Questions tagged [derived-algebraic-geometry]

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38 votes
4 answers
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What is a simplicial commutative ring from the point of view of homotopy theory?

Let $k$ be a field. There are two natural categories to consider: The category of simplicial commutative $k$-algebras. The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of $...
Akhil Mathew's user avatar
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68 votes
3 answers
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Derived algebraic geometry: how to reach research level math?

I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different. My goal is to study derived algebraic geometry, where derived ...
40 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/...
Peter Bonart's user avatar
34 votes
2 answers
2k views

What is the relationship between connective and nonconnective derived algebraic geometry?

"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", ...
Tim Campion's user avatar
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21 votes
1 answer
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Motivation and potential applications of spectral algebraic geometry

Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry. Now I'm curious what future is there for spectral algebraic ...
JDou9's user avatar
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4 votes
1 answer
1k views

Pushout schemes/stacks

I would like to know in what generality do we have pushouts for schemes/stacks/derived stacks. More precisely, let $f : X \rightarrow Y$ be a proper flat surjective morphism of schemes of finite type ...
Libli's user avatar
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42 votes
2 answers
2k views

What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
Anton Fetisov's user avatar
41 votes
5 answers
5k 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 ...
Tim Campion's user avatar
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23 votes
2 answers
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Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
Derived geometry's user avatar
20 votes
1 answer
793 views

Deligne's letter to Millson

The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan ...
Diane's user avatar
  • 201
13 votes
1 answer
928 views

Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks. For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
Klim Puhov's user avatar
12 votes
0 answers
834 views

$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
user40276's user avatar
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11 votes
1 answer
627 views

Thom Spectra and Hopf-Galois Extensions of Ring Spectra

So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...
Jonathan Beardsley's user avatar
10 votes
1 answer
816 views

$\infty$-categorical understanding of Bridgeland stability?

On triangulated categories we have a notion of Bridgeland stability conditions. Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
dumb's user avatar
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10 votes
1 answer
710 views

Why does passage to DG categories cure non-locality of derived categories?

In the famous book 'Residues and duality', the author notes that one of the principal difficulties in constructing the exceptional inverse image functor $f^{!}$ is that the derived category of ...
user avatar
9 votes
2 answers
1k views

Derived topological stacks?

I apologize for the vagueness of the following. Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory ...
zzz's user avatar
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8 votes
0 answers
423 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 ...
Stahl's user avatar
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8 votes
1 answer
495 views

Topological Hochschild homology and Hochschild homology of dg algebras

Topological Hochschild homology is a generalization of Hochschild homology from rings to $E_\infty$-ring spectra. On the other hand, there is a natural way to extend the notion of Hochschild homology ...
Harrison Chen's user avatar
8 votes
1 answer
923 views

Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$

All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
Mohan Swaminathan's user avatar
5 votes
1 answer
238 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 $\mathsf{Mod}_R$ ...
Emily's user avatar
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4 votes
1 answer
340 views

Does formation of the derived $\infty$-category preserve pushouts?

Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an ...
Stahl's user avatar
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4 votes
0 answers
403 views

Derived category of a fiber product

Let $X = Y \times_Z W$, where $X,Y,Z,W$ are Noetherian schemes, and consider the pullback diagram associated to $X, Y, Z, W$. We have a diagram $$ \require{AMScd} \begin{CD} D(Z) @>>> D(Y)\\ @...
Federico Barbacovi's user avatar