HOLONOMIC FILTERED MODULES IN THE CATEGORY OF MICRO-STRUCTURE SHEAVES

Since the late sixties, Various Auslander regularity conditions have been widely investigated in both commutative and non-commutative cases, [6]. J. E. Bjork studied the Auslander regularity on graded rings and positively filtered Noetherian Noetherian rings, [7]. In [7] the notion of a holonomic module over positively filtered rings has been introduced. Recently, Huishi, in his Ph. D. Thesis [12], investigate Auslander regularity condition and holonomity of graded and filtered modules over Zariski filtered rings. In this work, using the micro-structure sheaf techniques we characterize a generalized Holonomic sheaf theory. We introduce a general study of Auslander regularity on the micro-structure sheaves. We calculate the global dimension of modules over the micro- structure sheaves O . The main results are contained in Theorem (2.4), Theorem (3.6) and Theorem (3.7).

Triangular Automorphisms

This chapter focuses on triangular automorphisms, which can be analyzed by Lie techniques. Throughout the discussion K is a commutative ring containing ℚ as a subring. A formalism is introduced to analyze triangular automorphisms of such a polynomial algebra by means of their logarithms, the triangular derivations. After presenting some definitions and simple facts about filtered modules, filtered algebras, and graded algebras, the chapter considers triangular linear maps and the Lie algebra of an algebraic unitriangular group. It then describes derivations on the ring of column-finite matrices, along with iteration matrices and Riordan matrices. It also explains derivations on polynomial rings and concludes by applying triangular automorphisms to differential polynomials.

On arithmetic families of filtered -modules and crystalline representations

We consider stacks of filtered$\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize$p$-adic Galois representations of the absolute Galois group of a$p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.