Let $G$ be a finite $p$-group, $p \neq 2$. We construct a map from the space $J_G$, defined as the fiber of $\psi^k-1: B_G O \to B_G Spin$, to the space $(SF_G)_p$, defined as the 1-component of the zeroth space of the equivariant $p$-complete sphere spectrum. Our map produces the same splitting of the $G$-connected cover of $(SF_G)_p$ as we have described in previous work, but it also induces a natural splitting of the $p$-completions of the component groups of fixed point subspaces.

We extend Geisser and Hesselholt’s result on “bi-relative $K$-theory” from discrete rings to connective ring spectra. That is, if $\mathcal{A}$ is a homotopy cartesian n-cube of ring spectra (satisfying connectivity hypotheses), then the $(n + 1)$-cube induced by the cyclotomic trace $$K(\mathcal{A}) \to TC(\mathcal{A})$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.

The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one.

The paper considers the FPQC stacks which are associated to affine groupoid schemes. Using a formulation of a descent datum in terms of morphisms of affine groupoid schemes, explicit arguments are given which avoid appeal to the general principle of faithfully flat descent. This theory is applied to consider the notion of affine morphism.

The deformation bicomplex of a module algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module coalgebras, comodule (co)algebras, and (co)module bialgebras are also considered.