Syzygies of differentials of forms

作者：Bermejo Isabel; Gimenez Philippe; Simis Aron^*

来源：Journal of Algebra, 2013, 375: 41-58.

DOI：10.1016/j.jalgebra.2012.11.008

• Syzygies
• Kahler differentials
• Polarizability
• Jacobian ideal
• Homological dimension
• Polar map

Given a standard graded polynomial ring R = k[x(1) , . . . , x(n)] over a field k of characteristic zero and a graded k-subalgebra A = k[f(1) , . . . , f(m)] subset of R, one relates the module Omega(A/k) of Kahler k-differentials of A to the transposed Jacobian module D subset of Sigma(n)(i=1) Rdx(i) of the forms f(1) , . . . , f(m) by means of a Leibniz.map Omega(A/k) -%26gt; D whose kernel is the torsion of Omega(A/k). Letting D denote the R-submodule generated by the (image of the) syzygy module of Omega(A/k) and 3 the syzygy module of D, there is a natural inclusion D subset of 3 coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality - in which case one says that the forms f(1) , . . . , f(m) are polarizable. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious Connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in R and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.

• 出版日期2013-2-1