All Questions
Tagged with derived-algebraic-geometry reference-request
24 questions
3
votes
1
answer
296
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Derived Koszul complex
Let $X$ be a projective variety over $\mathbb{C}$ and $V$ be a vector bundle over $X$. Let $\pi: V\to X$ be the natural projection.
Let $i: X\to V$ be the zero section map. Let $V^\vee$ be the dual ...
- 653
3
votes
0
answers
390
views
Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective
A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.
Primary question:
Have there been any recent developments/advances on the above question? If not,...
6
votes
1
answer
274
views
Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
- 6,962
6
votes
0
answers
170
views
New investigations on Homotopical Algebraic Contexts
Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract ...
- 177
4
votes
2
answers
413
views
“Geometric” vs Homotopical completion
There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.
The first one is the “homotopical” (or maybe it should be called ...
- 1,374
2
votes
0
answers
91
views
Formal neighborhood of isolated singularity via DAG
I work over a field of characteristic $0$, denoted $k$. Let $f:\mathbf{A}^{d+1}\rightarrow\mathbf{A}^{1}$ have an isolated singularity at $0$, and let $\widehat{Z}$ denote the formal neighborhood of $...
- 121
3
votes
0
answers
220
views
Formal loop space in algebraic geometry
Does anyone have a reference or an explanation about the relationship between the formal loop space defined for affine schemes via $LX\left(R\right) = X\left(R\left(\left(t\right)\right)\right)$ (or ...
- 834
10
votes
0
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420
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What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?
The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...
- 1,969
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
2
votes
1
answer
390
views
The stack of equivariant local system is quasi-smooth
Let $G$ be a (connected ?) algebraic group and $X$ a smooth, projective, and connected algebraic curve, both over an algebraically closed field $k$ of characteristic $0$.
My questions are then as ...
- 1,516
8
votes
1
answer
578
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D-modules as ind-coherent sheaves over positive characteristics?
There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....
- 1,516
2
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0
answers
482
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About derived divided power envelope
Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree.
In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
- 121
5
votes
1
answer
836
views
Categorical-geometric Langlands for tori
Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?...
- 1,516
3
votes
0
answers
417
views
Derived geometry and theoretical physics
Is there any link between derived geometry and theoretical physics?
for example with particle physics or quantum mechanics?
Specifically something that included the obstruction bundle.
If possible I ...
- 272
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
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 ...
- 83
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
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
2
votes
0
answers
157
views
Why is the stabilization of augmented $\mathbb{E}_\infty$-algebras equivalent to $k$-module spectra?
(I have already asked this on Math.SE, but it didn't draw much attention there, so I am reposting it here.)
Example 1.1.4 of Jacob Lurie's DAGX says that the stabilization $\operatorname{Stab}((\...
- 95
4
votes
1
answer
168
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? ...
- 1,180
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 ...
- 1,593
54
votes
2
answers
4k
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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
0
votes
0
answers
173
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 ...
- 10.8k
6
votes
2
answers
686
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 ...
- 54.6k