Abstract. We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions are known, i.e., for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.
We use results of Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination of Betti diagrams of modules with a pure resolution. This implies the multiplicity conjecture of Herzog, Huneke, and Srinivasan for modules that are not necessarily Cohen-Macaulay and also implies a generalized version of these inequalities. We also give a combinatorial proof of the convexity of the simplicial fan spanned by pure diagrams.
We prove that a sequence of positive integers (h 0 , h 1 , . . . , h c ) is the Hilbert function of an artinian level module of embedding dimension two if and only ifwhere we assume that h −1 = h c+1 = 0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.
quelques pages suivantes je me propose de prSsenter une demonstration nouvelle et tr6s simple de l'important thdor6me de GALOlS sur l'existenee du groupe de substitutions appel6 9roupe d'une ~quation al.qdbrique. Elle a dr6 publide'en su6dois dans ma th&e inaugurale Deduktion af n6dvdndiga och tillr(icMiqa vilkoret four m(Ol~qheten af algebraiska eqvationers solution reed radikaler, Upsala Universitets Arsskrift, I886. Je la prdsente iei avee de ldg6res modifications.~. Avant d'en commencer l'exposition nous aurons h. nous expliquer sur le sens partieulier que nous attribuerons h eertaines expressions. Nous eonviendrons de regarder, avee GALOlS, comme rationnelle toute quantit6 qui pent s'exprimer par une fonetion rationnelle aux coefficients eommensurables g l'unit6 de eertaines quantitds donn&'s k priori et que nous regarderons eomme connues. Pour qu'une fonction soit appelde rationnelle nous entendrons que tousles coefficients en soient rationnelles.Si une fonetion rationnelle des quantit6s XO ~ ~1 , ~ • • ' ~ ~n--I reste inwMable par les substitutions d'un certain groupe, mdme en sup-Acta mathematiea.
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