Taylor and Lyubeznik Resolutions via Gröbner Bases

Abstract

Taylor presented an explicit resolution for arbitrary monomial ideals. Later, Lyubeznik found that a subcomplex already 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. c 2002 Elsevier Science Ltd. All rights reserved. The Taylor and the Lyubeznik ResolutionLet M = {m 1 , . . . , m r } ⊂ P = k[x 1 , . . . , x n ] 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 {v 1 , . . . , v r }. If k = (k 1 , . . . , k q ) is a sequence of integers with 1 ≤ k 1 < k 2 < · · · < k q ≤ r, we set m k = lcm(m k1 , . . . , m kq ). The P-module T q = P ⊗ Λ q V is then freely generated by all wedge products v k = v k1 ∧ · · · ∧ v kq . Finally, we introduce on the algebra T = P ⊗ ΛV the following P-linear differential δ:where k denotes the sequence k with the entry k removed. Obviously, the differential δ respects the grading of T by the form degree, as it maps the component T q into T q−1 (however, in general δ does not respect the natural bi grading of T given by T rq = P r ⊗ Λ q V). One can show that (T , δ) is a complex representing a free resolution of the ideal J 0 = M . Obviously, δv i = m i and the length of the resolution is given by the number r of monomials. This implies immediately that the resolution is rarely minimal.‡ Note that the ordering of the monomials m i in the set M has no real influence on the result: the arising resolutions are trivially isomorphic. †

A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

“…For monomial ideals I the situation is much more favourable. It follows immediately from Taylor's explicit resolution of such ideals [68] (see [59] for a derivation via Gröbner bases) that here a linear bound reg I ≤ n(q − 1) + 1 (69) holds where n is again the number of variables. Indeed, this resolution is supported by the lcm-lattice of the given basis and the degree of its kth term is thus trivially bounded by kq.…”

Differential Equations, Spencer Cohomology, and Computing Resolutions

“…we have dψ + ψd = 1 on elements of positive degree. One can show [81] that the Taylor resolution is a special case of the resolutions obtainable via Schreyer's construction. The contracting homotopy ψ is then related to normal form computations with respect to a Gröbner basis.…”

“…In [81] it was shown that this corresponds to repeated applications of Buchberger's chain criterion [12] for avoiding redundant S-polynomials in the construction of Gröbner bases.…”