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

抄録

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. This is the main result of this two-part series of papers, which is also a partial extension of the result of Watanabe and Harima, see Ref(1). In the first part of papers, here we give some preparatory results, especially, we introduce the weak regular sequence in the category of finitely generated graded modules over a polynomial ring whose coefficient field is an infinite field and give some fundamental results concerning the weak regular sequence.