Paper detail

Attempts on SGA for non-commutative rings

We define non-commutative schemes by using prime ideals of non-commutative rings, and discuss the étale 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 étale cohomology theory, and the comparison theorem between étale 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 étale cohomology with compact supports.