Computations of Buchsbaum–Rim multiplicities

✍ Scribed by Elizabeth Jones

• Book ID: 104152343
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 225 KB
• Volume: 162
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00108-0

✦ Synopsis

The Buchsbaum-Rim multiplicity is a generalization of the Samuel multiplicity and is deÿned on submodules of free modules M ⊂ F of a local Noetherian ring A such that M ⊂ mF and F=M has ÿnite length. Let A = k[x; y] (x; y) be a localization of a polynomial ring over a ÿeld. When F=M is isomorphic to a quotient of monomial ideals there is a region of the (x; y)-plane which corresponds to F=M . We wish to compute Buchsbaum-Rim multiplicity using the areas of pieces of this region in a manner similar to that used to compute the Samuel multiplicity of a monomial ideal. We carry out these computations in the case where F has rank 2 and F=M ∼ = I=J where I and J are monomial ideals, with the further restriction that I is generated by two elements and J is generated by at most three elements. We ÿnd that the Buchsbaum-Rim multiplicity is at most the di erence of the Samuel multiplicities of J and I with equality often holding. When equality does not hold the Buchsbaum-Rim multiplicity is the di erence of the Samuel multiplicities minus a term that can be expressed in terms of areas.