Leray's theorem

The Leray's theorem , named after Jean Leray , is a mathematical theorem in the field of algebraic topology and complex analysis . It's a way of easily finding sheaf cohomologies .

Formulation of the sentence

The following is a sheaf of Abelian groups over a paracompact Hausdorff space . As is well known, the cohomology groups result as an inductive limit of groups , with the open coverages of passing through, which are directed with respect to the refinement . This raises the question of whether there are open overlaps with there, so the inductive limit does not have to be performed. This is indeed the case because: ${\displaystyle {\mathcal {G}}}$ ${\displaystyle M}$ ${\displaystyle H^{q}(M,{\mathcal {G}})}$ ${\displaystyle H^{q}({\mathcal {U}},{\mathcal {G}})}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle H^{q}(M,{\mathcal {G}})=H^{q}({\mathcal {U}},{\mathcal {G}})}$

Theorem of Leray : Let it be a sheaf of Abelian groups over a paracompact Hausdorff space . Let there be an open coverage of , so that for all and coverage sets with the equation ${\displaystyle {\mathcal {G}}}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle M}$ ${\displaystyle q\geq 1}$ ${\displaystyle U_{0},\ldots ,U_{q}\in {\mathcal {U}}}$ ${\displaystyle U_{0}\cap \ldots \cap U_{q}\not =\emptyset }$

${\displaystyle H^{q}(U_{0}\cap \ldots \cap U_{q},{\mathcal {G}})=0}$

applies. Then

${\displaystyle H^{q}(M,{\mathcal {G}})=H^{q}({\mathcal {U}},{\mathcal {G}})}$for everyone . ${\displaystyle q\geq 0}$

If the coverage is such that the sheaf over the intersections of the coverage sets is cohomologically trivial, then the cohomology over the entire space already agrees with the cohomology of the coverage . The proof uses the existence of fine resolutions of a sheaf.

application

A typical example is intended to show how Leray's theorem can be used to calculate cohomology groups. Let it be the complex plane without the zero point. Then applies ${\displaystyle \mathbb {C} ^{*}:=\mathbb {C} \setminus \{0\}}$

${\displaystyle H^{1}(\mathbb {C} ^{*},\mathbb {Z} )\cong \mathbb {Z} }$,

where this is on the left side of the equation for the sheaf of -valent functions. In addition be ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

${\displaystyle U_{1}:=\mathbb {C} ^{*}\setminus \{x\in \mathbb {R} :x<0\},\quad U_{2}:=\mathbb {C} ^{*}\setminus \{x\in \mathbb {R} :x>0\}}$.

Then there is an open cover of . The overlap sets are star-shaped as slotted planes , i.e. simply connected , that is, homotopic and therefore also cohomologically trivial. This fulfills the requirements of Leray's theorem and one obtains . The latter can now be recognized as isomorphic to easily because of the finiteness, as explained in FIG. The cohomology is thus determined with the help of Leray's theorem. ${\displaystyle {\mathcal {U}}:=\{U_{1},U_{2}\}}$ ${\displaystyle \mathbb {C} ^{*}}$ ${\displaystyle H^{1}(\mathbb {C} ^{*},\mathbb {Z} )\cong H^{1}({\mathcal {U}},\mathbb {Z} )}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \mathbb {Z} }$

Individual evidence

1. ^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall 1965, chap. VI, Section D, Theorem 4
2. ^ O. Forster : Riemannsche surfaces , Springer Verlag Heidelberg 1977, ISBN 3-540-08034-1 , Chapter II, §12, example 12.9