DG Indschemes

We develop the notion of indscheme in the context of derived algebraic geometry, and study the categories of quasi-coherent sheaves and ind-coherent sheaves on indschemes. The main results concern the relation between classical and derived indschemes and the notion of formal smoothness

Derived Mackey functors and CpnC_{p^n}-equivariant cohomology

We establish a novel approach to computing $G$-equivariant cohomology for a finite group $G$, and demonstrate it in the case that $G = C_{p^n}$. For any commutative ring spectrum $R$, we prove a symmetric monoidal reconstruction theorem for genuine $G$-$R$-modules, which records them in terms of their geometric fixedpoints as well as gluing maps involving their Tate cohomologies. This reconstruction theorem follows from a symmetric monoidal stratification (in the sense of \cite{AMR-strat}); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from genuine $G$-spectra to genuine $G$-$\mathbb{Z}$-modules (a.k.a. derived Mackey functors) provides a convenient intermediate category for calculating equivariant cohomology. Indeed, as $\mathbb{Z}$-linear Tate cohomology is far simpler than $\mathbb{S}$-linear Tate cohomology, the above reconstruction theorem gives a particularly simple algebraic description of genuine $G$-$\mathbb{Z}$-modules. We apply this in the case that $G = C_{p^n}$ for an odd prime $p$, computing the Picard group of genuine $G$-$\mathbb{Z}$-modules (and therefore that of genuine $G$-spectra) as well as the $RO(G)$-graded and Picard-graded $G$-equivariant cohomology of a point.Comment: improved introduction; minor notational changes and reorganizatio

DG Indschemes

We develop the notion of indscheme in the context of derived algebraic geometry, and study the categories of quasi-coherent sheaves and ind-coherent sheaves on indschemes. The main results concern the relation between classical and derived indschemes and the notion of formal smoothness.Mathematic

Stratified noncommutative geometry

We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable $\infty$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as $E_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction, which is closely related to reflection functors and Verdier duality. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra, that expresses them in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory: the adelic stratification of the $\infty$-category of spectra.Comment: Added material on: reflection functors; Verdier duality; t-structures; alignment ("noncommutative general position"); the pullback and refinement operations; central co/augmented idempotents; non-presentable stratifications; categorical fixedpoints; gluing functors for $G$ nonabelian; naive $G$-spectra. (A version with improved formatting is available at https://etale.site/writing/strat.pdf.