Dirac geometry II: Coherent cohomology

Abstract

Whatever it is that animates anima and breathes life into higher algebra, this something leaves its trace in the structure of a Dirac ring on the homotopy groups of a commutative algebra in spectra. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we first embed this category in the larger infinity-category of Dirac stacks, which also contains formal Dirac schemes. We next develop the coherent cohomology of Dirac stacks, which amounts to a functor that to a Dirac stack X assigns a presentably symmetric monoidal stable infinity-category QCoh(X) of quasi-coherent sheaves together with a compatible t-structure. Finally, as applications of the general theory to stable homotopy theory, we use Quillen's theorem on complex cobordism and Milnor's theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to MU and F_p in terms of their functors of points. In the appendix, we develop a rudimentary theory of accessible presheaves of anima on coaccessible infinity-categories.