14 research outputs found
Normalization of Rings
We present a new algorithm to compute the integral closure of a reduced
Noetherian ring in its total ring of fractions. A modification, applicable in
positive characteristic, where actually all computations are over the original
ring, is also described. The new algorithm of this paper has been implemented
in Singular, for localizations of affine rings with respect to arbitrary
monomial orderings. Benchmark tests show that it is in general much faster than
any other implementation of normalization algorithms known to us.Comment: Final version, to be published in JSC 201
Local analysis of Grauert-Remmert-type normalization algorithms
Normalization is a fundamental ring-theoretic operation; geometrically it
resolves singularities in codimension one. Existing algorithmic methods for
computing the normalization rely on a common recipe: successively enlarge the
given ring in form an endomorphism ring of a certain (fractional) ideal until
the process becomes stationary. While Vasconcelos' method uses the dual
Jacobian ideal, Grauert-Remmert-type algorithms rely on so-called test ideals.
For algebraic varieties, one can apply such normalization algorithms
globally, locally, or formal analytically at all points of the variety. In this
paper, we relate the number of iterations for global Grauert-Remmert-type
normalization algorithms to that of its local descendants.
We complement our results by an explicit study of ADE singularities. This
includes the description of the normalization process in terms of value
semigroups of curves. It turns out that the intermediate steps produce only ADE
singularities and simple space curve singularities from the list of
Fruehbis-Krueger.Comment: 22 pages, 7 figure
Parallel algorithms for normalization
Given a reduced affine algebra A over a perfect field K, we present parallel
algorithms to compute the normalization \bar{A} of A. Our starting point is the
algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de
Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a
way which is compatible with normalization, apply a local version of the
normalization algorithm at each stratum, and find \bar{A} by putting the local
results together. Second, in the case where K = Q is the field of rationals, we
propose modular versions of the global and local-to-global algorithms. We have
implemented our algorithms in the computer algebra system SINGULAR and compare
their performance with that of the algorithm of Greuel, Laplagne, and Seelisch.
In the case where K = Q, we also discuss the use of modular computations of
Groebner bases, radicals, and primary decompositions. We point out that in most
examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and
Seelisch by far, even if we do not run them in parallel.Comment: 19 page
Non-commutative crepant resolutions: scenes from categorical geometry
Non-commutative crepant resolutions are algebraic objects defined by Van den
Bergh to realize an equivalence of derived categories in birational geometry.
They are motivated by tilting theory, the McKay correspondence, and the minimal
model program, and have applications to string theory and representation
theory. In this expository article I situate Van den Bergh's definition within
these contexts and describe some of the current research in the area.Comment: 57 pages; final version, to appear in "Progress in Commutative
Algebra: Ring Theory, Homology, and Decompositions" (Sean Sather-Wagstaff,
Christopher Francisco, Lee Klingler, and Janet Vassilev, eds.), De Gruyter.
Incorporates many small bugfixes and adjustments addressing comments from the
referee and other