Taylor and Lyubeznik Resolutions via Gröbner Bases

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.