Dirac Geometry I: Commutative Algebra

Abstract

The purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded by anima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of $$\mathbb {G}_m$$ G m -equivariant algebraic geometry.

Permutation modules and cohomological singularity

“…Our main results are the following. On the one hand, we show that the space Spc(K(G)) is a colimit of Spc(K(E)) over a suitable category of elementary abelian p-groups E that appear as subquotients of G. On the other hand, when E is elementary abelian, we describe the spectrum Spc(K(E)) as a 'Dirac scheme' in the sense of Hesselholt-Pstrągowski [HP23]. Combining these results yields a description of the topological space Spc(K(G)) for all G. Let us now explain these ideas.…”

“…Let E be an elementary abelian p-group.Let O • E be the sheaf of Z-graded rings on Spc(K(E)) obtained by sheafifying U → End • K(E) | U (1). Then (Spc(K(E)), O • E) is a Dirac scheme in the sense of[HP23]. Proof.…”