Attempts on SGA for non-commutative rings

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.