All Questions
Tagged with derived-algebraic-geometry ag.algebraic-geometry
159 questions
3
votes
0
answers
365
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Construction of derived Quot schemes
I am studying the construction of derived Quot schemes in the paper Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”.
Derived quot stacks are constructed from ...
- 479
2
votes
0
answers
230
views
Lie bracket on the unshifted tangent complex?
My problem is as follows: if $X$ is a derived scheme, or derived stack, or any kind of a space where tangent complex makes sense, I guess there should be a lie bracket on its tangent complex, ...
- 121
1
vote
0
answers
202
views
Is the cotangent complex sensitive to truncation?
$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Spec{Spec}$If $V$ is a dg vector space (in positive degrees), we can view it as a derived scheme $V = \Sym V^*$. It has (co)tangent complex $V$ (and $...
- 5,711
4
votes
0
answers
195
views
Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one
I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
- 351
4
votes
0
answers
282
views
The dual abelian scheme in derived algebraic geometry
$\def\Pic{\mathcal{Pic}}\def\Gm{\mathbb{G}_m}\def\Hom{\mathop{Hom}}\def\HOM{\mathcal{Hom}}$
If $A/S$ is an abelian scheme, the fppf sheaf $\Pic^0_{A/S}$ is representable by an abelian scheme $\hat{A}$....
- 1,196
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
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
3
votes
0
answers
246
views
Derived category and L-function
For abelian varieties over $\mathbb{Q}$ $\mathscr{A}$ and $\mathscr{A}'$, if derived categories $D(\mathscr{A})$ and $D(\mathscr{A}')$ are equivalent then L-functions are same $L(s,\mathscr{A})=L(s,\...
- 519
6
votes
0
answers
201
views
Smoothness of a variety implies homological smoothness of DbCoh
I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions.
It ...
5
votes
0
answers
587
views
When is the cotangent complex perfect?
Let $X\rightarrow S$ be a proper flat morphism of schemes.
When is the cotangent complex $L_{X/S}$ perfect ?
It is well known, that for local complete intersections the cotangent complex is perfect, ...
- 51
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
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
4
votes
0
answers
310
views
Dimension of derived Artin stacks and perfect complexes
I am interested in the concept of dimension of derived and $n$-Artin stacks. Take for example the definition 4.10 of From HAG to DAG: derived moduli stacks. in which they define the dimension of a ...
- 787
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
1
vote
0
answers
224
views
Two definitions of cotangent complex
I have reading a paper by Professor Pridham(https://arxiv.org/abs/0905.4044v4). Page 47-48 contains a comparison of the two definitions of the cotangent complex, but there is a part I don't understand....
- 479
4
votes
0
answers
159
views
Relation between $\mathbb{A}^1$-homotopy theory and derived algebraic geometry [duplicate]
I've often heard that one of the benefits of derived algebraic geometry, next to a cleaner intersection theory, is that "provides natural settings " for the $\mathbb{A}^1$-homotopy theory (...
- 4,418
1
vote
0
answers
281
views
Étale homotopy type of (derived) loop space
A feature of derived algebraic geometry is that we have internal homs. Furthermore, we can think of $B\mathbb{Z}$ as the derived algebraic geometric analogue of $S^1$. Thus we have an analogue of the ...
- 4,418
3
votes
0
answers
162
views
Homotopy Kan extensions, formally coherent functors and derived Schlessinger criterion
Let $k$ be a finite field. Denote by $discArt_k$ the category of Artinian rings with residue field $k$ and $Art_k$ the category of Artinian simplicial rings. Consider a functor $\mathcal{F}:disArt_k\...
- 4,418
1
vote
0
answers
202
views
Schlessinger criterion and finiteness of tangent space
Schlessinger's criterion allows us to study whether or not a functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$ from the category of local Artin $\Lambda$-algebras to sets is representable. One ...
- 4,418
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
1
vote
0
answers
137
views
Open problems for shifted symplectic structures
I am now interested in shifted symplectic structures.
