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The paper has two goals: the study of the associated graded ring of contracted homogeneous ideals in and the study of the Gröbner fan of the ideal of the rational normal curve in . These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with Cohen–Macaulay associated graded ring in terms of the numerical invariants arising from Zariski’s factorization. We determine explicitly the initial ideals (monomial or not) of , that are Cohen–Macaulay.
We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a -adic field. This gives a definition of a conductor for -adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse–Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups are abelian and killed by .
We show that the mapping cone of a morphism of differential graded Lie algebras, , can be canonically endowed with an -algebra structure which at the same time lifts the Lie algebra structure on and the usual differential on the mapping cone. Moreover, this structure is unique up to isomorphisms of -algebras.
On étudie les anneaux (notamment noethériens) dans lesquels l’ensemble des éléments non nuls est existentiel positif (réunion finie de projections d’ensembles « algébriques »). Dans le cas noethérien intègre, on montre notamment que cette condition est vérifiée pour tout anneau qui n’est pas local hensélien, et qu’elle ne l’est pas pour un anneau local hensélien excellent qui n’est pas un corps.
Ces résultats apportent au passage une réponse à une question de Popescu sur l’approximation forte pour les couples henséliens.
We investigate rings in which the set of nonzero elements is positive-existential (i.e., a finite union of projections of “algebraic” sets). In the case of Noetherian domains, we prove in particular that this condition is satisfied whenever the ring in question is not local Henselian, while it is not satisfied for any excellent local Henselian domain which is not a field.
As a byproduct, we obtain an answer to a question of Popescu on strong approximation for Henselian pairs.