Sheaf cohomology

Sheaf cohomology in mathematics , mainly in algebraic geometry and complex analysis , is a technique with which one can study global properties of topological spaces and sheaves defined on them . In the simplest case, the first cohomology group describes the difficulties in obtaining a global solution from local solutions.

definition

Specifically, a sheaf cohomology on a topological space is a delta functor from the category of sheaves of Abelian groups to the category of Abelian groups. This means: Each sheaf of Abelian groups is functionally assigned a sequence of Abelian groups for , and for each short exact sequence ${\displaystyle X}$ ${\displaystyle F}$ ${\displaystyle H^{k}(X,F)}$ ${\displaystyle k=0,1,2,\dots }$

${\displaystyle 0\to F'\to F\to F''\to 0}$

of sheaves of Abelian groups there is a natural long exact sequence

${\displaystyle 0\to H^{0}(X,F')\to H^{0}(X,F)\to H^{0}(X,F'')\to }$

${\displaystyle \to H^{1}(X,F')\to H^{1}(X,F)\to H^{1}(X,F'')\to }$

${\displaystyle \to H^{2}(X,F')\to \dots }$

In addition, the group of global cuts is from . ${\displaystyle H^{0}(X,F)=\Gamma (X,F)}$ ${\displaystyle F}$

Application examples

Logarithm of a holomorphic function

Problem: Let it be an area and a holomorphic function that does not vanish anywhere. We are looking for a holomorphic function so that applies to all . ${\displaystyle U\subseteq \mathbb {C} }$ ${\displaystyle f\colon U\to \mathbb {C} }$ ${\displaystyle g\colon U\to \mathbb {C} }$ ${\displaystyle f(z)=e^{g(z)}}$ ${\displaystyle z\in U}$

Such a thing always exists locally : If it is chosen to be fixed and so small that , then one can, because the integral is independent of the path ${\displaystyle g}$ ${\displaystyle z_{0}\in U}$ ${\displaystyle \epsilon >0}$ ${\displaystyle U_{\epsilon }(z_{0})\subseteq U}$

${\displaystyle g(z)=a+\int _{z_{0}}^{z}{\frac {f'(\zeta )}{f(\zeta )}}\,d\zeta }$ For ${\displaystyle z\in U_{\epsilon }(z_{0})}$

set, where it is chosen that applies. If you want to define globally according to the same principle, you need that ${\displaystyle a}$ ${\displaystyle f(z_{0})=e^{a}}$ ${\displaystyle g}$

${\displaystyle \int _{\gamma }{\frac {f'(\zeta )}{f(\zeta )}}\,d\zeta }$

disappears for every closed path . If you still divide through , you get a homomorphism ${\displaystyle \gamma }$ ${\displaystyle 2\pi i}$

${\displaystyle c_{f}\colon \pi _{1}(U)\to \mathbb {Z} ,}$

whose disappearance is necessary and sufficient for the existence of a global solution (where is the fundamental group of ). ${\displaystyle g}$ ${\displaystyle \pi _{1}(U)}$ ${\displaystyle U}$

Expressed with the term sheaf, local solvability means that the homomorphism of sheaves from the sheaf of holomorphic functions (with addition as a link) to the sheaf of non-vanishing holomorphic functions (with multiplication) is surjective. Its core is the sheaf of functions, which are locally constant integer multiples of , i.e. except for the multiplication with the constant sheaf . Together, the short exact sequence results ${\displaystyle \exp :{\mathcal {O}}\to {\mathcal {O}}^{*}}$ ${\displaystyle 2\pi i}$ ${\displaystyle 2\pi i}$ ${\displaystyle {\underline {\mathbb {Z} }}}$

${\displaystyle 0\to {\underline {\mathbb {Z} }}{\xrightarrow {2\pi i}}{\mathcal {O}}{\xrightarrow {\exp }}{\mathcal {O}}^{*}\to 0.}$

The given function is now an element of , and an archetype is sought under in . The sheaf cohomology provides an exact sequence ${\displaystyle f}$ ${\displaystyle \Gamma (U,{\mathcal {O}}^{*})}$ ${\displaystyle \exp }$ ${\displaystyle \Gamma (U,{\mathcal {O}})}$

${\displaystyle \Gamma (U,{\mathcal {O}})\to \Gamma (U,{\mathcal {O}}^{*})\to H^{1}(U,{\underline {\mathbb {Z} }}).}$

