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.
Read more about Chow Ring: Rational Equivalence, Definition of The Chow Ring, Geometric Interpretation, Functoriality, Cohomological Connections, Details of The Construction, Variants, History
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