8,669 research outputs found
Normal transversality and uniform bounds
For two ideals and of a noetherian ring, we characterize, in terms of
the vanishing of Tor modules, when the associated graded ring of the sum
is isomorphic to the tensor product of the associated graded ring of and
the associated graded ring of . It is shown that the relation type of the
tensor product of two standard algebras is bounded above by the maximum of the
relation type of each algebra. As a consequence, we deduce a uniform bound for
the relation type of maximal ideals of an excellent ring and a classical result
of Duncan and O'Carroll on the strong uniform Artin-Rees property.Comment: 12 pages, Late
The relation type of affine algebras and algebraic varieties
We introduce the notion of relation type of an affine algebra and prove that
it is well defined by using the Jacobi-Zariski exact sequence of
Andr\'e-Quillen homology. In particular, the relation type is an invariant of
an affine algebraic variety. Also as a consequence of the invariance, we show
that in order to calculate the relation type of an ideal in a polynomial ring
one can reduce the problem to trinomial ideals. When the relation type is at
least two, the extreme equidimensional components play no role. This leads to
the non existence of affine algebras of embedding dimension three and relation
type two
The primary components of positive critical binomial ideals
A natural candidate for a generating set of the (necessarily prime) defining
ideal of an -dimensional monomial curve, when the ideal is an almost
complete intersection, is a full set of critical binomials. In a somewhat
modified and more tractable context, we prove that, when the exponents are all
positive, critical binomial ideals in our sense are not even unmixed for , whereas for they are unmixed. We further give a complete
description of their isolated primary components as the defining ideals of
monomial curves with coefficients. This answers an open question on the number
of primary components of Herzog-Northcott ideals, which comprise the case
. Moreover, we find an explicit, concrete description of the irredundant
embedded component (for ) and characterize when the hull of the ideal,
i.e., the intersection of its isolated primary components, is prime. Note that
these last results are independent of the characteristic of the ground field.
Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and
Laurent polynomial rings, together with theory of Smith Normal Form and of
Fitting ideals. This gives a more transparent and completely general approach,
replacing the theory of multiplicities used previously to treat the particular
case .Comment: 21 page
The equations of Rees algebras of equimultiple ideals of deviation one.
We describe the equations of the Rees algebra R(I) of an equimultiple
ideal I of deviation one provided that I has a reduction generated
by a regular sequence x1, . . . , xs such that the initial forms x∗
1, . . . , x∗
s−1 are
a regular sequence in the associated graded ring. In particular, we prove that
there is a single equation of maximum degree in a minimal generating set of
the equations of R(I), which recovers some previous known results.Postprint (published version
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