So it has a holomorphic logarithm if and only if the image of in disappears. This image can be identified with the homomorphism explained above . ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle H^{1}(U,{\underline {\mathbb {Z} }})}$ ${\displaystyle c_{f}}$

Existence of functions with given values

Problem: A sequence of complex numbers without an accumulation point and another sequence of arbitrary complex numbers are given. Then there exists an entire function with for all ? ${\displaystyle a_{1},a_{2},\dots }$ ${\displaystyle b_{1},b_{2},\dots }$ ${\displaystyle f}$ ${\displaystyle f(a_{k})=b_{k}}$ ${\displaystyle k}$

Let it be , and the constant sheaf on be identified with its direct image on . Then the homomorphism , which is given by the evaluation of a function in the points in , is surjective. Because there are no other points in a sufficiently small area of , so that one can choose to use the constant function with value as an archetype for a given value . Let the core of be denoted by, so that we have the short exact sequence ${\displaystyle A=\{a_{1},a_{2},\dots \}}$ ${\displaystyle {\underline {\mathbb {C} }}_{A}}$ ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathcal {O}}\to {\underline {\mathbb {C} }}_{A}}$ ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle a_{k}}$ ${\displaystyle A}$ ${\displaystyle b_{k}}$ ${\displaystyle \Gamma (U,{\mathcal {O}})}$ ${\displaystyle b_{k}}$ ${\displaystyle {\mathcal {O}}\to {\underline {\mathbb {C} }}_{A}}$ ${\displaystyle {\mathcal {O}}(-A)}$

${\displaystyle 0\to {\mathcal {O}}(-A)\to {\mathcal {O}}\to {\underline {\mathbb {C} }}_{A}\to 0}$

receive. An exact sequence is obtained from the sheaf cohomology

${\displaystyle \Gamma (\mathbb {C} ,{\mathcal {O}})\to \Gamma (A,{\underline {\mathbb {C} }})\to H^{1}(\mathbb {C} ,{\mathcal {O}}(-A)).}$

One can now show that vanishes, so every element has an archetype in , i.e. H. every value distribution is implemented by a whole function. ${\displaystyle H^{1}(\mathbb {C} ,{\mathcal {O}}(-A))}$ ${\displaystyle b\in \Gamma (A,{\underline {\mathbb {C} }})}$ ${\displaystyle \Gamma (\mathbb {C} ,{\mathcal {O}})}$ ${\displaystyle b_{1},b_{2},\dots }$

Constructions

Let there be a fixed topological space and a sheaf of Abelian groups , with the stalk from over . ${\displaystyle X}$ ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle F_{x}}$ ${\displaystyle F}$ ${\displaystyle x}$

The Godement dissolution

Define a sheaf on by ${\displaystyle C^{0}(F)}$ ${\displaystyle X}$

${\displaystyle C^{0}(F)(U)=\prod _{x\in U}F_{x}}$

with the projections as restriction images. There is a canonical injective homomorphism that assigns the family of its germs to a cut . The definition of sheaves as etale spaces explains the term "sheaf of discontinuous cuts" for . Sit now ${\displaystyle F\to C^{0}(F)}$ ${\displaystyle s}$ ${\displaystyle (s_{x})}$ ${\displaystyle C^{0}(F)}$

${\displaystyle Z^{1}(F)=C^{0}(F)/F,\qquad C^{1}(F)=C^{0}(Z^{1}(F))}$

and iterative

${\displaystyle Z^{k+1}(F)=C^{k}(F)/Z^{k}(F),\qquad C^{k+1}(F)=C^{0}(Z^{k+1}(F))}$

We get a dissolution

${\displaystyle 0\to F\to C^{0}(F)\to C^{1}(F)\to C^{2}(F)\to \dots }$

Then the sheaf cohomology is defined as the -th cohomology of the complex . ${\displaystyle H^{k}(X,F)}$ ${\displaystyle k}$ ${\displaystyle \Gamma (X,C^{k}(F))}$

The Godement resolution has the advantage that it is easy to define and does not require any choices. However, it is mostly unsuitable for concrete calculations.

