Computing syzygies with Grobner bases
Steven V Sam · 2008
The aim of this article is to motivate the inclusion of Gröbner bases in algebraic geometry via the computation of syzygies. In particular, we will discuss Gröbner bases for finite modules over polynomial rings, and mention how this tool can be used to compute minimal free resolutions of graded finite modules. We won’t prove any results. Instead, we refer the reader to the references at the end for proofs. This article was written as a final project for the program on computational algebraic geometry at the University of Utah during June 16 to July 3, 2008. We begin with an example from [BM]. 1.1 The example of the twisted cubic in P 3. We work in projective space P 3 over a field k, and let S = k[w, x, y, z] be the homogeneous coordinate ring of P 3. Define polynomials f1 = w 2 − xy, f2 = wy − xz, and f3 = wz − y 2; the homogeneous ideal I = (f1, f2, f3) defines a twisted cubic curve X ⊂ P 3. Note also that X has a parameterization (r, s) ↦ → (r 2 s, r 3, rs 2, s 3). We would somehow like to be able deform I to some monomial ideal J, motivated by the fact that computations are much easier to perform for monomial ideals, and with the hopes that such computations could be used to obtain information about our original ideal I. We define an action of k × on the monomials of S of degree < 4 by t · w a x b y c z d = t −(16a+4b+c) w a x b y c z d. On a polynomial p with degree < 4, we get an operation by having k × act on the monomials of p, and then “clear denominators ” for t. In our case, this means the following: