Almost regular sequences on graded modules associated with a subfunctor of the 0-th local cohomology functor II

抄録

This is the last part of this two-part series of papers, here, we introduce the regular sequence in the category of finitely generated graded modules over a polynomial ring whose coefficient field is an infinite field. For a nonzero principal graded module, its first syzygy ideal being a completely full, is equivalent to that the principal module has a regular sequence whose length is equal to the Kull dimension of the base polynomial ring. This is the result of Watanabe and Harima, see Ref(1). We extend the result as follows. For graded modules whose first syzygies are generated by elements of the same degree, to have a component-wise linear syzygy so actually to have linear syzygy is equivalent to that there exists a regular sequence whose length is equal to the Kull dimension of the base ring.