Cohomology of a coverage

Let it be a family of open subsets of such that . For and put . This gives a cosimplicial topological space and, by using a simplicial Abelian group, which, according to the Dold-Kan correspondence, corresponds to a coquette complex in non-negative degrees. Its cohomology is the cohomology of with respect to coverage . ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle \textstyle X=\bigcup _{i\in I}U_{i}}$ ${\displaystyle k\geq 0}$ ${\displaystyle j=(i_{0},\dots ,i_{k})\in I^{k+1}}$ ${\displaystyle \textstyle U_{j}=\bigcap _{\nu =0}^{k}U_{i_{\nu }}}$ ${\displaystyle F}$ ${\displaystyle H^{*}(U,F)}$ ${\displaystyle F}$ ${\displaystyle U}$

Specifically, the complex is given by

${\displaystyle C^{k}(U,F)=\prod _{j\in I^{k+1}}\Gamma (U_{j},F)}$

with the differential

${\displaystyle (d^{k}s)(i_{0},\dots ,i_{k+1})=\sum _{\nu =0}^{k+1}(-1)^{k}\rho _{U_{i_{0},\dots ,i_{k+1}}}^{U_{i_{0},\dots ,{\widehat {i_{\nu }}},\dots ,i_{k+1}}}s(i_{0},\dots ,{\widehat {i_{\nu }}},\dots ,i_{k+1})}$

where the restriction of cuts denotes. ${\displaystyle \rho }$ ${\displaystyle F}$

1-coccycles are families with on (with implicit restrictions). Two 1-cocycle are kohomolog when a family is having for all . ${\displaystyle s_{ij}\in \Gamma (U_{ij},F)}$ ${\displaystyle s_{ik}=s_{ij}+s_{jk}}$ ${\displaystyle U_{ijk}}$ ${\displaystyle t_{i}\in \Gamma (U_{i},F)}$ ${\displaystyle s_{ij}=t_{i}-t_{j}}$ ${\displaystyle i,j}$

If there is an overlap with for all , then the canonical homomorphism is bijective for all . This statement, known as Leray's theorem, applies in particular to open affine coverages of separated schemas if, in addition, there is a quasi-coherent module sheaf . ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle H^{k}(U_{i_{1}\dots i_{m}},F)=0}$ ${\displaystyle k\geq 1}$ ${\displaystyle H^{k}(U,F)\to H^{k}(X,F)}$ ${\displaystyle k\geq 0}$ ${\displaystyle F}$

Čech cohomology

If a cover is as in the previous section, then a refinement of a cover is combined with a mapping , so that applies to all . Then we get homomorphisms for all . In principle, Čech cohomology is the direct limit of these refinements. For technical reasons, however, coverages are considered with for all and refinements with for all . Then is called ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle U}$ ${\displaystyle (V_{j})_{j\in J}}$ ${\displaystyle h\colon J\to I}$ ${\displaystyle V_{j}\subseteq U_{h(j)}}$ ${\displaystyle j\in J}$ ${\displaystyle H^{k}(U,F)\to H^{k}(V,F)}$ ${\displaystyle k}$ ${\displaystyle (U_{x})_{x\in X}}$ ${\displaystyle x\in U_{x}}$ ${\displaystyle x\in X}$ ${\displaystyle (V_{x})_{x\in X}}$ ${\displaystyle V_{x}\subseteq U_{x}}$ ${\displaystyle x\in X}$

${\displaystyle {\check {H}}^{k}(X,F)=\varinjlim _{U}H^{k}(U,F)}$

the Čech cohomology of . ${\displaystyle F}$

There are canonical homomorphisms that are for bijective and for injective. If a paracompact Hausdorff space is , they are bijective for everyone . ${\displaystyle {\check {H}}^{k}(X,F)\to H^{k}(X,F)}$ ${\displaystyle k=0,1}$ ${\displaystyle k=2}$ ${\displaystyle X}$ ${\displaystyle k}$

Sheaf cohomology as a derived functor

Let it be a sheaf of rings and a sheaf of modules. (The case of sheaves of Abelian groups is covered by.) Then the category of -module sheaves has enough injective objects that the derived functor for the functor of the global intersections can be formed. In general it is true that the derived functor can be calculated using acyclic resolutions, and one can show that wilted sheaves are acyclic. (A sheaf is called wilted if all open subsets are surjective.) The Godement resolution consists of wilted -module-sheaves, so it is a derived functor, and it doesn't matter whether the derived functor for -modular sheaves or for sheaves is abelscher Forms groups. ${\displaystyle O}$ ${\displaystyle F}$ ${\displaystyle O}$ ${\displaystyle O={\underline {\mathbb {Z} }}}$ ${\displaystyle O}$ ${\displaystyle \Gamma (X,{-})}$ ${\displaystyle F}$ ${\displaystyle \Gamma (X,F)\to \Gamma (U,F)}$ ${\displaystyle U\subseteq X}$ ${\displaystyle O}$ ${\displaystyle H^{k}(X,{-})}$ ${\displaystyle O}$

