Intersection multiplicity of Serre on regular schemes

The study of the intersection multiplicity function χOX(F,G) over a regular scheme X for a pair of coherent OX-modules F and G is the main focus of this paper. We mostly concentrate on projective schemes, vector bundles over projective schemes, regular local rings and their blow-ups at the closed point. We prove that (a) vanishing holds in all the above cases, (b) positivity holds over Proj of a graded ring finitely generated over its 0th component which is artinian local, when one of F and G has a finite resolution by direct sum of copies of O(t) for various t, and (c) non-negativity holds over , R regular local, and over arbitrary smooth projective varieties if their tangent bundles are generated by global sections. We establish a local–global relation for χ for a pair of modules over a regular local ring via χ of their corresponding tangent cones and χ of their corresponding blow-ups. A new proof of vanishing and a special case of positivity for Serre's Conjecture are also derived via this approach. We also demonstrate that the study of non-negativity is much more complicated over blow-ups, particularly in the mixed characteristics.