Proper map (algebraic geometry)

Proper map (algebraic geometry)
Name	Proper map (algebraic geometry)
Field	Algebraic geometry
Introduced	Grothendieck era
Key people	Alexander Grothendieck, Jean-Pierre Serre, David Mumford, Pierre Deligne, Michel Demazure

Proper map (algebraic geometry)

A proper map is a morphism of schemes or algebraic spaces that generalizes the topological notion of a proper continuous map to the setting of algebraic geometry. In the Grothendieck school it serves as the algebraic analogue of compactness and plays a central role in theorems of finiteness, cohomology, and moduli problems. Properness is intimately connected to notions developed by figures such as Alexander Grothendieck, Jean-Pierre Serre, David Mumford, Pierre Deligne, and Michel Demazure and appears throughout the work of institutions like the Institut des Hautes Études Scientifiques and projects such as the Séminaire de Géométrie Algébrique.

Definition

A morphism f: X → Y of schemes (or algebraic spaces or algebraic stacks) is defined to be proper if it is separated, of finite type, and universally closed. The definition was formalized in Grothendieck's work in the context of the Éléments de géométrie algébrique and is used in texts by Jean-Pierre Serre, Robin Hartshorne, Alexander Grothendieck, and David Mumford. Equivalent formulations use valuative criteria and projective embeddings as in results due to Nagata and refinements by Raynaud and Gruson.

Basic properties

Proper morphisms are stable under base change, composition, and products; these stability properties are essential in the treatments by Grothendieck and in stacks literature developed at Harvard University and University of Paris-Sud. For proper f: X → Y the direct image of a coherent sheaf is coherent, a finiteness result associated with the work of Jean-Pierre Serre and proven in frameworks used at École Normale Supérieure. Properness implies that for any complete curve or compact variety studied by researchers at Princeton University or University of Cambridge the image behaves analogously to compact images in Cambridge topology. Separatedness, finite type, and universal closedness appear in the treatments by Michel Demazure and in expositions from École Polytechnique seminars.

Examples and non-examples

Standard examples include projective morphisms such as projective space P^n_k → Spec k and closed immersions; classical examples studied by David Mumford and in texts from Harvard and Princeton include projective curves, Abelian varieties, and moduli spaces constructed in the work of Pierre Deligne and Nicholas Katz. Proper non-projective examples arise via Nagata compactification results associated with Masayoshi Nagata and later refinements by Olivier Rolin and Raynaud. Non-examples include affine space A^1_k → Spec k (unless base is proper) and open immersions such as complements of divisors, which feature in counterexamples in seminars at University of Bonn and University of Chicago.

Valuative criterion

The valuative criterion gives an equivalent description: a morphism f: X → Y is proper if and only if for every valuation ring R with field of fractions K and every commutative diagram Spec K → X over Spec R → Y there exists a unique extension Spec R → X. This criterion was articulated in Grothendieck's seminars and used in arguments by Jean-Pierre Serre, Alexander Grothendieck, and Pierre Deligne for uniqueness and existence problems in moduli theory. The valuative criterion is central in compactification results such as those of Nagata and in studies of degenerations by researchers at Institute for Advanced Study and Max Planck Institute for Mathematics.

Stability under operations

Properness is preserved under base change, composition, fiber products, and closed immersions; these closure properties are employed in the construction of stacks and spaces at Princeton and Harvard. Finite morphisms and projective morphisms are proper, and results about properness under limits and descent reflect contributions from the seminars at Institute des Hautes Études Scientifiques and the work of Raynaud and Gruson. Properness is local on the base for the Zariski topology in many standard treatments used at École Normale Supérieure and University of Cambridge.

Properness in related categories

The notion extends to algebraic spaces, Deligne–Mumford stacks, and Artin stacks; foundational treatments appear in the work of Pierre Deligne, David Mumford, and in lecture series at Institute for Advanced Study. Proper algebraic spaces and proper stacks satisfy analogues of the valuative criterion and cohomological finiteness theorems used in the study of moduli of curves and sheaves in projects associated with Moduli of Curves programs and institutions such as Imperial College London and ETH Zurich.

Applications and significance

Properness underpins theorems like the proper base change theorem and cohomology finiteness statements central to arithmetic geometry research at Princeton University, Harvard University, Institut des Hautes Études Scientifiques, and in the proof strategies of the Weil conjectures by Pierre Deligne. Proper morphisms are indispensable in compactification techniques, construction of moduli spaces in Geometric Invariant Theory developed by David Mumford, and in arithmetic applications involving Shimura varieties and Abelian varieties studied at Institute for Advanced Study and Max Planck Institute for Mathematics.

Category:Algebraic geometry