Fine sheaf

A fine sheaf is a mathematical term from the field of algebraic topology and function theory . It is a sheaf with an additional property. With the help of such sheaves, the sheaf cohomology can also be calculated for general sheaves on paracompact Hausdorff spaces .

definition

Let there be a topological space and a sheaf of Abelian groups above . ${\displaystyle M}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle M}$

Is a locally finite open cover of so called a family of Garbenmorphismen one of the overlap subordinate partition of unity if the following applies: ${\displaystyle {\mathcal {U}}=(U_{a})_{a\in A}}$ ${\displaystyle M}$ ${\displaystyle (\eta _{a})_{a\in A}}$ ${\displaystyle \eta _{a}:{\mathcal {G}}\rightarrow {\mathcal {G}}}$

• For all there is an open environment of , so that for all where the stalk is over and the morphisms induced on the stalks are also indicated with. ${\displaystyle a\in A}$ ${\displaystyle V_{a}}$ ${\displaystyle M\setminus U_{a}}$ ${\displaystyle \eta _{a}(x)=0}$ ${\displaystyle x\in {\mathcal {G}}_{p},p\in V_{a}}$ ${\displaystyle {\mathcal {G}}_{p}}$ ${\displaystyle p\in M}$ ${\displaystyle \eta _{a}}$
• ${\displaystyle \sum _{a\in A}\eta _{a}(x)=x}$for everyone . ${\displaystyle x\in {\mathcal {G}}_{p},p\in M}$

Note that the sum in the above definition, because of the local finiteness of the coverage, always only has finitely many summands different from 0 and is therefore well defined.

The sheaf above is called fine if there is a subordinate partition of one for every locally finite, open covering of . ${\displaystyle {\mathcal {G}}}$ ${\displaystyle M}$ ${\displaystyle M}$

Examples

• If a normal Hausdorff space is , then the sheaf of continuous functions is over fine. ${\displaystyle M}$ ${\displaystyle M}$
• If there is a C ^∞ -manifold , then the sheaf of infinitely often differentiable functions is fine. ${\displaystyle M}$ ${\displaystyle {\mathcal {C}}^{\infty }}$
• The sheaf of holomorphic functions over a Riemann surface is not fine. ${\displaystyle {\mathcal {O}}}$

Sentences and Applications

Since the paracompact Hausdorff spaces, by definition, have a sufficient number of locally finite overlaps, it stands to reason that one can prove strong theorems about fine sheaves in such spaces.

• If a fine sheaf is above a paracompact Hausdorff space, then the sheaf cohomology applies to all . ${\displaystyle {\mathcal {G}}}$ ${\displaystyle H^{q}(M,{\mathcal {G}})=0}$ ${\displaystyle q>0}$

That doesn't apply to, because that's the group of global cuts . This can be used to show the following sentence ${\displaystyle q=0}$ ${\displaystyle H^{0}(M,{\mathcal {G}})}$

• Is a sheaf over a paracompact Hausdorff space and ${\displaystyle {\mathcal {G}}}$

${\displaystyle 0\rightarrow {\mathcal {G}}\rightarrow {\mathcal {G}}_{0}\,{\xrightarrow {d_{0}}}\,{\mathcal {G}}_{1}\,{\xrightarrow {d_{1}}}\,{\mathcal {G}}_{2}\,{\xrightarrow {d_{2}}}\,\ldots }$

a fine sheaf resolution, that is, all sheaves are fine and all sheaf morphisms are exact , whereby exactness should apply here for every stalk, so each induces a mapping between the groups of global sections, and it applies ${\displaystyle {\mathcal {G}}_{q}}$ ${\displaystyle d_{q}}$ ${\displaystyle d_{q}}$ ${\displaystyle d_{q}^{*}:\Gamma (M,{\mathcal {G}}_{q})\rightarrow \Gamma (M,{\mathcal {G}}_{q+1})}$

${\displaystyle H^{q}(M,{\mathcal {G}})\cong \mathrm {ker} (d_{q}^{*})/\mathrm {im} (d_{q-1}^{*})}$.

One can further show that there is a fine resolution for every sheaf over a paracompact Hausdorff space, so that the above theorem can in principle always be used to calculate cohomology groups. A typical application example is the fine resolution

${\displaystyle 0\rightarrow {\mathcal {O}}\rightarrow {\mathcal {C}}^{\infty }\,{\xrightarrow {\overline {\partial }}}\,{\mathcal {C}}^{\infty }\rightarrow 0}$

the sheaf of holomorphic functions over a domain , where be the differential operator . This results in ${\displaystyle M\subset \mathbb {C} }$ ${\displaystyle {\overline {\partial }}}$ ${\displaystyle \textstyle {\frac {1}{2}}({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}})}$

• ${\displaystyle H^{0}(M,{\mathcal {O}})=\Gamma (M,{\mathcal {O}})}$
• ${\displaystyle H^{1}(M,{\mathcal {O}})=\Gamma (M,{\mathcal {C}}^{\infty })/{\overline {\partial }}\Gamma (M,{\mathcal {C}}^{\infty })}$
• ${\displaystyle H^{q}(M,{\mathcal {O}})=0}$for everyone . ${\displaystyle q\geq 2}$

Since, according to the so-called lemma Dolbeault the differential equation for predetermined functions on in releasably is valid , and therefore even for all of areas . ${\displaystyle {\overline {\partial }}f=g}$ ${\displaystyle C^{\infty }}$ ${\displaystyle g}$ ${\displaystyle M}$ ${\displaystyle C^{\infty }}$ ${\displaystyle {\overline {\partial }}\Gamma (M,{\mathcal {C}}^{\infty })=\Gamma (M,{\mathcal {C}}^{\infty })}$ ${\displaystyle H^{q}(M,{\mathcal {O}})=0}$ ${\displaystyle q\geq 1}$ ${\displaystyle M\subset \mathbb {C} }$

Individual evidence

1. ↑ Robert C. Gunning: Lectures on Riemann surfaces , BI university pocket books , Volume 837 (1972), ISBN 3-411-00837-7 , Chapter III.4: Fine sheaves
2. ↑ Robert C. Gunning: Lectures on Riemann surfaces , BI university pocket books , Volume 837 (1972), ISBN 3-411-00837-7 , Chapter III.4, 2nd sentence
3. ↑ Robert C. Gunning: Lectures on Riemann surfaces , BI university pocket books , Volume 837 (1972), ISBN 3-411-00837-7 , Chapter III.4, 2nd sentence
4. ↑ Robert C. Gunning: Lectures on Riemann surfaces , BI university pocket books , Volume 837 (1972), ISBN 3-411-00837-7 , Chapter III.5
5. ^ O. Forster: Riemannsche surfaces , Springer Verlag Heidelberg 1977, ISBN 3-540-08034-1 , Chapter II, §13