MODULES THAT ARE FINITE BIRATIONAL ALGEBRAS

摘要

Let A be a commutative ring and let B be a faithful A-module with a distinguished element e ∈ B. It would be nice to understand in terms of the theory of A-modules whether B supports the structure of an A-algebra with identity element e. In general there is of course nothing unique about such an algebra structure. But there is at most one such structure if B is a finite birational A-module in the sense that there is an element d ∈ A, which is a nonzerodivisor on B, such that d B is contained in Ae is contained in B. In this case, indeed, the algebra structure of B is determined by the fact that it is a subalgebra of B[d~(-1)] = A[d~(-1)].