Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
170
result(s) for
"Hornell, James"
Sort by:
The multiplicity function of a local ring
Let A be a local ring with maximal ideal m. Let f∈Af \\in A, and define μA(f){\\mu _A}(f) to be the multiplicity of the A-module A/AfA/Af with respect to m. Under suitable conditions μA(fg)=μA(f)+μA(g){\\mu _A}(fg) = {\\mu _A}(f) + {\\mu _A}(g). The relationship of μA{\\mu _A} to reduction of A, normalization of A and a quadratic transform of A is studied. It is then shown that there are positive integers n1,…,ns{n_1}, \\ldots ,{n_s} and rank one discrete valuations v1,…,vs{v_1}, \\ldots ,{v_s} of A centered at m such that μA(f)=n1v1(f)+⋯+nsvs(f){\\mu _A}(f) = {n_1}{v_1}(f) + \\cdots + {n_s}{v_s}(f) for all regular elements f of A.
Journal Article
A note on the relationship between Weil and Cartier divisors
Using a generalized equivalence relation, a subquotient of the group of Weil divisors is shown to be isomorphic to the group of Cartier divisors modulo linear equivalence for a reduced subscheme of a projective space over a field. A difficulty of the nonreduced case is discussed.
Journal Article
A note on the geometric criteria for the factoriality of an affine ring
Let RR be a normal integral domain finitely generated over a field kk, let UU be an affine algebraic variety with coordinate ring RR, and let VV be a projective completion of UU over kk. RR is factorial if and only if the class group of VV over kk is generated by the kk-rational cycles at infinity with respect to UU. If kk is algebraically closed, RR is shown to be factorial if and only if the Picard group of VV is zero and the Néron-Severi group of VV is generated by the kk-rational cycles at infinity. If kk is finitely generated over its prime field, some well-known arithmetic results are applied to show the existence of affine localizations of VV which have a factorial coordinate ring over kk. The relationship between the existence of an affine localization of VV with a factorial coordinate ring, and the birationality of VV is also discussed.
Journal Article
The Role of Birds in Early Navigation
It is not generally recognized how important a part was played by birds in the guidance of early voyagers when sailing on the uncharted seas of the ancient world, and how their regular migrations year by year and season by season, along definite routes have contributed to the discovery of many lands and isolated islands. We know that Columbus made his first American landfall by noting the south-westerly direction taken by great flocks of small field-birds seen flying overhead in the evening towards land as yet out of sight of those aboard the ships; we know, too, that the significance of their flight was at once recognized, for did not Pinzon point out its meaning to his crew by his remark ‘Those birds know their business’ (Morison, 1841, 664-5)? How true his words were we now know, for it is about the time of year (October) when Columbus saw these battalions of birds darkening the sky, that certain species of landbirds set out from the eastern shores of North America by a southerly route, via Bermuda, to seek a warmer climate and a more plentiful supply of food.
Journal Article