# Attempts on SGA for non-commutative rings

## TL;DR

This paper introduces a framework for non-commutative schemes based on prime ideals, explores their cohomology theories, and establishes finiteness and comparison theorems, extending classical algebraic geometry concepts to non-commutative settings.

## Contribution

It defines non-commutative schemes via prime ideals, develops their cohomology theories, and proves key finiteness and comparison theorems for these schemes.

## Key findings

- Finiteness theorem for higher direct images in étale cohomology.
- Comparison theorem between étale and Betti cohomology.
- Expression of L-functions via étale cohomology with compact supports.

## Abstract

We define non-commutative schemes by using prime ideals of non-commutative rings, and discuss the \'etale cohomology, the Betti cohomology, and the fundamental groups of non-commutative schemes. For non-commutative schemes which are finite over centers, we prove the finiteness theorem for the higher direct images in \'etale cohomology theory, and the comparison theorem between \'etale cohomology and Betti cohomology. In Appendix, for non-commutative schemes over finite fields which are finite over centers and satisfy a certain condition, $L$-functions are expressed by using \'etale cohomology with compact supports.