Chow Ring

In algebraic geometry, the Chow ring (named after W. L. Chow) of an algebraic variety is an algebraic-geometric analogue of the cohomology ring of the variety considered as a topological space: its elements are formed out of actual subvarieties (so-called algebraic cycles) and its multiplicative structure is derived from the intersection of subvarieties. In fact, there is a natural map from one to the other which preserves the geometric notions which are common to the two (for example, Chern classes, intersection pairing, and a form of Poincaré duality). The advantage of the Chow ring is that its geometric definition allows it to be defined without reference to non-algebraic concepts; in addition, using algebraic techniques that are not available in the purely topological case, certain constructions that exist for both rings are simpler in the Chow ring.

There is also a bivariant version of the Chow theory (often referred to as the "operational Chow theory") introduced by William Fulton and Robert MacPherson.