844 research outputs found
Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals
We prove a theorem unifying three results from combinatorial homological and
commutative algebra, characterizing the Koszul property for incidence algebras
of posets and affine semigroup rings, and characterizing linear resolutions of
squarefree monomial ideals. The characterization in the graded setting is via
the Cohen-Macaulay property of certain posets or simplicial complexes, and in
the more general nongraded setting, via the sequential Cohen-Macaulay property.Comment: 31 pages, 1 figure. Minor changes from previous version. To appear in
Advances in Mathematic
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
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