This paper computes the sizes (cardinalities) of many algebraic K‑theory groups attached to connective Morava K‑theory, a family of important cohomology theories in stable homotopy theory. The authors give complete size formulas in infinitely many degrees for primes p>2, and after changing coefficients to the algebraic closure of the finite field F_p they determine sizes in all degrees and show the even‑degree groups vanish.

Morava K‑theories are higher analogues of the prime fields familiar from algebra. Algebraic K‑theory is a deep invariant that encodes arithmetic and geometric information. The direct computations of integral algebraic K‑groups of ring spectra that are not ordinary rings are rare. Here the authors focus on connective Morava K‑theory k(n) and on a class of connective E1 ring spectra whose homotopy groups look like a polynomial ring F[x_{2m}] over a perfect field F of characteristic p.

Their method uses trace methods. Instead of computing K‑theory directly, they study topological cyclic homology (TC) and related theories (TC− and TP). These are easier to compute and, by a theorem of Dundas, Goodwillie and McCarthy, they approximate K‑theory closely in the contexts considered. A key technical tool is a new “orbit filtration” on TC that comes from the classical May filtration on topological Hochschild homology (THH). Combining the orbit filtration with calculations of TC for formal polynomial differential graded algebras F[x_{2m}] lets them count the size of homotopy groups of TC and then deduce information about K‑theory.

Concrete results include: for connective Morava K‑theory over an algebraically closed field F of characteristic p, all even algebraic K‑groups vanish; the odd groups K_{2k−1} are countably infinite with bounded p‑torsion for k≥1, and K_0 is countably infinite as well. For finite fields F_q they obtain precise quotient formulas relating the sizes of consecutive TC homotopy groups to Witt vector groups; these feed into cardinality formulas for K‑groups of k(n) over F_q in many degrees. As an application they compute the sizes of algebraic K‑groups of the truncated Witt vectors W( overline{F}_p )/p^n and prove these groups vanish in even degrees.