Serre's Intersection Multiplicity

Serre's intersection multiplicity of two finitely generated modules and over a regular local ring of Krull dimension is the alternating sum

provided their module tensor product has finite module length. Here is module length, and Tor is computed over . Regularity ensures that the Tor modules in degrees greater than vanish. For nonzero , the positivity conjecture asserts when . The Krull dimension of a module is the supremum of over prime ideals at which its localization is nonzero.

For example, in , the modules and have for every integer . Their module tensor product is , of module length , and their Tor in positive degrees vanishes.