Coherent sheaf

In the mathematical sub-areas of algebraic geometry and complex analysis , coherent sheaves are the analog of finitely generated modules over Noetherian rings .

definition

It is a small space , i.e. H. a topological space together with a sheaf of rings . Then a - module sheaf is called coherent , if ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}_{X}}$ ${\displaystyle {\mathcal {O}}_{X}}$ ${\displaystyle {\mathcal {M}}}$

1. ${\displaystyle {\mathcal {M}}}$is finitely generated, d. H. each point of has an open environment on which a surjection exists, and ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {O}}_{U}^{n}\to {\mathcal {M}}_{|U}}$
2. for every open subset of and every morphism the kernel is finitely generated ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}_{U}^{n}\to {\mathcal {M}}_{|U}}$

properties

• The coherent sheaves form an Abelian category that is stable under extensions . This means in particular: Is

${\displaystyle 0\to {\mathcal {M}}'\to {\mathcal {M}}\to {\mathcal {M}}''\to 0}$

a short exact sequence of modular sheaves, and if two of the three sheaves are coherent, so is the third.

• The bearer of a coherent sheaf is complete. (This applies more generally to any finitely generated module sheaves.)

Coherent sheaves in algebraic geometry

• If there is a locally Noetherian scheme , then the coherent sheaves are precisely those quasi-coherent sheaves which locally correspond to the finitely generated modules. ${\displaystyle X}$
• Theorem of coherence : direct images and higher direct images of coherent sheaves under actual morphisms are coherent, provided that the target schema is locally Noetherian . If, in particular, a Noetherian ring and an actual -scheme, then the cohomology groups of coherent sheaves are finitely generated as -modules. ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle A}$

Coherent sheaves in complex analysis

• Oka's coherence theorem : In contrast to algebraic geometry, the fact that it is itself coherent is not trivial. ${\displaystyle {\mathcal {O}}_{X}}$
• Direct images and higher direct images of coherent sheaves under actual holomorphic images are coherent.

literature

• Hans Grauert, Reinhold Remmert, Coherent Analytic Sheaves . Springer-Verlag, Berlin 1984. ISBN 3-540-13178-7
  General: Appendix, §3; Coherence of the structural grain: Chap. 2, §5; direct images: chap. 10, §4
• A. Grothendieck , J. Dieudonné : Éléments de géométrie algébrique . Publications mathématiques de l'IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967)
  General: 0 _I , 5.3; direct images: III, 3.2