تقرير
On the shape of a pure O-sequence
| العنوان: | On the shape of a pure O-sequence |
|---|---|
| المؤلفون: | Boij, M., Migliore, J., Miro'-Roig, R., Nagel, U., Zanello, F. |
| المصدر: | Memoirs AMS 218 (2012), no. 2024, vii + 78 pp |
| سنة النشر: | 2010 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Commutative Algebra, Mathematics - Algebraic Geometry, Primary: 13D40, 05E40, 06A07, 13E10, 13H10. Secondary: 05A16, 05B35, 14M05, 13F20 |
| الوصف: | An order ideal is a finite poset X of (monic) monomials such that, whenever M is in X and N divides M, then N is in X. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree. Equivalently, in the language of commutative algebra, pure O-sequences are the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of Richard Stanley's early works in this area, and have since played a significant role in at least three disciplines: the study of simplicial complexes and their f-vectors, level algebras, and matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, making an extensive use of algebraic and combinatorial techniques, includes: (i) A characterization of the first half of a pure O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) characterization; (v) A pithy connection of the ICP with Stanley's matroid h-vector conjecture; (vi) A specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 in characteristic zero. As a corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over any field); (vii) An analysis of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii) Some observations about pure f-vectors, an important special case of pure O-sequences. Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly minor revisions with respect to last year's version |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1003.3825 |
| رقم الأكسشن: | edsarx.1003.3825 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |