130 research outputs found
On the cycle structure of hamiltonian k-regular bipartite graphs of order 4k
It is shown that a hamiltonian -regular bipartite graph of order
contains a cycle of length . Moreover, if such a cycle can be
chosen to omit a pair of adjacent vertices, then is bipancyclic.Comment: 3 page
Flatness testing over singular bases
We show that non-flatness of a morphism f of complex-analytic spaces with a
locally irreducible target Y of dimension n manifests in the existence of
vertical components in the n-fold fibred power of the pull-back of f to the
desingularization of Y. An algebraic analogue follows: Let R be a locally
(analytically) irreducible finite type complex-algebra and an integral domain
of Krull dimension n, and let S be a regular n-dimensional algebra of finite
type over R (but not necessarily a finite R-module), such that the induced
morphism of spectra is dominant. Then a finite type R-algebra A is R-flat if
and only if the tensor product of S with the n-fold tensor power of A over R is
a torsion-free R-module.Comment: Published versio
A fast flatness testing algorithm in characteristic zero
We prove a fast computable criterion that expresses non-flatness in terms of
torsion: Let R be a regular algebra of finite type over a field K of
characteristic zero and let F be a module finitely generated over an R-algebra
of finite type. Given a maximal ideal m in R, let S be the coordinate ring of
the blowing-up of Spec(R) at the closed point m. Then F is flat over R
localized in m if and only if the tensor product of F with S over R is a
torsion-free module over R localized in m. If K is the field of reals or
complex numbers, we give a stronger criterion - without the regularity
assumption on R. We also show the corresponding results in the real- and
complex-analytic categories.Comment: Published versio
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