Achim Krause

• Krause, Achim (2025). The Picard group in equivariant homotopy theory via stable module categories. Journal of Topology. ISSN 1753-8416. 18(2). doi: 10.1112/topo.70020. Full text in Research Archive Show summary

  We develop a mechanism of “isotropy separation for compact objects” that explicitly describes an invertible G-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module category. As an application, we carry out a complete analysis of possible combinations of geometric fixed points of invertible G-spectra in the case G=A5. A further application is given by showing that the Picard groups of Sp^G and a category of derived Mackey functors agree.

• Dotto, Emanuele; Krause, Achim; Nikolaus, Thomas & Patchkoria, Irakli (2025). Witt vectors with coefficients and TR. Proceedings of the London Mathematical Society. ISSN 0024-6115. 130(5). doi: 10.1112/plms.70047. Full text in Research Archive Show summary

  We give a new construction of 𝑝-typical Witt vectors with coefficients in terms of ghost maps and show that this construction is isomorphic to the one defined in terms of formal power series from the authors’ previous paper. We show that our construction recovers Kaledin’s polynomial Witt vectors in the case of vector spaces over a perfect field of characteristic 𝑝. We then identify the components of the 𝑝-typical TR with coefficients, originally defined by Lindenstrauss and McCarthy and later reworked by the second and third authors in joint work with McCandless, with the 𝑝-typical Witt vectors with coefficients. This extends a celebrated result of Hesselholt and Hesselholt–Madsen relating the components of TR with the Witt vectors. As an application, we give an algebraic description of the components of the Hill–Hopkins–Ravenel norm for cyclic 𝑝-groups in terms of 𝑝-typical Witt vectors with coefficients.

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