What are the open problems of shifted symplectic structures regarding the moduli space of sheaves ?
Especially now I am interested in moduli ...
- 479
1
vote
0
answers
185
views
Is there a stacky definition of irreducible symplectic manifold?
I am now interested in studying symplectic structures in the field of stacks.
In particular, is there a stacky definition of irreducible symplectic manifold ?
I'm also interested in similar things in ...
- 479
2
votes
1
answer
470
views
Two results about (shifted) symplectic structures
I am now interested in shifted symplectic structures.
I found Zhang's results about symplectic structures (2011, p.3-4, arXiv link,
Comm. Anal. Geom. 2017) and Pantev–Toen–Vaquié–Vezzosi's results ...
- 479
3
votes
0
answers
317
views
Reference request: Derived structure on the moduli stack of Higgs bundles
I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that
It is often better to put
derived ...
- 385
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
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
4
votes
0
answers
503
views
Derived category of a fiber product
Let $X = Y \times_Z W$, where $X,Y,Z,W$ are Noetherian schemes, and consider the pullback diagram associated to $X, Y, Z, W$. We have a diagram
$$
\require{AMScd}
\begin{CD}
D(Z) @>>> D(Y)\\
@...
- 1,325
4
votes
1
answer
676
views
Jordan–Hölder sequence for $\mu$-semi stable sheaves
Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class.
I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{...
- 1,286
18
votes
0
answers
1k
views
What is the relationship between Artin and Lurie representability?
Artin's representability theorem gives conditions for a functor from commutative rings to sets (or groupoids) to be representable by an algebraic space (stack). The conditions are largely expressed ...
- 8,723
1
vote
0
answers
148
views
Perfect complexes on affine schemes
I'm reading a paper on algebraic stacks and in some part is stated the following:
Let $X$ be an algebraic stack and let
$P\in D_{qc}(X)$ be a perfect complex. Then, for every $x\in |X|$, there ...
- 41
1
vote
1
answer
201
views
Computing units in a dg-algebra
Let $\mathbb{G}_m= Spec(k[z,z^{-1}])$ be the usual multiplicative group over a field $k$ viewed as a discrete commutative dg-algebra, and let $A$ be some arbitrary commutative dg-algebra concentrated ...
- 3,019
2
votes
0
answers
213
views
Do dg schemes have derived points?
Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...
- 7,972
11
votes
0
answers
607
views
The virtual fundamental class as derived intersection
Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
- 7,972
11
votes
1
answer
2k
views
Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 28.8k
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
1
vote
1
answer
398
views
Is the pushforward of a closed immersion of spectral Deligne-Mumford stacks conservative?
Let $ X \hookrightarrow Y$ be a closed immersion of (connective) spectral Deligne-Mumford stacks, is $ i_* : Qcoh(X) \rightarrow Qcoh(Y)$ conservative? Somehow I couldn't find the statement in SAG...
- 595
2
votes
1
answer
400
views
Confusion about DAG terminology
This question refers to higher and derived algebraic geometry as developed by Toen-Vezzosi, not by Lurie. I have seen two expository documents by Toen.
In the first text, there is a definition:
A ...
user141006
3
votes
1
answer
304
views
Local quotient covers for derived Deligne-Mumford geometric stacks of Toen-Vezzosi
Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X_{\alpha} \rightarrow X$, such that for each $\alpha$, there ...
- 31
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
4
votes
0
answers
477
views
DAG applied to homotopy theory: how to reach research level?
It is my dream to do research on applications of spectral algebraic geometry in homotopy theory one day. Specifically, giving a more uniform treatment for the results proved via scary computations (of ...
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
3
votes
0
answers
336
views
DAG vs Classical algebraic geometry
I have a very vague question, but also a fairly specific wish. Namely, I'm wondering what the similarities and differences are between the theory of ordinary schemes on the one hand, and the theory of ...
- 213
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
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 ...
- 121
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
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
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