In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of THH (resp. logarithmic THH) of complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.

In this paper we compute the topological Hochschild homology of quotients of discrete valuation rings (DVRs). Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of $\operatorname {THH}$ (respectively, logarithmic $\operatorname {THH}$ ) of complete DVRs originally due to Lindenstrauss and Madsen (respectively, Hesselholt and Madsen).

We present Bianchi's proof on the classification of real (and complex) 3-dimensional Lie algebras in a coordinate free version from a strictly representation theoretic point of view. Nearby we also compute the automorphism groups and from this the orbit dimensions of the corresponding orbits in the algebraic variety X ⊆ Λ 2 V * ⊗ V describing all Lie brackets on a fixed vector space V of dimension 3. Moreover we clarify which orbits lie in the closure of a given orbit and therefore the topology on the orbit space X/G with G = Aut(V ).

For a not-necessarily commutative ring R we define an abelian group W (R; M ) of Witt vectors with coefficients in an R-bimodule M . These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is thatFor an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element χ f ∈ W (R). This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment f ↦ χ f induces an isomorphism between a suitable completion of cyclic K-theory K cyc 0 (R) and W (R).Finally, Kaledin defines in [Kal18a] (see also [Kal18b]) abelian groups W n (V ) of 'polynomial Witt vectors' for a vector space V over a perfect field k of characteristic p. We will also show in the forthcoming paper [DKNP] that his group W n (V ) is isomorphic to our group W p,n (k; V ) of truncated, p-typical Witt vectors with coefficients.