Torsion functors with monomial support
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TORSION FUNCTORS WITH MONOMIAL SUPPORT Fred Rohrer
Received: 10 October 2011 / Published online: 11 April 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology. ˇ Keywords Torsion functor · Local cohomology · Monomial ideal · Cech cohomology · Flatness · Cox scheme Mathematics Subject Classification (2010) Primary 13D45 · Secondary14M25
1 Introduction Over a Noetherian ring (where rings are always understood to be commutative), torsion functors (and hence local cohomology functors) depend only on the radical of their supporting ideals. Without Noetherianness this need not hold. Supposing the supporting ideals to be monomial ideals of finite type of a polynomial algebra one might hope for it to still hold— after all, monomial ideals behave quite independently of the base ring and hence might be unaffected by its potential lack of some nice properties. We will see that this is indeed true. However, the torsion functors with respect to a monomial ideal of finite type and its radical may be different, unless the nilradical of the base ring is nilpotent. But using a monomial variant of the notion of radical it is possible to get a satisfying result that is independent of the base ring and moreover can be generalised to a class of subalgebras of polynomial algebras.
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F. Rohrer ( ) Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam e-mail: [email protected] Present address: F. Rohrer Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
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F. ROHRER
There were two main reasons for writing this note. First, the recent preprint [2] by Botbol and Chardin on Castelnuovo–Mumford regularity over not necessarily Noetherian polynomial algebras contains the claim that local cohomology with respect to monomial ideals of finite type remains the same when we replace the supporting ideal by its radical. This aroused suspicions and turned out to be indeed wrong. Luckily, the aforementioned monomial variant of radical can save the day, and the moral might be that working over not necessarily Noetherian rings requires some meticulousness even in a monomial setting. The second reason comes from the study of toric schemes, i.e., “toric varieties over arbitrary base rings”. An overview of this continuing project including its motivation can be found in [8]. With this approach to toric geometry one gets a relation between flatness of ˇ quasicoherent sheaves on a toric scheme and graded Cech cohomology over a corresponding restricted Cox ring with respect to a certain sequence of monomials. The ideal J gener
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