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Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ …

The $p$-adic completion of $K( \mathbb{F}_p)$ is known (by Quillen's calculation) to be $H \mathbb{Z}_p$; in particular, $K(\mathbb{F}_p)$ is acyclic with respect to all Morava $K$-theories $K(n), 0 < …

Consider a Moore spectrum $S^n/p$. Has the $K$-theory of the extended powers of $S^n/p$ been computed? For some context: equivalently, I'd like to have the free $E_\infty$-algebra over $KU$ on the $ …

Let $G$ be a group and let $V$ be a complex projective representation of $G$, so that $G$ acts on the projectivization $\mathbb{P}(V)$. Is there any way to calculate the $G$-equivariant complex $K$-th …

Let $E \to F$ be a $G$-Galois extension of fields. What is the simplest example where the natural map $K(E) \to K(F)^{hG}$ is not an equivalence on connective covers (i.e., where finite Galois desce …

Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with …

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with …