398 research outputs found
Powers are Golod
Let be a proper graded ideal in a positively graded polynomial ring
over a field of characteristic 0. In this note it is shown that is
Golod for all
Monomial localizations and polymatroidal ideals
In this paper we consider monomial localizations of monomial ideals and
conjecture that a monomial ideal is polymatroidal if and only if all its
monomial localizations have a linear resolution. The conjecture is proved for
squarefree monomial ideals where it is equivalent to a well-known
characterization of matroids. We prove our conjecture in many other special
cases. We also introduce the concept of componentwise polymatroidal ideals and
extend several of the results, known for polymatroidal ideals, to this new
class of ideals
Stability properties of powers of ideals over regular local rings of small dimension
Let be a regular local ring or a polynomial ring over a
field, and let be an ideal of which we assume to be graded if is a
polynomial ring. Let astab resp. be the smallest
integer for which Ass resp. Ass stabilize, and
dstab be the smallest integer for which depth stabilizes. Here
denotes the integral closure of . We show that
astab if dim, while
already in dimension , astab and may differ
by any amount. Moreover, we show that if dim, then there exist ideals
and such that for any positive integer one has and .Comment: 9 pages, Comments are welcom
Bounds for the regularity of local cohomology of bigraded modules
Let be a finitely generated bigraded module over the standard bigraded
polynomial ring , and let . The
local cohomology modules are naturally bigraded, and the components
H^k_Q(M)_j=\Dirsum_iH^k_Q(M)_{(i,j)} are finitely generated graded
-modules. In this paper we study the regularity of
, and show in several cases that \reg H^k_Q(M)_j is linearly
bounded as a function of
The face ideal of a simplicial complex
Given a simplicial complex we associate to it a squarefree monomial ideal
which we call the face ideal of the simplicial complex, and show that it has
linear quotients. It turns out that its Alexander dual is a whisker complex. We
apply this construction in particular to chain and antichain ideals of a finite
partially ordered set. We also introduce so-called higher dimensional whisker
complexes and show that their independence complexes are shellable
Depth stability of edge ideals
Let be a connected finite simple graph and let be the edge ideal of
. The smallest number for which \depth S/I_G^k stabilizes is denoted
by \dstab(I_G). We show that \dstab(I_G)<\ell(I_G) where
denotes the analytic spread of . For trees we give a stronger upper bound
for \dstab(I_G). We also show for any two integers there exists a
tree for which \dstab(I_G)=a and
On the fiber cone of monomial ideals
We consider the fiber cone of monomial ideals. It is shown that for monomial
ideals of height , generated by elements, the fiber
cone of is a hypersurface ring, and that has positive depth
for interesting classes of height monomial ideals , which
are generated by elements. For these classes of ideals we also show that
is Cohen--Macaulay if and only if the defining ideal of is
generated by at most 3 elements. In all the cases a minimal set of generators
of is determined
Matching numbers and the regularity of the Rees algebra of an edge ideal
The regularity of the Rees ring of the edge ideal of a finite simple graph is
studied. We show that the matching number is a lower and matching number~
is an upper bound of the regularity, if the Rees algebra is normal. In general
the induced matching number is a lower bound for the regularity, which can be
shown by applying the squarefree divisor complex
Expansions of monomial ideals and multigraded modules
We introduce an exact functor defined on multigraded modules which we call
the expansion functor and study its homological properties. The expansion
functor applied to a monomial ideal amounts to substitute the variables by
monomial prime ideals and to apply this substitution to the generators of the
ideal. This operation naturally occurs in various combinatorial contexts
Freiman ideals
In this paper we study the Freiman inequality for the minimal number of
generators of the square of an equigenerated monomial ideal. Such an ideal is
called a Freiman ideal if equality holds in the Freiman inequality. We classify
all Freiman ideals of maximal height, the Freiman ideals of certain classes of
principal Borel ideals, the Hibi ideals which are Freiman, and classes of
Veronese type ideals which are Freiman
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