TY - JOUR

T1 - Dirac geometry II

T2 - coherent cohomology

AU - Hesselholt, Lars

AU - Pstrągowski, Piotr

N1 - Publisher Copyright: © The Author(s), 2024.

PY - 2024

Y1 - 2024

N2 - Dirac rings are commutative algebras in the symmetric monoidal category of Z-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger ∞-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and 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 Fp in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.

AB - Dirac rings are commutative algebras in the symmetric monoidal category of Z-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger ∞-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and 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 Fp in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.

U2 - 10.1017/fms.2024.2

DO - 10.1017/fms.2024.2

M3 - Journal article

AN - SCOPUS:85186888728

VL - 12

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

SN - 2050-5094

M1 - e27

ER -