Combinatorial Theory

Extremal ideals are a class of square-free monomial ideals which dominate and determine many algebraic invariants of powers of all square-free monomial ideals. For example, the \(r^{th}\) power \({\mathcal{E}_q}^r\) of the extremal ideal on \(q\) generators has the maximum betti numbers among the \(r^{th}\) power of any square-free monomial ideal with \(q\) generators. In this paper we study the combinatorial and geometric structure of the (minimal) free resolutions of powers of square-free monomial ideals via the resolutions of powers of extremal ideals. Although the end results are algebraic, this problem has a natural interpretation in terms of polytopes and discrete geometry. Our guiding conjecture is that all powers \({\mathcal{E}_q}^r\) of extremal ideals have resolutions supported on their Scarf simplicial complexes, and thus their resolutions are as small as possible. This conjecture is known to hold for \(r \leq 2\) or \(q \leq 4\). In this paper we prove the conjecture holds for \(r=3\) and any \(q\geq 1\) by giving a complete description of the Scarf complex of \({\mathcal{E}_q}^3\). This effectively gives us a sharp bound on the betti numbers and projective dimension of the third power of any square-free monomial ideal. For large \(i\) and \(q\), our bounds on the \(i^{th}\) betti numbers are an exponential improvement over previously known bounds. We also describe a large number of faces of the Scarf complex of \({\mathcal{E}_q}^r\) for any \(r,q \geq 1\).

Mathematics Subject Classifications: 13D02, 13F55, 05E40, 52B20

Keywords: Free resolutions, bounds on Betti numbers, extremal ideals, monomial ideals, powers of ideals, Scarf complex, discrete Morse theory