The Hilbert Function of a Level Algebra

Level algebras were first introduced and investigated by R.P. Stanley in the 1970s, and have since attracted the attention of more and more researchers in commutative algebra, because of both their intrinsic interest and the many applications they have to several other branches of mathematics. These include algebraic geometry, invariant theory, algebraic combinatorics and, surprisingly, even complexity theory. Thus---especially over the last few years---the theory of level algebras has been the object of a remarkable amount of research, and now can most definitely be considered a central topic within commutative algebra. A level algebra is defined as a Cohen-Macaulay (standard) graded algebra whose socle is concentrated in one degree, or, equivalently, whose minimal graded free resolution only has one shift in the last module (there are other equivalent definitions, but those two are perhaps the most common). The simplest case, when the socle is one-dimensional, is that of a Gorenstein algebra. The memoir under review extensively investigates the properties of level algebras---including their Hilbert functions, the Weak Lefschetz Property (WLP), etc.---mainly in Krull dimension 0, which is the Artinian case, and Krull dimension 1, the case of points in the projective space. The authors employ and successfully combine a number of different techniques. These range from homological algebra to combinatorial methods, which are especially useful to prove the non-existence of certain classes of possible level Hilbert functions, and from the theory of Macaulay's inverse systems (also known as Matlis duality)---which is also used in the study of the WLP---to a more geometric approach, which includes liaison (or linkage) theory. In particular, this provides a very powerful tool to construct, in an explicit fashion, new classes of level algebras and level sets of points. The long appendix is devoted to the classification of the Hilbert functions of Artinian level algebras having codimension 3 and small socle degree. The tables are constructed with the help of a computer program and the techniques introduced in the previous part. It is important to remark that the above classification is not yet explicitly known for all socle degrees (this reviewer believes that it never will be, but this is perhaps a less important remark). However, the main and most natural property which may be suggested by the characterization for small socle degrees provided in the memoir (that all Artinian level Hilbert functions of codimension 3 are unimodal, as asked in Question 4.4) is now known to be false, as recently proved by the reviewer (see [J. Algebra 305 (2006), no. 2, 949--956; MR2266862 (2007h:13026)], where the WLP is also shown not to hold in some instances, answering negatively another point of Question 4.4. This paper has already been taken up or improved on in subsequent articles or preprints by Migliore, Shin and J. Ahn, M. Boij and the reviewer, and in the Ph.D. Thesis of A. Weiss ["Some new non-unimodal level algebras'', Tufts Univ., Medford, MA, 2007], who successfully developed ideas of A. Iarrobino). The memoir under review deserves huge credit not only because it contributes substantially to the theory of level algebras, but also since it is the first work which finally presents, or at least mentions and includes in the bibliography, basically all the relevant research done in this area over the years---until about the middle of 2003, when the memoir was completed. Therefore, this work unarguably qualifies as a must-read---and as the main source of information---for all researchers who today work, or want to begin working, on level algebras. It is unfortunate, however, that the memoir has been published almost four years after it was completed, since\u2014as we saw above in what is perhaps the most telling instance, the very recent series of papers on non-unimodal level Hilbert functions\u2014lots of important work (often inspired by questions or ideas contained in the memoir) has already been done since 2003. Thus, if the work under review contains the most accurate historical and bibliographical account of level algebras up to the year 2003, today it is already not up to date. It would be great, therefore, if the authors would decide to update their memoir periodically (perhaps in the form of a book), so as to allow their precious work to remain the chief reference in this very active area of research for many years to come