Codimension, multiplicity and integral extensions
Let A [subset or is implied by] B be a homogeneous inclusion of standard graded algebras with A0 = B0. To relate properties of A and B we intermediate with another algebra, the associated graded ring G = grA1B(B). We give criteria as to when the extension A [subset or is implied by] B is integral or birational in terms of the codimension of certain modules associated to G. We also introduce a series of multiplicities associated to the extension A [subset or is implied by] B. There are applications to the extension of two Rees algebras of modules and to estimating the (ordinary) multiplicity of A in terms of that of B and of related rings. Many earlier results by several authors are recovered quickly.(Received July 20 1999)
(Revised October 4 1999)
1 Partially supported by CNPq, Brazil.
2 Partially supported by the NSF.
3 Partially supported by the NSF.