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Unformatted text preview: NAGATA COMPACTIFICATION FOR ALGEBRAIC SPACES BRIAN CONRAD, MAX LIEBLICH, AND MARTIN OLSSON Abstract. We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes. To the memory of Masayoshi Nagata 1. Introduction 1.1. Motivation. The Nagata compactification theorem for schemes is a very useful and fundamental result. It says that if S is a quasi-compact and quasi-separated scheme (e.g., any noetherian scheme) and if f : X S is a separated map of finite type from a scheme X then f fits into a commutative diagram of schemes (1.1.1) X j / f @ @ @ @ @ @ @ @ X f S with j an open immersion and f proper; we call such an X an S-compactification of X . Nagatas papers ([N1], [N2]) focused on the case of noetherian schemes and unfortunately are difficult to read nowadays (due to the use of an older style of algebraic geometry), but there are several available proofs in modern language. The proof by Lutkebohmert [L] applies in the noetherian case, and the proof of Deligne ([D], [C2]) is a modern interpretation of Nagatas method which applies in the general scheme case. The preprint [Vo] by Vojta gives an exposition of Delignes approach in the noetherian case. Temkin has recently introduced some new valuation-theoretic ideas that give yet another proof in the general scheme case. The noetherian case is the essential one for proving the theorem because it implies the general case via approximation arguments [C2, Thm. 4.3]. An important application of the Nagata compactification theorem for schemes is in the definition of etale cohomology with proper supports for any separated map of finite type f : X S between arbitrary schemes. Since any algebraic space is etale-locally a scheme, the main obstacle to having a similar construction of such a theory for etale cohomology of algebraic spaces is the availability of a version of Nagatas theorem for algebraic spaces. Strictly speaking, it is possible to develop the full six operations formalism even for non- separated Artin stacks ([LO1], [LO2]) despite the lack of a compactification theorem in such cases. However, the availability of a form of Nagatas theorem simplifies matters tremendously, and there are cohomological applications for which the approach through compactifications seems essential, such as the proof of Fujiwaras theorem for algebraic spaces [Va] (from which one can deduce the result for DeligneMumford stacks via the Date : October 24, 2009....
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