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# 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.