Let K be field, S = K[x1, . . . , xn] the polynomial ring over K in the indeterminates x1, . . . , xn and I ⊂ S a graded ideal. Let (𝔽, ∂) be a graded free S-resolution of R = S/I . Each free module 𝔽a in the resolution is of the form 𝔽a = ⊕j S(−j)baj . We set

ta (𝔽) = max{ j : baj ≠ 0}.

In the case that 𝔽 is the graded minimal free resolution of I we write ta(I ) instead of ta (𝔽).

We say 𝔽 satisfies the subadditivity condition, if ta+b(𝔽) ≤ ta(𝔽)+tb(𝔽).

Remark 1. The Taylor complex and the Scarf complex satisfy the subadditivity condition. Indeed, both complexes are cellular resolutions supported on a simplicial complex. From this fact the assertion follows immediately.