# John Rognes: Algebraic K-theory of elliptic cohomology

**Time: **
Tue 2022-04-12 14.15 - 16.00

**Location: **
Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

**Video link: **
Meeting ID: 921 756 1880

**Participating: **
John Rognes (University of Oslo)

**Abstract:** Joint work with G. Angelini-Knoll, Ch. Ausoni, D.L. Culver and E. Höning. We calculate the mod \((p, v_1, v_2)\) homotopy \(V(2)_* TC(BP\langle 2\rangle)\) of the topological cyclic homology of the truncated Brown–Peterson spectrum \(BP\langle 2\rangle\), at all primes \(p\ge7\), and show that it is a finitely generated and free \(\mathbb{F}_p[v_3]\)-module on \(12p+4\) generators in explicit degrees within the range \(-1 \le * \le 2p^3+2p^2+2p-3\). At these primes \(BP\langle 2\rangle\) is a form of elliptic cohomology, and our result also determines the mod \((p, v_1, v_2)\) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure \(v_2\)-periodicity to pure \(v_3\)-periodicity in a precise quantitative manner.