Cohomological Kernels, Division Algebras, Quadratic Forms and the deRham Witt Complex - Aravire - Flores; Roberto (2023)
Cohomological Kernels, Division Algebras, Quadratic Forms and the de Rham Witt Complex The main Goals of this Project are the following: G1 Construct new families indecomposable division algebras using cohomological kernels of di¤erential forms (Brauer classes) of some non-normal multiquadratic extensions of a …eld and valuation theory. G2 Prove that certain division algebras of index pm whose Galois closure is of certain type, are cyclic. To this end we will be using the de Rham Witt complex. G3 Study the behavior of Hn+1 pm under certain …eld extensions. For instance …nd ker Hn+1 pm ! Hn+1 pm F pmpx1; : : : pmpxs where x1; : : : xsare p-independent in F and m 2. PROJECT RESULTS: 1. We have made advances in the theory Division Algebras constructing new fam- ilies of some non-normal multiquadratic extensions of a …eld. In particular, we published the paper Cohomological kernels of non-normal extensions in characteristic two and indecomposable division algebras of index eight, Roberto Aravire, Bill Jacob and Manuel O’Ryan; Communications in Algebra, where we computed the cohomological kernels ker Hn+1 2 F ! Hn+1 2 E of …eld extensions E=F in characteristic two where the separable part of E is a quadratic extension F() and E is quadratic or quartic and purely inseparable over F(). Explicit generators for these kernel are described in both the normal and non- normal cases. The second part of this article gives a construction of an indecomposable division algebra D of index 8 and exponent 2 in characteristic two. This paper is related with G1. 1 2. Also we have made advances the study of the behavior of Hn+1pm F under theextension F ! L = Fpmpx1; : : : ;pmpxswhere x1; : : : xs are p-independentin F and m 2. The articleCohomological Kernels of purely inseparable …eld extensions,Roberto Aravire, Bill Jacob and Manuel O’Ryan; Paci…c Journal ofAlgebra,deals about this subject. When L has exponent 1, the kernel is the sum ofthe kernels of the simple subextensions, but when L has larger ...