Minimal homogeneous bases for polynomial ideals (Q1762555)

The authors consider the problem to find a minimal generating set of the homogenization \(I^h\) of an ideal \(I\) generated by \(\{f_1,\dots,f_t\}\), \(k\) a field. The results are standard and well known for a long time. Moreover most of them can be proved much faster taking into account that \(R\) is a H-local ring. E.g., the result that every minimal homogeneous basis of \(I\) has the same number of elements -- proved with much effort as (2.6) -- is an easy consequence of the H-local version of Nakayama's lemma. The results about syzygies of (homogeneous) leading terms may be traced back (at least) to \textit{F. S. Macaulay} [Proc. Lond. Math. Soc. (2) 26, 531--555 (1927; JFM 53.0104.01)]