Multiplicities and degree functions in local rings via intersection products

We prove a theorem on the intersection theory over a Noetherian local ring , which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings , using the theory of rational equivalence developed by Thorup, and the Snapper–Mumford–Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self‐contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert–Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over 2‐dimensional excellent local rings in terms of our intersection product.