Koszul algebras and Gröbner bases of quadrics

Let R be a standard graded K-algebra, that is, an algebra of the form R = K[x1, . . . , xn]/I where K[x1, . . . , xn] is a polynomial ring over the field K and I is a homogeneous ideal with respect to the grading deg(xi) = 1. Let M be a finitely generated graded R-module. Consider the (essentially unique) minimal graded Rfree resolution of M · · · → Ri → Ri−1 → · · · → R1 → R0 → M → 0 The rank βi of the i-th module in the minimal free resolution of M is called the i-th Betti number of M . One can also keep track of the graded structure of the resolution. It follows that the free modules in the resolution are indeed direct sums of “shifted” copies of R: Ri = ⊕jR(−j) βij