2013/01/17 by Craig Huneke, Paolo Mantero, Huneke, Craig +5
Mathematics · #13D02 #13D05 #14M06 #14M07 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13D02 #msc:13D05 #msc:14M06 #msc:14M07
paper · pdf · doi:10.48550/arxiv.1301.4147
21 pages (fixed typo in statement of main theorem from version 1)
arxiv created 2013/01/19 · arxiv updated 2013/01/22
Let K be an algebraically closed field. There has been much interest in characterizing multiple structures in ¶nK defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no such finite characterization of multiple structures is possible if one only assumes Serre's (S1) property holds. Specifically, we prove that for any positive integers h, e ≥ 2 with (h,e) ≠ (2,2) and p ≥ 5 there is a homogeneous ideal I in a polynomial ring over K such that (1) the height of I is h, (2) the Hilbert-Samuel multiplicity of R/I is e, (3) the projective dimension of R/I is at least p and (4) the ideal I is primary to a linear prime (x1,..., xh). This result is in stark contrast to Manolache's characterization of Cohen-Macaulay multiple structures in codimension 2 and multiplicity at most 4 and also to Engheta's characterization of unmixed ideals of height 2 and multiplicity 2.