Hilbert Series and Hilbert Polynomial - Graded Algebra and Polynomial Rings

Graded Algebra and Polynomial Rings

Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g₁, ..., g_n of degree 1, then the map which sends X_i onto g_i defines an homomorphism of graded rings from onto S. Its kernel is a homogeneous ideal I and this defines an isomorphism of graded algebra between and S.

Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.