On a scheme , the functor can be restricted to the category of quasi-coherent module sheaves. If is quasi-compact and separated , has enough injective objects, and the derived functor calculated on is the same as that calculated on the category of all modules. ${\displaystyle \Gamma (X,{-})}$ ${\displaystyle \mathrm {Qcoh} (X)}$ ${\displaystyle X}$ ${\displaystyle \mathrm {Qcoh} (X)}$ ${\displaystyle \mathrm {Qcoh} (X)}$ ${\displaystyle O_{X}}$

More resolutions

Further classes of acyclic sheaves that can thus be used for resolutions are soft sheaves and especially fine sheaves in (complex) analysis .

Non-Abelian H ¹

If a sheaf does not necessarily have to be Abelian groups (in the following written multiplicatively), one can transfer the cover construction at least for . 1 coccycles for a coverage are families that satisfy for everyone . Two cocycles and are called kohomolog if there are such that applies to all . To be cohomologous is an equivalence relation on the 1-coccycles, and the set of equivalence classes is again denoted by. It contains the class of the trivial cocycle as an excellent element. In the direct Limes you get a dotted amount . ${\displaystyle F}$ ${\displaystyle H^{1}}$ ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle c_{ij}\in F(U_{ij})}$ ${\displaystyle c_{ij}c_{jk}=c_{ik}}$ ${\displaystyle i,j,k\in I}$ ${\displaystyle (c_{ij})}$ ${\displaystyle (d_{ij})}$ ${\displaystyle f_{i}\in F(U_{i})}$ ${\displaystyle c_{ij}=f_{i}d_{ij}f_{j}^{-1}}$ ${\displaystyle i,j}$ ${\displaystyle H^{1}(U,F)}$ ${\displaystyle H^{1}(X,F)}$

In the non-Abelian case, there are still exact sequences under different assumptions that generalize the long exact sequence for Abelian sheaves. There is also one for non-Abelian sheaves. See Giraud. ${\displaystyle H^{2}(X,F)}$

Comparison with singular cohomology

If a topological space and an Abelian group, one can form the singular cohomology on the one hand, and the sheaf cohomology of the constant sheaf on the other . The condition that there is a CW complex is sufficient to obtain canonically isomorphic groups, but weaker conditions also suffice. ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle H^{*}(X,A)}$ ${\displaystyle H^{*}(X,{\underline {A}})}$ ${\displaystyle X}$

H ¹ and torsors

Is a small space , i. H. a topological space together with a sheaf of rings , then there is a bijection between canonical and the amount of isomorphism of line bundles on . ${\displaystyle X}$ ${\displaystyle O_{X}}$ ${\displaystyle H^{1}(X,O_{X}^{\times })}$ ${\displaystyle X}$

This statement allows a far-reaching generalization: For every sheaf of groups there is a canonical bijection between and the set of isomorphism classes of torsors. The relation to line bundles arises as follows: If an object is on , then there is a correspondence between locally to isomorphic objects and torsors. The correspondence assigns the torsor to an object . ${\displaystyle G}$ ${\displaystyle H^{1}(X,G)}$ ${\displaystyle G}$ ${\displaystyle E}$ ${\displaystyle X}$ ${\displaystyle E}$ ${\displaystyle E'}$ ${\displaystyle \mathrm {Aut} (E)}$ ${\displaystyle E'}$ ${\displaystyle \mathrm {Isom} (E,E')}$

A torsor for a sheaf of (not necessarily Abelian) groups in a space is a sheaf of sets on together with a - (left) operation, so that there is an open covering of on which becomes trivial. In more detail, this means: there is a sheaf morphism which induces an operation from to for every open subset . For each one should now be isomorphic as a sheaf with -operation to with the left translation as operation. A torsor is trivial if and only then. H. globally isomorphic to with the left translation if not empty. ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle G\times P\to P}$ ${\displaystyle U\subseteq X}$ ${\displaystyle G(U)}$ ${\displaystyle P(U)}$ ${\displaystyle i\in I}$ ${\displaystyle P|_{U_{i}}}$ ${\displaystyle G|_{U_{i}}}$ ${\displaystyle G|_{U_{i}}}$ ${\displaystyle G}$ ${\displaystyle P(X)}$

