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From triangulated categories to module categories via localisation II: Calculus of fractions
We show that the quotient of a Hom-finite triangulated category C by the
kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We
further show that the class of regular morphisms in the quotient admit a
calculus of left and right fractions. It follows that the Gabriel-Zisman
localisation of the quotient at the class of regular morphisms is abelian. We
show that it is equivalent to the category of finite dimensional modules over
the endomorphism algebra of T in C.Comment: 21 pages; no separate figures. Minor changes. To appear in Journal of
the London Mathematical Society (published version is different
Denominators of cluster variables
Associated to any acyclic cluster algebra is a corresponding triangulated
category known as the cluster category. It is known that there is a one-to-one
correspondence between cluster variables in the cluster algebra and exceptional
indecomposable objects in the cluster category inducing a correspondence
between clusters and cluster-tilting objects.
Fix a cluster-tilting object T and a corresponding initial cluster. By the
Laurent phenomenon, every cluster variable can be written as a Laurent
polynomial in the initial cluster. We give conditions on T equivalent to the
fact that the denominator in the reduced form for every cluster variable in the
cluster algebra has exponents given by the dimension vector of the
corresponding module over the endomorphism algebra of T.Comment: 22 pages; one figur
Detailed analysis of radiation data from the Gemini 4 and Gemini 7 proton-electron spectrometer experiments Final report, 13 Jun. 1967 - 30 Dec. 1968
Detailed analysis of radiation data from Gemini 4 and 7 proton-electron spectrometer experiment
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