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      10
      votes
      2answers
      857 views

      Results relying on higher derived algebraic geometry

      Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-...
      3
      votes
      1answer
      203 views

      Geometry of 2-arrows

      It is frequently said that one of the contributions of Grothendieck to geometry was to systematically think about the properties of morphisms, as opposed to the properties of spaces themselves (the ...
      3
      votes
      0answers
      103 views

      Descent for the cotangent complex along faithfully flat SCRs

      By Theorem 3.1 of Bhatt-Morrow-Scholze II (https://arxiv.org/pdf/1802.03261.pdf), we know that for $R$ a commutative ring, $\wedge^{i}L_{(-)/R}$ satisfies descent for faithfully flat maps $A \...
      8
      votes
      0answers
      546 views

      Visualization and new geometry in higher stacks (soft question)

      I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...
      2
      votes
      0answers
      108 views

      Compact generation of quasicoherent sheaves on mapping stack

      Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
      3
      votes
      0answers
      179 views

      Tannaka duality for $DG$ indschemes

      In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism $$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$ where $X$ and ...
      8
      votes
      1answer
      476 views

      Spectral and derived deformations of schemes

      I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is. Let $S = (X, ...
      3
      votes
      0answers
      147 views

      A naive question about representations of group stacks

      For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...
      10
      votes
      0answers
      242 views

      Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack

      I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8. ...
      14
      votes
      0answers
      1k views

      What to expect from spectral algebraic geometry

      So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
      4
      votes
      0answers
      344 views

      The lisse-etale site and derived algebraic geometry

      If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
      7
      votes
      0answers
      226 views

      Identifying and reconstructing the derived category from its auto-equivalences

      Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
      3
      votes
      0answers
      180 views

      What is a proper n-etale morphism?

      Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$? If $n=1$, I think this should be a finite etale ...
      2
      votes
      1answer
      228 views

      Is the property of being a dg generator open?

      Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}_\mathcal{C}X$-modules (...
      13
      votes
      3answers
      1k views

      Is it always possible to write a scheme as a colimit of affine schemes?

      My question is: Is it possible to write any scheme as a (1-categorical) colimit of a diagram of affines? If no, what are some examples? I ask this question because I have read that one can write any ...

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