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
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 ).
In this monograph we will consider the following extension of question iv) to
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 . Question
iv)' has also been considered in , , , , . In particular, Iarrobino
solved iv)' in  for hi = 2 (see  and  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
), 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 ).
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
have homogeneous coordinate ring which is level (see ).
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 ). This simple idea becomes a powerful tool (thanks to a recent result of
Peeva ) 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 ).
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 ). Several of
his other papers (, ) amply support this view. We have always considered