8,031 research outputs found
Ideals of general forms and the ubiquity of the Weak Lefschetz property
Let be positive integers and let be an
ideal generated by general forms of degrees , respectively, in a
polynomial ring with variables. When all the degrees are the same we
give a result that says, roughly, that they have as few first syzygies as
possible. In the general case, the Hilbert function of has been
conjectured by Fr\"oberg. In a previous work the authors showed that in many
situations the minimal free resolution of must have redundant terms which
are not forced by Koszul (first or higher) syzygies among the (and hence
could not be predicted from the Hilbert function), but the only examples came
when . Our second main set of results in this paper show that further
examples can be obtained when . We also show that if
Fr\"oberg's conjecture on the Hilbert function is true then any such redundant
terms in the minimal free resolution must occur in the top two possible degrees
of the free module. Related to the Fr\"oberg conjecture is the notion of Weak
Lefschetz property. We continue the description of the ubiquity of this
property. We show that any ideal of general forms in has
it. Then we show that for certain choices of degrees, any complete intersection
has it and any almost complete intersection has it. Finally, we show that most
of the time Artinian ``hypersurface sections'' of zeroschemes have it.Comment: 24 page
The Weak and Strong Lefschetz Properties for Artinian K-Algebras
Let A = bigoplus_{i >= 0} A_i be a standard graded Artinian K-algebra, where
char K = 0. Then A has the Weak Lefschetz property if there is an element ell
of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has
maximal rank, for every i, and A has the Strong Lefschetz property if times
ell^d : A_i --> A_{i+d} has maximal rank for every i and d.
The main results obtained in this paper are the following.
1) EVERY height three complete intersection has the Weak Lefschetz property.
(Our method, surprisingly, uses rank two vector bundles on P^2 and the
Grauert-Mulich theorem.)
2) We give a complete characterization (including a concrete construction) of
the Hilbert functions that can occur for K-algebras with the Weak or Strong
Lefschetz property (and the characterization is the same one).
3) We give a sharp bound on the graded Betti numbers (achieved by our
construction) of Artinian K-algebras with the Weak or Strong Lefschetz property
and fixed Hilbert function. This bound is again the same for both properties.
Some Hilbert functions in fact FORCE the algebra to have the maximal Betti
numbers.
4) EVERY Artinian ideal in K[x,y] possesses the Strong Lefschetz property.
This is false in higher codimension.Comment: To appear in J. Algebr
On the shape of a pure O-sequence
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 versio
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