All Questions
Tagged with derived-algebraic-geometry ag.algebraic-geometry
159 questions
70
votes
3
answers
22k
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 ...
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 ...
- 63.1k
44
votes
5
answers
6k
views
What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
- 64k
41
votes
1
answer
3k
views
Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry
Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
- 511
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{...
29
votes
2
answers
3k
views
What (or how) are the new spaces of derived algebraic geometry?
I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena ...
- 787
27
votes
0
answers
1k
views
Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
24
votes
0
answers
730
views
What is the status of a result of Kontsevich and Rosenberg?
In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...
- 331
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 ...
- 7,799
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 ...
- 433
20
votes
1
answer
1k
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 ...
- 35.5k
20
votes
1
answer
859
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 ...
- 3,017
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 ...
- 595
18
votes
1
answer
2k
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 ...
- 1,889
18
votes
1
answer
2k
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 deformation-...
- 281
18
votes
0
answers
1k
views
What is the relationship between Artin and Lurie representability?
Artin's representability theorem gives conditions for a functor from commutative rings to sets (or groupoids) to be representable by an algebraic space (stack). The conditions are largely expressed ...
- 8,723
17
votes
1
answer
1k
views
How would you organize a cycle of seminars aimed at learning together some basics of Derived Algebraic Geometry?
This question is similar to this one because it's asking about a possible roadmap towards learning some derived algebraic geometry (DAG). But it's also different, because the goal is not to form a ...
- 23.4k
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 ...
- 16.2k
15
votes
1
answer
786
views
Why do we say IndCoh(X) is analogous to the set of distributions on X?
$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as ...
15
votes
1
answer
805
views
When does QCoh have 'enough perfect complexes'?
Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a ...
- 19.6k
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 $\...
- 7,799
15
votes
0
answers
284
views
Solution spaces of algebraic differential equations and derived geometry
We consider potentially non-linear differential equations on the formal one dimensional disc $\Delta$. Such equations are given by expressions $$P(z,f,f',f'',...)=0,$$ where $P$ is an element of the ...
user108998
15
votes
0
answers
3k
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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
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 ...
user100272
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}(...
- 7,799
13
votes
1
answer
2k
views
Proj construction in derived algebraic geometry
The question
My question is easy to state:
Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’...
- 133
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) ...
- 525
12
votes
2
answers
2k
views
Results relying on higher derived algebraic geometry
Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-...
user138661
12
votes
2
answers
978
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 ...
- 1,126
12
votes
0
answers
606
views
What's a holonomic D-module from the point of view of de Rham spaces?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...
- 773
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
11
votes
1
answer
789
views
Connectedness, loops and formal moduli problems
Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...
- 7,972
11
votes
1
answer
2k
views
Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 28.8k
11
votes
0
answers
607
views
The virtual fundamental class as derived intersection
Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
- 7,972
10
votes
1
answer
883
views
$\infty$-categorical understanding of Bridgeland stability?
On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
- 103
9
votes
2
answers
803
views
Explanation for Lurie's SAG Remark 25.1.3.7
I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry:
Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\...
- 93
9
votes
2
answers
2k
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 $\text{Coh}(S)...
- 2,216
9
votes
1
answer
748
views
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
- 301
9
votes
0
answers
288
views
Every Spectral Deligne-Mumford stack satsifies fpqc descent?
In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too ...
- 19.6k
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
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
8
votes
1
answer
720
views
Milnor excision for algebraic stacks
Recall that a commutative square of commutative rings
$$\begin{matrix}
A&\to&B\\
\downarrow &&\downarrow\\
A^\prime&\to&B^\prime\end{matrix}$$
is called a Milnor square if the ...
- 19.6k
8
votes
1
answer
2k
views
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 ...
- 83
8
votes
1
answer
324
views
$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$
In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...
- 1,349
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
0
answers
279
views
Motivating derived stacks via Euclidean geometry
Here (see Section 3) triangles in Euclidean plane are used to motivate the notion of DM stack (an equilateral triangle has more symmetries than a generic triangle).
Can something similar be done to ...
user138661
7
votes
1
answer
590
views
If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?
I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos ...
- 1,969
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 ...
- 1,423
7
votes
2
answers
1k
views
derived critical locus
I am looking for discussion in the literature that properly formalizes the heuristic idea that a BV-BRST complex is a model for the "derived critical locus of a function on an $\infty$-Lie algebroid".
...
- 19.9k
7
votes
1
answer
424
views
Kapustin-Witten branes and the derived moduli stack of Higgs bundles
A lot has been discussed on overflow regarding geometric Langlands and the physics of Kapustin and Witten's groundbreaking paper https://arxiv.org/abs/hep-th/0604151. I would like to add my two cents ...
- 163