4 1. INTRODUCTION Another integer vector that we can associate to A is its h-vector, h, defined by h = h(A) = (l,fti,...,ft5) = (l,dimfc Ai,...,dim fe i4 a ) which encodes the Hilbert function of A as a vector. So, question iv) above becomes: if s(A) (0,..., 0,1) what are the possibilities for h(A)? We have a complete answer to question iv) in only two cases: when hi 2 (well known) and when h\ 3 (see [64]). In this monograph we will consider the following extension of question iv) to question iv)': iv)' Let A be an algebra with socle degree s. If s(A) = (0,..., 0, c), with c 1, what are the possibilities for h(A) = (1, hi,..., hs)l Algebras A for which s(A) = (0,..., 0, c), c 1, are referred to in the literature as level algebras of type c, and their study was initiated by Stanley in [63]. Question iv)' has also been considered in [14], [20], [23], [43], [44]. In particular, Iarrobino solved iv)' in [44] for hi = 2 (see [23] and [13] for further references). Thus, our interest in iv)' is in the case where hi 3. Level algebras have been studied in several different contexts. E.g., there is a strong connection between level algebras and pure simplicial complexes. More precisely, if A is a simplicial complex with n-vertices (xi,... ,x n ), let fc[A] denote the Stanley-Reisner ring associated to A. Set A A = fc[A]/(xf,... ,x^). Then the algebra A A is level if and only if A is pure (see [7]). Certain simplicial complexes also have level Stanley-Reisner rings. E.g., skele- tons of Cohen-Macaulay complexes, triangulations of spheres and matroid com- plexes. Other examples come by considering the ideals of minors of a generic matrix. Also, for any d 1, n 2 and any t such that fd + n\ fd+l + n\ _^(d + n\ \ n J ~ \ n J n\n lj t general points in P n have homogeneous coordinate ring which is level (see [49]). This monograph is organized in the following way. In Chapter 2 we make some preliminary definitions, recall some standard re- sults about level algebras, and give our first results. Our first main result is to prove a decomposition for finite O-sequences which are the h-vectors of algebras with a given socle vector. This result (Theorem 2.10) extends and improves an analogous theorem of Stanley (Corollary 2.11). In Chapter 3 we reinterpret the notion of a level Artinian algebra homologically. Using this point of view we explain the com- binatorial notion of Cancelation in Resolutions (first considered for level algebras in [24]). This simple idea becomes a powerful tool (thanks to a recent result of Peeva [61]) which we explore. In this chapter we also give some of our principal "non-level sequence" results. In Chapter 4 we use the homological point of view to define standard level algebras of any Krull dimension. We also recall the definition of the Weak Lefschetz Property (see [71]). The weak and strong Lefschetz properties for Artinian algebras have an inter- esting history. Although R. Stanley has said (in a private communication) that he never explicitly mentions this property for arbitrary Gorenstein rings, he does assert that he was "morally aware of the concept since 1975" (see [66]). Several of his other papers ([67], [68]) amply support this view. We have always considered
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