596 research outputs found
Finiteness in derived categories of local rings
New homotopy invariant finiteness conditions on modules over commutative
rings are introduced, and their properties are studied systematically. A number
of finiteness results for classical homological invariants like flat dimension,
injective dimension, and Gorenstein dimension, are established. It is proved
that these specialize to give results concerning modules over complete
intersection local rings. A noteworthy feature is the use of techniques based
on thick subcategories of derived categories.Comment: 40 pages. Minor revisions. To appear in Commentarii Math. Helvetic
The flag f-vectors of Gorenstein* order complexes of dimension 3
We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently
the flag f-vectors of Gorenstein* order complexes of dimension 3. As a
corollary, we characterize the f-vectors of Gorenstein* order complexes in
dimensions 3 and 4. This characterization rise a speculated intimate connection
between the f-vectors of flag homology spheres and the f-vectors of Gorenstein*
order complexes
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
Homology over local homomorphisms
The notions of Betti numbers and of Bass numbers of a finite module N over a
local ring R are extended to modules that are only assumed to be finite over S,
for some local homomorphism f: R --> S. Various techniques are developed to
study the new invariants and to establish their basic properties. In several
cases they are computed in closed form. Applications go in several directions.
One is to identify new classes of finite R-modules whose classical Betti
numbers or Bass numbers have extremal growth. Another is to transfer ring
theoretical properties between R and S in situations where S may have infinite
flat dimension over R. A third is to obtain criteria for a ring equipped with a
`contracting' endomorphism -- such as the Frobenius endomorphism -- to be
regular or complete intersection; these results represent broad generalizations
of Kunz's characterization of regularity in prime characteristic.Comment: To appear in the American Journal of Mathematics; new version has
minor changes in the presentation; table of content removed; 52 page
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