A minimal free resolution of a finitely generated graded module M over a commutative Noetherian Z-graded ring R in which all maps are homogeneous module homomorphisms, i.e., they map every homogeneous element to a homogeneous element of the same degree. It is usually written in the form ...->⊕_(j element Z) R(-j)^(β_(s j))->...->⊕_(j element Z) R(-j)^(β_(1j))->⊕_(j element Z) R(-j)^(β_(0j))->M->0, where R(-j) indicates the ring R with the shifted graduation such that, for all a element Z, R(-j)_a = R_(a - j).

Betti number | graded module | Hilbert function