Questions tagged [derived-algebraic-geometry]

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3 votes
1 answer
125 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
181 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
270 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 ...
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4 votes
1 answer
293 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
149 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 ...
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7 votes
1 answer
430 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
88 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
70 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) ...
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4 votes
0 answers
235 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 ...
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9 votes
1 answer
631 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}...
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6 votes
0 answers
131 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
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101 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}(\...
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4 votes
0 answers
160 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 ...
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3 votes
0 answers
275 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 ...
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2 votes
0 answers
156 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 ...
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3 votes
0 answers
145 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 ...
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6 votes
1 answer
290 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
262 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....
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4 votes
1 answer
158 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 $\mathsf{Mod}_R$ ...
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3 votes
0 answers
122 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>&...
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7 votes
0 answers
341 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 ...
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9 votes
0 answers
485 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
238 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 ...
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31 votes
1 answer
2k views

Connes'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
267 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 ...
22 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 ...
4 votes
1 answer
477 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?...
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1 vote
0 answers
84 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}_\...
26 votes
0 answers
988 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 ...
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2 votes
1 answer
174 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-...
7 votes
0 answers
288 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 ...
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4 votes
0 answers
139 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 ...
3 votes
0 answers
325 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 ...
2 votes
0 answers
153 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, ...
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1 vote
0 answers
146 views

Is the cotangent complex sensitive to truncation?

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Spec{Spec}$If $V$ is a dg vector space (in positive degrees), we can view it as a derived scheme $V = \Sym V^*$. It has (co)tangent complex $V$ (and $...
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2 votes
0 answers
185 views

Structure sheaf of derived intersection

Everything is over a field $k$ of characteristic $0$. Let $X$ and $Y$ two closed dg subschemes over a dg scheme $Z$. I am trying to understand the structure sheaf of the derived intersection of $X$ ...
4 votes
0 answers
167 views

Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one

I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
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5 votes
1 answer
248 views

Questions about $\text{Perf}(A)$ of dg algebra $A$

[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$. [...
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4 votes
0 answers
215 views

The dual abelian scheme in derived algebraic geometry

$\def\Pic{\mathcal{Pic}}\def\Gm{\mathbb{G}_m}\def\Hom{\mathop{Hom}}\def\HOM{\mathcal{Hom}}$ If $A/S$ is an abelian scheme, the fppf sheaf $\Pic^0_{A/S}$ is representable by an abelian scheme $\hat{A}$....
22 votes
0 answers
634 views

What is the status of a result of Kontsevich and Rosenberg?

In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...
14 votes
0 answers
228 views

Solution spaces of algebraic differential equations and derived geometry

We consider potentially non-linear differential equations on the formal one dimensional disc $\Delta$. Such equations are given by expressions $$P(z,f,f',f'',...)=0,$$ where $P$ is an element of the ...
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5 votes
1 answer
241 views

Is there a definition of reduced $E_\infty$ ring?

[Edit: I have completely changed the question in response to the replies given] I am curious if there is well defined notion of reduced $E_\infty$-ring. Let $CAlg$ denote the $\infty$-category of $E_\...
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3 votes
0 answers
190 views

Derived category and L-function

For abelian varieties over $\mathbb{Q}$ $\mathscr{A}$ and $\mathscr{A}'$, if derived categories $D(\mathscr{A})$ and $D(\mathscr{A}')$ are equivalent then L-functions are same $L(s,\mathscr{A})=L(s,\...
9 votes
2 answers
320 views

How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?

Suppose $A$ and $B$ are $E_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E_n$ colimit-preserving ...
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4 votes
0 answers
244 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 ...
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5 votes
0 answers
160 views

Smoothness of a variety implies homological smoothness of DbCoh

I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions. It ...
5 votes
0 answers
346 views

When is the cotangent complex perfect?

Let $X\rightarrow S$ be a proper flat morphism of schemes. When is the cotangent complex $L_{X/S}$ perfect ? It is well known, that for local complete intersections the cotangent complex is perfect, ...
36 votes
5 answers
4k views

What is the cotangent complex good for?

The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
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8 votes
0 answers
233 views

Every Spectral Deligne-Mumford stack satsifies fpqc descent?

In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too ...
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3 votes
0 answers
175 views

Dimension of derived Artin stacks and perfect complexes

I am interested in the concept of dimension of derived and $n$-Artin stacks. Take for example the definition 4.10 of From HAG to DAG: derived moduli stacks. in which they define the dimension of a ...
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