# Questions tagged [derived-algebraic-geometry]

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266
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### 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.
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### $\varinjlim_{k}\Omega^{k}\circ \Sigma^{k}$ as 1-excisive approximation of the identity functor

In chapter 6 of HA by Lurie, for any functor $F:C\rightarrow D$ between 'nice' categories (like differentiable infinity categories), there is an $n$ excisive approximation to this functor behaving ...

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### Square zero extension in the derived setting

Here we take the infinity category of simplicial ring $SCRing=Fun^{\prod}(Poly^{op},Spc)$ and follow the construction 25.3.1.1 in DAG by Lurie, where we extend the construction of square zero ...

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### Infinite suspension is cotangent complex

In Higher Algebra by Lurie, we define the absolute cotangent complex $L_{A}$ through the composition $C\stackrel{\triangle}{\longrightarrow} Fun(\triangle^{1},C)\stackrel{F}{\longrightarrow}T_{C}$ ...

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### The spectrum object in the $\infty$ category $CAlg_{R}$

Consider the infinity category of simplicial rings $CAlg=Fun^{\prod}(Poly^{op},Spc)$, and the under-category over $R$: $CAlg_{A/}$ is equivalent to $CAlg_{A}=Fun^{\prod}(Poly^{op}_{R},Spc)$ by 25.1.4....

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### The space of virtual Cartier divisors on a classical scheme over a closed immersion is discrete

I am currently reading the paper Virtual Cartier divisors and blow-ups where the virtual Cartier divisor on an $X$ scheme $S$ over a quasi-smooth closed immersion $Z\rightarrow X$ is defined to be the ...

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### Derived b-calculus and logarithmic tangent sheaves

Melrose's b-calculus provides a powerful framework for analyzing elliptic operators on manifolds with boundary. In the context of log geometry, log smooth manifolds offer a natural generalization of ...

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### Construction of Weil restriction in the derived setting

I am currently reading 19.1,2 in SAG by Lurie about Weil restrictions. In the classical case, Weil restriction is the right adjoint to the pullback functor along some morphism $f:X\rightarrow Y$, and ...

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### Existence of Kan extension for the functor with codomain a complete infinity category

I am currently reading this paper on derived blow up, in definition 2.4.1, I am faced with such situation:
if we denote the infinity category of simplicial ring as $Alg$ and the 1 category of ...

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### Does derived tensor product preserve fiber sequence？

In lemma 3.1.5 of this paper I read, there is a fiber sequence of the underlying spaces of simplicial commutative rings $A\stackrel{f}{\longrightarrow} A\rightarrow A/\!\!/f$. Here we define the "...

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### 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{...

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### Classical schemes as derived schemes are discrete valued

$\newcommand\Spc{\mathrm{Spc}}\newcommand\SCRing{\mathrm{SCRing}}\DeclareMathOperator\Map{Map}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hom{Hom}\...

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### What is the Isomorphism subspace of the mapping space in an infinity category

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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^...

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### Animated rings and $\mathbb E_{\infty}$-rings

Let $R$ be a discrete commutative ring. The Proposistion 25.1.2.4 in Lurie's Spectral Algebraic Geometry says that the natural functor from the $\infty$-category of animated $R$-algebras to the $\...

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### What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?

Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\...

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### 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\...

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### Derived $\infty$-category of quasi-coherent sheaves on schemes

Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...

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### 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 ...

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### 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 ...

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### The criterion for dimensional conjecture for universal Galois deformation rings

I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...

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### “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 ...

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### Quasicompact quasiaffine classical schemes are nonconnectively-affine

In this answer to What is the relationship between connective and nonconnective derived algebraic geometry? I learned that any quasicompact open subscheme of an affine scheme is affine in the sense of ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Derived flat bundles

I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### Dualizing sheaf for classifying stack and duality

For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...

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### 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)...

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### 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 ...

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### 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-...

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### 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:/...

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### 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 ...

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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{...

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### 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 ...

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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 ...

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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 ...

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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\...

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### 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 ...

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### 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 ...

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### $\infty$-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 $\infty$-...

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### 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 ...

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### 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 ...