Taylor and Lyubeznik Resolutions via Gröbner Bases

Taylor presented an explicit resolution for arbitrary monomial ideals. Later, Lyubeznik found that already a subcomplex defines a resolution. We show that the Taylor resolution may be obtained by repeated application of the Schreyer Theorem from the theory of Gröbner bases, whereas the Lyubeznik resolution is a consequence of Buchberger’s chain criterion. Finally, we relate Fröberg’s contracting homotopy for the Taylor complex to normal forms with respect to our Gröbner bases and use it to derive a splitting homotopy that leads to the Lyubeznik complex. 1. The Taylor and the Lyubeznik Resolution Let M = {m1, . . . , mr} ⊂ P = k[x1, . . . , xn] be a finite set of monomials. Taylor [1960] constructed in her Ph.D. thesis an explicit free resolution of the monomial ideal J = 〈M〉. The associated complex consists essentially of an exterior algebra and a differential defined via the least common multiples of subsets of M. Let V be some r-dimensional k-vector space with the basis {v1, . . . , vr}. If k = (k1, . . . , kq) is a sequence of integers with 1 ≤ k1 < k2 < · · · < kq ≤ r, we set mk = lcm(mk1 , . . . , mkq). The P-module Tq = P ⊗ Λ V is then freely generated by all wedge products vk = vk1 ∧ · · · ∧ vkq . Finally, we introduce on the algebra T = P ⊗ ΛV the following P-linear differential δ: