All Questions
Tagged with derived-algebraic-geometry ag.algebraic-geometry
159 questions
5
votes
1
answer
301
views
Operations on filtered / graded vector spaces via $\mathbb{A}^1 / \mathbb{G}_m$
A well known fact (e.g. Moulinos - The geometry of filtrations) is that one can describe the category of filtered vector spaces as $\operatorname{FilVect} \simeq \operatorname{QCoh}(\mathbb{A}^1/\...
- 834
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
3
votes
1
answer
294
views
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
389
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,...
3
votes
0
answers
179
views
Étale morphisms of derived schemes and stacks
Conventions: In the below,
unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali.
an algebraic stack will be a stack $\mathscr{S}$ over a base ...
- 1,349
4
votes
0
answers
149
views
'Naive cotangent complex' as 1-truncation of cotangent complex
In the stacks project, there is a 'naive version' of cotangent complex $NL_{S/R}$ for a ring morphism $R\rightarrow S$, given by the chan complex $(I/I^{2}\rightarrow \Omega_{R[S]/R}\otimes_{R[S]}S)$ ...
- 618
1
vote
0
answers
62
views
Geometric stability conditions on calabi-yau's fibred over Fano always identical to geometric stability conditions on Fano
I apologize in advance for the long title. This question is motivated primarily by [2], with the explicit example of $\mathbb{P}^2$ and $\omega_{\mathbb{P}^2}$ computed in [3] and [1], respectively.
...
- 317
2
votes
1
answer
304
views
The exact sequence for a derived zero locus
For a locally free sheaf of rank one $L$ on a derived scheme and a morphism $s:L\rightarrow O_{X}$, we consider the derived zero locus of $s$ defined by the following derived fiber product
$$\require{...
- 618
3
votes
0
answers
196
views
Divided power structure on $E_\infty$-algebras?
Let $A$ be a simplicial commutative ring, then it is known that the ideal of elements of degree $\ge1$ in the associated CDGA has a "DG divided power structure," which induces a divided ...
- 371
5
votes
0
answers
223
views
What do we know about effective epimorphisms of derived affine schemes/manifolds?
By default, all terms are understood in the infinity sense (“category” means “(∞,1)-category”, etc.)
Recall that the morphism $X \to Y$ is an effective epimorphism if the Čech diagram
$$ ... \to X \...
- 6,962
2
votes
1
answer
199
views
Does every Artin $\infty$-stack have a formally étale cover by semi-free non-connective CDGAs?
The title essentially says it all. Feel free to assume as many finite generation conditions as you want. For example, I'm pretty sure the Chevalley-Eilenberg complex $\simeq \wedge^\bullet \mathfrak g^...
- 563
7
votes
1
answer
629
views
Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,349
3
votes
1
answer
183
views
Pushforward of exceptional vector bundle is spherical for local P^2
I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical ...
- 317
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
3
votes
0
answers
151
views
Is a derived scheme determined by classical + formal points?
Say we have a derived scheme over an algebraically closed field $X/k$, viewed as a functor $X : \operatorname{Aff}_k^{\operatorname{op}} \to \infty\operatorname{-Grpd}$ and we know its formal ...
- 834
2
votes
0
answers
157
views
Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology
Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...
- 889
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 ...
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
3
votes
0
answers
187
views
Does "derived" make anything constant in non-flat families?
This is an extremely basic (and surely amateurish) question that might be about derived geometry.
In usual algebraic geometry, if we have a flat projective morphism $f:X \to S$ with $S$ integral, and ...
- 318
3
votes
0
answers
308
views
Algebraic Fukaya categories and mirror symmetry
Dominic Joyce and collaborators have outlined a programme to construct algebraic Fukaya categories on an algebraic symplectic manifold (“Fukaya categories” of complex Lagrangians in complex symplectic ...
- 163
5
votes
1
answer
445
views
What is the k-linear structure on the derived infinity category of quasi-coherent sheaves?
Let $f : X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} \mathrm{Spec} (k)$ be morphisms of schemes (feel free to add any hypothesis necessary). Let $\mathrm{QCoh}(Y)$ denote the derived (stable)...
- 53
1
vote
0
answers
109
views
Computing Grothendieck group of (unnodal) Enriques surface
Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two ...
