Some geometric results on K-theory with Z/kZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}/k{\mathbb{Z}}$$\end{document}-coefficients

We establish some geometric results on K-theory with coefficients in Z/kZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}/k{\mathbb{Z}}$$\end{document}. The first one is a new proof of the Atiyah–Patodi–Singer mod k index theorem (Math Proc Camb Philos Soc 79:71–99, 1976) in the case of Dirac operators, i.e. in a geometric situation. The second one is a Grothendieck–Riemann–Roch theorem for Z/kZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}/k{\mathbb{Z}}$$\end{document}-K-theory.