If there is a system of trivializations, one obtains through a 1-cocyclic, conversely one can use cocyclic to glue trivial torsors together. ${\displaystyle f_{i}:G|_{U_{i}}\to P|_{U_{i}}}$ ${\displaystyle c_{ij}=f_{i}|_{U_{ij}}^{-1}\circ f_{j}|_{U_{ij}}}$

In the logarithm example , the logarithms of form a torsor: For every logarithm on a subset and every integer is also a logarithm, and if is connected, there can be no others. The class of this torsor in is trivial if and only if it has a global logarithm. ${\displaystyle f}$ ${\displaystyle {\underline {\mathbb {Z} }}}$ ${\displaystyle g}$ ${\displaystyle V}$ ${\displaystyle k}$ ${\displaystyle g+2\pi ik}$ ${\displaystyle V}$ ${\displaystyle H^{1}(U,{\underline {\mathbb {Z} }})}$

Higher direct images

If there is a continuous mapping and a sheaf of Abelian groups , then the direct image is a left exact functor and the derived functor can be formed. He is the cortex of the prowl . ${\displaystyle f\colon X\to Y}$ ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle f_{*}}$ ${\displaystyle R^{k}f_{*}}$ ${\displaystyle U\mapsto H^{k}(f^{-1}(U),F)}$

The cohomology of and the cohomology of are related to one another via the Leray spectral sequence : There is also a spectral sequence that converges to. ${\displaystyle F}$ ${\displaystyle f_{*}F}$ ${\displaystyle E_{2}^{pq}=H^{p}(Y,R^{q}f_{*}F)}$ ${\displaystyle H^{*}(X,F)}$

Important sentences about sheaf cohomology

Algebraic Geometry

• If a quasi-coherent module sheaf is on an affine schema , then it is for all . ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle H^{k}(X,F)=0}$ ${\displaystyle k\geq 1}$
• If there is a schema, the underlying space of which is Noetherian and has dimensions , then there are abelian groups for and for each . ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle H^{k}(X,F)=0}$ ${\displaystyle k>n}$ ${\displaystyle F}$
• Grothendieck's coherence theorem: If it is actually above a Noetherian ring and a coherent module sheaf, then there is a finitely generated module for each . ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle F}$ ${\displaystyle H^{k}(X,F)}$ ${\displaystyle k}$ ${\displaystyle A}$
• Serre's vanishing theorem: For a coherent sheaf on a projective scheme is for and . ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle H^{k}(X,F(n))=0}$ ${\displaystyle k\geq 1}$ ${\displaystyle n\gg 0}$
• Serre duality
• Hans Grauert's theorem of semi-continuity
• Serres GAGA and Grothendiecks GFGA

Complex analysis

• Theorem B by Henri Cartan : For coherent sheaves on stone spaces , the higher cohomology vanishes
• Cartan-Serre's theorem of finiteness: coherence groups of coherent sheaves on compact complex spaces are finite-dimensional (as vector spaces), generalized in Grauert's coherence theorem ${\displaystyle \mathbb {C} }$
• Grauert's theorem of semi-continuity
• Hodge theory

literature

• Glen Bredon , Sheaf Theory. New York 1997
• Jean Giraud , cohomology non abélienne. Berlin 1971
• Roger Godement , Topologie algébrique et théorie des faisceaux. Paris 1958
• Phillip Griffiths , Joseph Harris , Principles of Algebraic Geometry. New York 1994
• Alexander Grothendieck , Jean Dieudonné , Éléments de géométrie algébrique . Publications mathématiques de l'IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967)
• Alexander Grothendieck, Sur quelques points d'algèbre homologique. Tohôku Math. J., II. Ser. 9, 119-221 (1957)
• Masaki Kashiwara , Pierre Schapira , Sheaves on Manifolds. Berlin 1990
• Robert Wayne Thomason , Thomas Trobaugh , Higher Algebraic K-Theory of Schemes and of Derived Categories. In: The Grothendieck Festschrift, Volume III. Boston 1990

Footnotes

1. ^ Godement, II, 5.9
2. ↑ See Thomason, Trobaugh, Appendix B
3. ↑ See Bredon, III, 1.
4. ↑ See Giraud, III, 2.5.1 for the exact requirements.