Questions tagged [derived-algebraic-geometry]
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276 questions
12
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0
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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
2
votes
0
answers
441
views
About an argument in absolute prismatic cohomology
In Bhatt-Lurie Absolute prismatic cohomology, proof of Corollary 4.1.15, it asserts that extension of scalars along the quotient map is conservative and preserves small limits:
I think the ...
- 1,906
10
votes
0
answers
420
views
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
3
votes
0
answers
451
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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
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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 $$\langle \text{Ku}(X), \mathcal{E}, \mathcal{O}_X\...
- 317
4
votes
0
answers
271
views
Confusion about definition of crystals
In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...
- 426
4
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0
answers
202
views
Cohomological methods in intersection theory and derived categories
Are there any enumerative questions akin to: “What is the number of planes containing a given line tangent to a given cubic surface in $\mathbb{P}^3$” that we can answer using derived categories? I've ...
- 511
8
votes
1
answer
324
views
$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$
In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...
- 1,349
2
votes
0
answers
123
views
Moduli stack of doubly periodic complexes?
Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
- 5,711
1
vote
0
answers
275
views
Fourier-Mukai transform is the derived functor
In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me.
Let $X$ be an abelian variety over an ...
- 615
0
votes
0
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170
views
Cone of morphism induced by Serre duality
For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
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)$$
and $\mathcal{F}$ any ind-coherent sheaf, where $\text{Sing}(...
- 5,711
3
votes
1
answer
422
views
Derived $\ell$-completion of $\mathbf{Q}_\ell$ sheaf?
I came across some notation that I’m having trouble understanding in Hansen-Scholze’s preprint ‘Relative Perversity.’ In the last paragraph of Proposition 3.4 there is the notation
$A\widehat{\otimes^{...
- 1,217
2
votes
1
answer
131
views
Right adjoint of subcollection of semi-orthogonal decomposition
Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case ...
- 317
2
votes
0
answers
354
views
Higher-order HKR theorems?
Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
- 64k
3
votes
0
answers
173
views
(Commutative) Algebras in $\mathsf{dgCat}_k$
Suppose $k$ is a fixed commutative ring, and let $\mathsf{dgCat}_k$ denote the category of $k$-linear dg-categories. We will equip $\mathsf{dgCat}_k$ with the Morita model structure (see theorem 2.27 ...
- 1,349
3
votes
1
answer
176
views
Left adjoint for nested admissible categories
This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}_X \...
- 317
7
votes
1
answer
498
views
Basic example of derived descent
I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example.
Given a ...
- 258
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
2
votes
1
answer
173
views
Monoidal colimit-preserving functor from spaces to $A$-modules
I am reading Lurie's Elliptic Cohomology II and it claims (Section 4.1.3) that for an $\mathbb{E}_\infty$-ring $A$ "there is an essentially unique symmetric monoidal functor $\mathcal{S} \to \...
3
votes
0
answers
90
views
Derived prestacks regarded as functors into spectra
If $k$ is a field (probably of characteristic zero), the usual definition of a derived prestack is a functor $ X \colon {\operatorname{CDGA}}_{k}^{\le 0} \to \operatorname{Spaces} $ from (graded) ...
- 295
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
6
votes
0
answers
219
views
Truncated Sphere Spectra and their Modules
I'm trying to use truncations $\tau_{\leq n}S$ of the sphere spectrum to ``interpolate'' between $\DeclareMathOperator{\H}{H} \H\mathbb{Z}$ and $S$, and I am struggling to find references for ...
- 455
1
vote
0
answers
209
views
Computing the cotangent complex of morphisms of perfect complexes
In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...
- 595
4
votes
0
answers
259
views
Cotangent complex of a formal thickening
Let $R$ be an (animated) commutative ring, with cotangent complex $L_R$ and let $\mathcal{C}(R) = \mathcal{D}(R)_{\Sigma^{-1}L_R/}$ be the category of nice square zero extensions of $R$. A typical ...
- 858
3
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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
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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
6
votes
1
answer
394
views
2-shifted Poisson bracket on Lie algebra cohomology
Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$...
- 819
8
votes
1
answer
578
views
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
5
votes
1
answer
298
views
Interpolating between the flat and smooth affine lines in spectral algebraic geometry
Consider the following construction (which came up recently in a question about "spectral exterior algebras"):
Pick a ring spectrum $R$ and consider the $\infty$-category $\mathsf{Mod}_R$ ...
- 11.8k
3
votes
0
answers
213
views
Base-change theorems for stable $\infty$-categories
Omitting some technicalities, the base-change theorem for quasicoherent sheaves says that if we have the following diagram of (derived) schemes
$\require{AMScd}$
\begin{CD}
X \times_S Y @>\pi_2>&...
- 2,356
8
votes
0
answers
751
views
What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
- 191
9
votes
1
answer
770
views
Coherent objects in a hypercomplete $\infty$-topos
In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\...
- 898
2
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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
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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
4
votes
1
answer
493
views
Intuition for points of the moduli of objects for a dg-category
Problem summary: I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in this paper by Brav and Dyckerhoff) actually looks like/how to give an alternative ...
- 185
21
votes
1
answer
2k
views
Why does elliptic cohomology fail to be unique up to contractible choice?
It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
- 423
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
1
vote
0
answers
137
views
A question about relative deformations of the structure sheaf of the diagonal
Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. Let us consider an associative
algebra $${p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \in \text{Alg}_\...
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
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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
8
votes
0
answers
482
views
Relationship between different definitions of the Hochschild homology
Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to ...
- 1,349
4
votes
0
answers
153
views
Preorientation of additive formal group
In "A Survey of Elliptic Cohomology", Section 3.2, Lurie asserts that the preorientations of the additive formal group $\widehat{\mathbf G}_a$ over $\mathbf Z$ are classified by the $\mathbb ...
- 2,011
3
votes
0
answers
365
views
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