Journal of the Ramanujan Mathematical Society

Brian Conrad ¹

Affiliations
1 Department of Mathematics, Stanford University, Stanford, CA 93405, US

Abstract

There are two small (but fixable) errors in [1] (entirely my fault) that we address here. I am grateful to Andrea Parri and Johan de Jong for pointing out these respective mistakes that I made.

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Brian Conrad ¹

Affiliations
1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, US

Abstract

We provide a proof of Nagata’s compactification theorem: any separated map of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism. This is a detailed exposition of private notes of Deligne that translate Nagata’s method into modern terms, and includes some applications of general interest in the theory of rational maps, such as refined versions of Chow’s Lemma and the elimination of indeterminacies in a rational map, as well as a blow-up characterization of when a proper morphism (to a rather general base scheme) is birational.

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Brian Conrad ¹

Affiliations
1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, US

Abstract

We use scheme-theoretic methods, particularly descent theory and the Nagata Compactification Theorem, to provide a modern proof of Chevalley’s structure theorem for abstract algebraic groups over perfect fields.

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