P.r.a.g.mat.i.c. 2008 Free Resolutions and Hilbert Series: Algebraic, Combinatorial and Geometric Aspects

Introduction: For a standard graded k-algebra A = ⊕ n≥0 An let H(A, t) = ∑ n≥0 dimk Ant n be its Hilbert-series. By classical results H(A, t) = hA(t) (1−t)d is a rational function, where d is the Krull dimension of A and hA(t) = h0 + · · ·+hrt a polynomial with integer coefficients with h0 = 1. Since the work of Stanley (see e.g. [41], [37]) in the 70’s enumerative properties of the coefficient series (h0, . . . , hr) have been studied intensively. This includes the classification the coefficient series of specific classes of combinatorially interesting algebras (see e.g. [7], [6]) and the explicit control over the coefficient series for classical algebras A (see e.g. [14], [19]). Despite many groundbreaking results some of the most fundamental questions remain open. Among them the g-theorem for Stanley-Reisner rings k[∆] of Gorenstein∗ simplicial complexes [46], the Charney-Davis conjecture for k[∆] of flag Gorenstein∗ simplicial complexes [13] and the unimodality property of the coefficient series for classical rings, such a quotients by determinantal ideals. Using Gröbner bases techniques (see [42] for aspects relevant to the school topic) it has been shown [35], [4], [12], [30] or conjectured that questions about the unimodality property for some classes of algebras can been resolved with the help of g-theorems for suitable classes of simplicial complexes. On the other hand new g-theorems continue to emerge. In addition flag-ness has been linked to the Koszul property of A and the question of real rootedness of the polynomial hA(t) [35], [34]. Alexander duality has become an important tool in the study of squarefree monomial ideals, due to the remarkable result of Eagon-Reiner [15] and extensions by Terai [44] , relating data of the resolution of the Stanley-Reisner ring of a simplicial complex to that of its Alexander dual. This fundamental result and further extensions to squarefree modules [36], [45], and even more generally to multigraded modules by Miller [33] has many applications. For example, by using this duality, a classification of all Cohen-Macaulay bipartite graphs has been given [20] and a relationship between the Hilbert-Burch theorem and Dirac’s theorem on chordal graphs has been exhibited [21]. In graph theory vertex covers are classical topic of research. In recent papers [22], [23], [24] vertex covers of higher order, not only for graphs but for