Given a weighted homogeneous ideal I in R, a polynomial ring in n variables over a field K, such that R/I has dimension d, and assuming that A, the polynomial ring in the last d variables over K, is a Noether normalization of R/I, we show how to construct the short resolution of R/I, i.e., a minimal graded free resolution of R/I as A-module. We exhibit, in particular, a Schreyer-like method to compute a (non-necessarily minimal) graded free resolution of R/I as A-module and describe how to prune the previous resolution to obtain a minimal one when R/I is a 3-dimensional simplicial toric ring.

This is a joint work with Ignacio García-Marco and Mario González-Sánchez.