# Generating algebras over commutative rings

`2021-03-10 10:30:00``2021-03-10 11:30:00``Generating algebras over commutative rings``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. ================================================= Topic: BIU Algebra Seminar -- First Time: Mar 10, 2021 10:30 AM Jerusalem Join Zoom Meeting https://us02web.zoom.us/j/87964715372 Meeting ID: 879 6471 5372``Zoom -- see invitation below``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`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.

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Topic: BIU Algebra Seminar -- First

Time: Mar 10, 2021 10:30 AM Jerusalem

Join Zoom Meeting

https://us02web.zoom.us/j/87964715372

Meeting ID: 879 6471 5372

Last Updated Date : 23/02/2021