Abstract
We give a procedure and describe an algorithm to compute the dimension of a module over Laurent polynomial ring. We prove the cancellation theorems for projective modules and also prove the qualitative version of Laurent polynomial analogue of Horrocks' Theorem.
Highlights
Hilbert 1 introduced free resolution on iterated syzygies of a finitely generated graded module M over polynomial ring R K x1, . . . , xn
We prove the cancellation theorems for projective modules and prove the qualitative version of Laurent polynomial analogue of Horrocks’ Theorem
Eisenbud 3 proved that a commutative Noetherian local ring R is regular if and only if it has finite global dimension, in which case the global dimension coincides with the Krull dimension of R
Summary
Hilbert 1 introduced free resolution on iterated syzygies of a finitely generated graded module M over polynomial ring R K x1, . . . , xn. Eisenbud 3 proved that a commutative Noetherian local ring R is regular if and only if it has finite global dimension, in which case the global dimension coincides with the Krull dimension of R This theorem opened the door for application of homological methods to commutative algebra. Auslander and Buchsbaum 5 realized that Serre’s method could be used to study the close connection between the dimension and multiplicity over a local ring This led them to the Auslander-Buchsbaum equality 6 : let M be a finitely generated module over a Noetherian local ring R, m. We give a procedure and describe an algorithm to compute the projective dimension of a module which is valid over the Laurent polynomial ring. We prove the qualitative version of Horrocks’ theorem over the Laurent polynomial ring
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