$-algebras
I work in algebraic geometry. Lately, the answer to most of my questions seems to be "you should read Lurie's Higher Algebra." I took this advice seriously, however it turned out not to be an easy ...
CommunityBot
- 1
6
votes
0
answers
517
views
relative spectrum in derived algebraic geometry
I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks.
More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's ...
- 121
5
votes
1
answer
569
views
Derived completion of complexes
Suppose $K$ is a bounded above complex of free abelian groups, and take its derived $\ell$-adic completion $K^{\wedge,\ell} = R\lim (K/\ell^n)$ in the derived category, for $\ell$ a prime.
If $K\to L$...
- 52.7k
6
votes
1
answer
513
views
Is every algebraic space a 1-geometric stack?
In many references (Toen, Higher and derived stacks: a global overview, Toen, Vezzosi, Homotopical algebraic geometry II, and so on), the definition of $n$-geometric stack appears.
In the non-derived ...
- 1,088
1
vote
0
answers
204
views
How can we construct a derived scheme as a gluing of derived schemes?
More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch
In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of ...
- 421
4
votes
0
answers
106
views
Global functions algebra of formal (infinity) groupoid associated to Lie (infinity) algebroid
I was wondering if there is a smooth (sophisticated) way to associate the algebra of global functions of formal groupoid associated to Lie-Rinehart algebra (considered as 1-stack) to its Chevalley-...
- 121
15
votes
0
answers
3k
views
What to expect from spectral algebraic geometry
So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
- 3,019
14
votes
2
answers
781
views
Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I'll be using homological grading throughout this question.
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}(...
- 63.9k
15
votes
1
answer
1k
views
Can "ampleness" be detected inside the derived category?
Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.
One of the possible definitions of an ample line bundle goes as follows:
Def 1: A line bundle $\...
CommunityBot
- 1
4
votes
0
answers
552
views
The lisse-etale site and derived algebraic geometry
If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
- 4,008
14
votes
1
answer
953
views
Reference for symplectic structures on schemes?
My original goal was to read the PTVV paper Shifted Symplectic Structures https://arxiv.org/pdf/1111.3209v4.pdf. I was quickly humbled!
Being told the theory ought to generalize symplectic structures ...
- 4,713
3
votes
0
answers
180
views
vanishing of higher homotopy sheaves of cotangent complex
Let $X$ be a $\mathbb{C}$-scheme and suppose that there is an isomorphism (in the derived category of qc-sheaves on $X$) between the cotangent complex, $\mathbb{L}_{X}$, and its 0th homotopy sheaf, ...
CommunityBot
- 1
9
votes
0
answers
507
views
Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
- 7,799
4
votes
0
answers
538
views
Why do motivic stacks make sense?
In the paper "Motivic model categories and motivic derived algebraic geometry", Yuki Kato, whose email-address I sadly couldn't find out, describes a procedure to "motivy" the objects of any $(\infty,...
- 13.6k
3
votes
1
answer
354
views
Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?
Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...
- 1,505
7
votes
1
answer
954
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 ...
- 8,445
5
votes
1
answer
711
views
Closed symmetric monoidal structure on the derived category of modules whose unit is a dualizing complex?
Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to ...
- 66
3
votes
0
answers
188
views
What are projective morphisms in derived algebraic geometry
I was curious if someone could point me to a reference or tell me if there is any notion of projective morphism between derived schemes? What about a notion of Proj? Also I am interested in what blow-...
- 431
9
votes
0
answers
287
views
derived schemes and perfect obstruction theories
In a survey article of Toen's it is claimed that that there is forgetful $\infty$-functor between the $\infty$-category of derived schemes locally of finite presentation over a field $k$ and the $\...
- 131
5
votes
0
answers
316
views
Derived stack 2-perfect complexes and derived equivalences
Let $X$ be a scheme of finite type over $\mathbb{C}$. Toën and Vaquié construct the derived stack of perfect complexes on $X$, which I will denote $\mathcal{DP}er(X)$. They prove that this derived ...
- 7,320
54
votes
2
answers
4k
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 ...
CommunityBot
- 1
3
votes
0
answers
286
views
Exterior tensor of derived categories of coherent sheaves
Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor
$$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \...
- 1,423
19
votes
3
answers
2k
views
Is it always possible to write a scheme as a colimit of affine schemes?
My question is: Is it possible to write any scheme as a (1-categorical) colimit of a diagram of affines? If no, what are some examples?
I ask this question because I have read that one can write any ...
- 5,482
5
votes
1
answer
438
views
Twisted derived Morita theory of schemes
It has been proved by Toën and Lunts-Schnürer that the dg category $\mathrm{L}_{qcoh}(X\times Y)$ of quasi-coherent sheaves over the product of two quasi-compact, quasi separated (and flat over a ...
- 2,327
7
votes
0
answers
279
views
Is there a derived geometric interpretation of morse functions?
Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\...
- 1,716
1
vote
0
answers
86
views
derived invariants, perversity and modular coefficients
Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$.
Let $n$ an integer such that it is not prime with the order of $\Gamma$.
Then $\pi_{*}\mathbb{Z}/n\...
- 3,472
4
votes
0
answers
232
views
Motivic Interpretation of Rationally Trivial Cycles
The Chow groups are defined by taking groups of cycles modulo rationally trivial cycles. One then has a cycle class map to etale cohomology (over the base field), and for a number field, one expects ...
CommunityBot
- 1
5
votes
0
answers
225
views
Does the Amitsur complex have a universal property?
The question is essentially the title. In other words, is there some universal property that the Amitsur complex for a morphism of rings $\phi:A\to B$ satisfies as a cosimplicial ring, or cosimplicial ...
- 10.5k
21
votes
1
answer
3k
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 ...
- 30.3k
4
votes
0
answers
248
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 $\...
- 16.9k
6
votes
0
answers
608
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 ...
- 3,101
2
votes
1
answer
246
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: \...
- 27.5k
2
votes
3
answers
480
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 ...
- 118k
1
vote
0
answers
300
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, ...
- 11
31
votes
1
answer
1k
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 $\mathcal{...
CommunityBot
- 1
2
votes
0
answers
113
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, i....
- 3,472
20
votes
1
answer
1k
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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 ...
- 6,239
13
votes
1
answer
961
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) ...
- 24.1k
6
votes
1
answer
684
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 $\Bbbk$, ...
- 1,254
3
votes
1
answer
468
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 ...
- 7,320
4
votes
1
answer
1k
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 ...
CommunityBot
- 1