Using Grothendieck Groups to Define and Relate the Hilbert and Chern Polynomials Axiomatically
نویسنده
چکیده
The Hilbert polynomial can be defined axiomatically as a group homomorphism on the Grothendieck group K(X) of a projective variety X, satisfying certain properties. The Chern polynomial can be similarly defined. We therefore define these rather abstract notions to try and find a nice description of this relationship. Introduction Let X = P be a complex projective variety over an algebraically closed field k with coordinate ring S = k[x0, . . . , xr]. Recall that the Hilbert function of a finitely generated graded S-module M is defined as
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