We consider derived invariants of varieties in positive characteristic arising from topological Hochschild homology. Using theory developed by Ekedahl and Illusie-Raynaud their study the slope spectral sequence, we examine behavior under equivalences various $p$-adic quantities related to Hodge-Witt crystalline cohomology groups, including numbers, domino numbers. As a consequence, obtain restr...

Given a scheme in characteristic p together with a lifting modulo p2, we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We use this functor to generalize the decomposition theorem of Deligne-Illusie to the case of de Rham cohomology with coefficients.

These are lecture notes for a 1–semester undergraduate course (in mathematics, physics, engineering, chemistry or biology) on the De Rham–Hodge theory. This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green’s theorem...

In elementary mathematics, the simplest variety – i.e., geometric object defined by polynomial equations – that one encounters is a line, i.e., the set of points in the Euclidean plane R defined by an equation of the form aX + bY = c (where a, b, c ∈ R). After translation, rotation, and dilation, such an equation may be written in the form X = 0. In this case, the variety in question passes thr...