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70 votes
3 answers
22k views

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 ...
41 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
21 votes
1 answer
3k views

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
  • 433
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
  • 7,320
44 votes
5 answers
6k 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
13 votes
1 answer
961 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
10 votes
1 answer
883 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
  • 103
8 votes
1 answer
980 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
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 $\infty$-...
Stahl's user avatar
  • 1,349
4 votes
0 answers
503 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