arXiv subjects

Andrea Lachmann

Publications and source records attributed to Andrea Lachmann.

3 recordsLinked to original sources

arXiv2025

Chow-Witt Rings of Classifying Spaces of Products of Multiplicative and Cyclic Groups

We compute the total Chow-Witt rings of the classifying space B\mu_n of the roots of unity, as well as the products BG_m x B\mu_n and B\mu_m x B\mu_n for all m,n greater than or equal to 1 based on the strategy by di Lorenzo and Mantovani (2023) for B\mu_n with n even. Moreover we compute the total I^j-cohomology and Chow-Witt rings of P^q x P^r for all q,r greater than or equal to 1 and of BG_m x BG_m.

math.AG↗

arXiv2024

Bounding the $K(p-1)$-local exotic Picard group at $p>3$

In this paper, we bound the descent filtration of the exotic Picard group $\kappa_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $\beta$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at $3n^2+1$ on the $E_{2n^2+2}$-page. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum $\mathrm{E}_n^{hN}$, where N is the normalizer in $\mathbb{G}_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,\pi_{2n}\mathrm{E}_n)$.

math.AT↗

arXiv2023

The localisation theorem for the $\mathrm{K}$-theory of stable $\infty$-categories

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\mathrm{K}$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\mathrm{K}$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and demonstrate that the cofinality theorem can be derived from the universal property alone.

math.KT↗