- 317
3
votes
1
answer
335
views
resolution property and perfect stacks
Recall that for a scheme $X$, it has the resolution property if every coherent sheaf $E$ on $X$, is the quotient of a finite locally free $\mathcal{O}_X$-module.
On the other hand, Ben-Zvi-Nadler-...
- 3,472
1
vote
1
answer
256
views
Examples when algebraic 1-stack = derived enhancement?
Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of https:/...
- 163
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
3
votes
0
answers
451
views
Infinite dimensional dg-manifolds
In Def 2.5.1 in " Derived Quot schemes" by Ciocan-Fontanine and Kapranov, we can find the notion of dg-manifolds.
In detail, let $X$ be a dg-scheme over $k$ ($k$ : an algebraic closed field ...
- 889
3
votes
1
answer
416
views
Should we expect Kuznetsov component to be independent of exceptional collection
As explained in the comments of this answer, given a smooth Fano 3-fold of index 1 and genus $g \geq 6$, we have two semiorthogonal decompositions $$\langle \text{Ku}(X), \mathcal{E}, \mathcal{O}_X\...
- 317
4
votes
0
answers
271
views
Confusion about definition of crystals
In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...
- 426
8
votes
1
answer
324
views
$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$
In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...
- 1,349
2
votes
0
answers
123
views
Moduli stack of doubly periodic complexes?
Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
- 5,711
1
vote
0
answers
275
views
Fourier-Mukai transform is the derived functor
In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me.
Let $X$ be an abelian variety over an ...
- 615
0
votes
0
answers
170
views
Cone of morphism induced by Serre duality
For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...
- 317
4
votes
1
answer
532
views
Two notions of singular support?
Arinkin-Gaitsgory have defined the notion of singular support for any quasismooth $Y$
$$\text{SS}(\mathcal{F})\ \subseteq\ \text{Sing}(Y)$$
and $\mathcal{F}$ any ind-coherent sheaf, where $\text{Sing}(...
- 5,711
3
votes
1
answer
422
views
Derived $\ell$-completion of $\mathbf{Q}_\ell$ sheaf?
I came across some notation that I’m having trouble understanding in Hansen-Scholze’s preprint ‘Relative Perversity.’ In the last paragraph of Proposition 3.4 there is the notation
$A\widehat{\otimes^{...
- 1,217
2
votes
1
answer
131
views
Right adjoint of subcollection of semi-orthogonal decomposition
Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case ...
- 317
2
votes
0
answers
354
views
Higher-order HKR theorems?
Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
- 64k
3
votes
1
answer
176
views
Left adjoint for nested admissible categories
This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}_X \...
- 317
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
4
votes
1
answer
499
views
Hard Lefschetz theorem in intersection cohomology
In [1,2] the authors proved the Hard Lefschetz theorem in intersection cohomology:
Let $Z$ be a complex projective variety of pure complex dimension $d$, with $\xi\in H^2(Z,\mathbb{Q})$ the first ...
- 828
2
votes
0
answers
189
views
Is the homotopy limit of derived schemes along affine maps a derived scheme?
The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd ...
- 301
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
4
votes
0
answers
352
views
What does the cotangent complex tell you when it takes animated inputs?
These two links: What is the cotangent complex good for? and Intuition about the cotangent complex? are quite helpful in giving intution for the cotangent complex in terms of deformations but I don't ...
- 301
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
3
votes
0
answers
398
views
Applications derived algebraic geometry in Morse theory
Have derived algebraic geometry been used to understand the topology of complex varieties? For example are there any applications in Morse theory?
The reason I am asking this is two fold. First one is ...
- 5,901
2
votes
0
answers
235
views
Formally étale maps of animated $k$-algebras
In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
- 301
3
votes
0
answers
213
views
Derived Chow varieties
I recently encountered the "Hidden Smoothness Principle" envisioned by Deligne, Drinfeld, Beilinson, Kontsevich that singularities occurring in certain moduli spaces is the consequence of ...
- 5,901
2
votes
0
answers
482
views
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
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
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
2
votes
1
answer
213
views
Derived quot schemes and the derived linearity locus
I am studying derived quot schemes in the paper Ciocan-Fontanine and Kapranov, “Derived Quot schemes” .
On page 36 ~ 37, the derived linearity locus is defined.
Let $S$ be a $\mathbb{Z}_-$-graded dg-...
- 479