Regularity jumps for powers of ideals

Let K be a field. Let R = K[x1, . . . ,xn] be the polynomial ring over K. Let I = ⊕i∈NIi be a homogeneous ideal. For every i, j ∈ N one defines the i jth graded Betti number of I as

βi j(I) = dimK TorRi (I,K) j and set

ti(I) = max{ j |βi j(I) = 0}

with ti(I) = −∞ if it happens that TorRi (I,K) = 0. The Castelnuovo-Mumford regularity reg(I) of I is defined as

reg(I) = sup{ti(I)− i : i ∈N}.