# Questions tagged [derived-algebraic-geometry]

The derived-algebraic-geometry tag has no usage guidance.

229
questions

2
votes

1
answer

231
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-...

1
vote

1
answer

181
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:/...

3
votes

0
answers

143
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 ...

8
votes

0
answers

344
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{...

2
votes

0
answers

218
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 ...

10
votes

0
answers

315
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 ...

3
votes

0
answers

396
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 ...

3
votes

1
answer

308
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\...

3
votes

0
answers

191
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 ...

4
votes

0
answers

163
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 ...

6
votes

0
answers

130
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 $\infty$-...

2
votes

0
answers

118
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 ...

1
vote

0
answers

238
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 ...

0
votes

0
answers

142
views

### Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...

4
votes

1
answer

278
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}(...

3
votes

1
answer

393
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^{...

2
votes

1
answer

102
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 ...

3
votes

0
answers

285
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 ...

3
votes

0
answers

146
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 ...

3
votes

1
answer

156
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 \...

7
votes

1
answer

333
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 ...

2
votes

1
answer

333
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 ...

4
votes

1
answer

369
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 ...

2
votes

0
answers

174
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 ...

7
votes

1
answer

531
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 ...

2
votes

1
answer

133
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

79
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) ...

4
votes

0
answers

294
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 ...

9
votes

1
answer

677
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}...

6
votes

0
answers

164
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 ...

1
vote

0
answers

149
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}(\...

4
votes

0
answers

199
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 ...

3
votes

0
answers

332
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 ...

2
votes

0
answers

189
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 ...

3
votes

0
answers

171
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 ...

6
votes

1
answer

327
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 $-2$...

8
votes

1
answer

390
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....

5
votes

1
answer

203
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 $\mathsf{Mod}_R$ ...

3
votes

0
answers

163
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>&...

7
votes

0
answers

498
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 ...

9
votes

0
answers

530
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 $\...

2
votes

0
answers

329
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 ...

38
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/...

3
votes

1
answer

335
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 ...

21
votes

1
answer

1k
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 ...

5
votes

1
answer

619
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
vote

0
answers

98
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 ...

2
votes

1
answer

203
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-...

8
votes

0
answers

359
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 ...