I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that …

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., …

I'm interested in the following construction. Start with derived category of coherent sheaves witch equivalent to derived category of representations of some dg-algebra. Quasi-iso …

The question is already in the title. Less succinctly, let's call a map $f:X \to Y$ of schemes $L$-trivial if its cotangent complex is quasi-isomorphic to $0$. Such maps have strik …

I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Sp …

Q1: My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of this paper, they are defined as being the ind-completion of the ca …

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general: $\bullet …

Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$): $Maps(S, Loc …

In the introduction to HAGII Toen and Vezzosi write that in brave new algebraic geometry (that is, algebraic geometry over the category of symmetric spectra) Z[T] is not smooth ove …

In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the sam …

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree? …

Is there a sensible notion of blowing up in any of the available frameworks for derived algebraic geometry? I am happy to remain in the affine setting, where I think the right ques …

Let $\iota_A:A\hookrightarrow X$ and $\iota_B:B\hookrightarrow X$ be subschemes of a smooth ambient variety $X$. Then the derived tensor product $$\mathcal O_A\stackrel{L}{\otimes} …

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there intere …

I am looking for discussion in the literature that properly formalizes the heuristic idea that a BV-BRST complex is a model for the "derived critical locus of a function on an $\in …