Search

3 results

V - Betti and de Rham Cohomology
Arthur Ogus, University of California, Berkeley
Book:

Lectures on Logarithmic Algebraic Geometry

Published online:

02 November 2018

Print publication:

08 November 2018, pp 442-528
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

Chapter V is devoted to the topology of log schemes. It begins with adiscussion of log analytic spaces and their “Betti realizations,” as invented by Kato and Nakayama, which reveal the geometric effect of adding a log structure to a log analytic space.TheBetti realization of a log analytic space is home to the “logarithmic exponential map” and carries a sheaf of rings in which can be found solutions to differential equations with log poles and unipotent monodromy.Then the de Rham complex of a log scheme is discussed, along with some of its canonical filtrations which come from the combinatorics of the underlying log structure. After a discussion of the Cartier operator, the chapter concludes with some comparison and finiteness theorems for analytic, algebraic, and Betti cohomologies.

Poincaré Lemma on Quaternion-like Heisenberg Groups
Der-Chen Chang, Nanping Yang, Hsi-Chun Wu
Journal:

Canadian Mathematical Bulletin / Volume 61 / Issue 3 / 01 September 2018

Published online by Cambridge University Press:

20 November 2018, pp. 495-508

Print publication:

01 September 2018
- Article
- - You have access
- PDF
- Export citation
For smooth functions ${{a}_{1}}\,,\,{{a}_{2}}\,,\,{{a}_{3}}\,,\,{{a}_{4}}\,$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system ${{X}_{1}}f\,=\,{{a}_{1}},\,{{X}_{2}}f=\,{{a}_{2}},\,{{X}_{3}}f\,=\,{{a}_{3}},\,\text{and}\,{{X}_{4}}f\,=\,{{a}_{4}}$. In addition, a formula for the solution function $f$ is deduced, assuming solvability of the system.

Cohomology of Crystalline Smooth Sheaves
Wiesł Nizioł
Journal:

Compositio Mathematica / Volume 129 / Issue 2 / November 2001

Published online by Cambridge University Press:

04 December 2007, pp. 123-147

Print publication:

November 2001
- Article
- - You have access
- PDF
- Export citation
We define and study a candidate for ‘arithmetic’ cohomology theory with values in crystalline p-adic smooth sheaves. We show that it injects into arithmetic étale cohomology and prove a duality result.