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Creators/Authors contains: "Pstrągowski, Piotr"

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  1. (, Journal of the European Mathematical Society)
    Inspired by the work of Hahn–Raksit–Wilson, we introduce a variant of the even filtration which is naturally defined on\mathbf{E}_{1}-rings and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn–Raksit–Wilson filtration on ring spectra of arithmetic interest, showing that various “motivic” filtrations are in fact invariants of the\mathbf{E}_{1}-structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective\mathbf{E}_{1}-rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights. 
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    Free, publicly-accessible full text available July 11, 2026
  2. ; (, Forum of Mathematics, Sigma)
    Abstract Dirac rings are commutative algebras in the symmetric monoidal category of$$\mathbb {Z}$$-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$$\infty $$-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$$\operatorname {MU}$$and$$\mathbb {F}_p$$in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves. 
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