12 janvier 2021

Mardi 12 janvier 14:00-15:15 Uriya First (Université de Haïfa)
Generating algebras over commutative rings

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Lieu : Séminaire en ligne

Résumé : Let R be a noetherian (commutative) ring of Krull dimension d. A classical theorem of Forster states that a rank-n locally free R-module can be generated by n+d elements. Swan and Chase observed that this upper bound cannot be improved in general. I will discuss a joint work with Zinovy Reichstein and Ben Williams where similar upper and lower bounds are obtained for R-algebras, provided that R is of finite type over an infinite field k. For example, every Azumaya R-algebra of degree n (i.e. an n-by-n matrix algebra bundle over Spec R) can be generated by floor(d/(n-1))+2 elements, and there exist degree-n Azumaya algebras over d-dimensional rings which cannot be generated by fewer than floor(d/(2n-2))+2 elements. The proof reinterprets the problem as a question on « how well » certain algebraic spaces approximate the classifying stack of the automorphism scheme of the algebra in question.

Generating algebras over commutative rings  Version PDF