Sheaf Cohomology – I. The Concept of a Sheaf

The tools of sheaves and sheaf cohomology were introduced to patch together local objects (holomorphic functions, holomorphic sections of holomorphic vector bundles) to form global objects. Let us consider first a simple example. Suppose we have a Riemann surface {M} and point {P} of {M}. We would like to produce a global meromorphic function on {f} on {M} with a given principal part {p} at {P}. Locally we can always do it. We cover {M} by open coordinate charts {\{U_\alpha\}} so that {P} belongs to only one chart {U_{\alpha_0}}. We can choose meromorphic functions {f_\alpha} on {U_\alpha} so that the principal part of {f_\alpha} at {P} is {p} if {P} belongs to {U_\alpha}. Of course, in general, these local meromorphic functions {f_\alpha} cannot be patched together to give a global meromorphic function with principal part {p} at {P}. The discrepancies are given by

\displaystyle f_{\alpha\beta} = f_\beta - f_\alpha

on {U_\alpha \cap U_\beta}, which is a holomorphic function on {U_\alpha \cap U_\beta}. Observe that {f_{\alpha\beta}} is skew-symmetric in {\alpha} and {\beta} and

\displaystyle f_{\alpha\beta} + f_{\beta\gamma} + f_{\gamma\alpha} = 0

on {U_\alpha \cap U_\beta \cap U_\gamma}. If we can modify {f_\alpha} on {U_\alpha} by a holomorphic function {g_\alpha} on {U_\alpha} to make the discrepancies vanish, then we can piece the local meromorphic functions {f_\alpha} together to form a global meromorphic function. In other words, we want to find a holomorphic function {g_\alpha} on {U_\alpha} so that

\displaystyle f_{\alpha\beta} = g_\beta - g_\alpha.

Then the meromorphic function which is equal to {f_\alpha - g_\alpha} is a global meromorphic function whose principal part at {P} is {p}.

Now when we choose the local meromorphic functions, it does not matter how big the open set is on which a local meromorphic function is defined as long as we can make the choice for some open neighborhood of each point. In order to minimize the role played by the size of the open neighborhood of a point on which the local object is defined, we introduce the concept of a germ. Fix a point {P} of a topological space {M}. Suppose we have two functions {f} and {g} defined respectively on open neighborhoods {U_f} and {U_g} of {P}. We introduce the following equivalence relation. The two functions {f} and {g} are equivalent if there exists some open neighborhood {W} of {P} in {U_f \cap U_g} so that {f|W = g|W}. By the germ of a function at {P}, we mean an equivalence class of functions defined on open neighborhoods of {P} in the equivalence relation given above. Our goal is to try to piece together germs of a certain class of functions (i.e. holomorphic functions) to form global functions in the same class. Let us conduct our discussion by looking at an example. Take a point {P} of {\mathbb{C}}. Consider the set of germs at {P} of all holomorphic functions of {\mathbb{C}} defined locally near {P}. This set is simply the set of all convergent power series centered at {P}. We denote this set by {\mathcal{O}_P} or {\mathcal{O}_{\mathbb{C},\,P}}. We denote the union of {\mathcal{O}_P} for all {P \in \mathbb{C}} by {\mathcal{O}} or {\mathcal{O}_\mathcal{C}}. There is a natural projection {\pi: \mathcal{O} \rightarrow \mathbb{C}} so that {\pi} maps the element of {\mathcal{O}_P} to {P}. Our goal of the whole process is to find global objects. We want to piece the germs of the holomorphic functions together to form global holomorphic functions. Suppose {G} is an open subset of {\mathbb{C}} and we want to find a global holomorphic function {f} on {G}. We have to find for every {P} in {G} an element of {\mathcal{O}_P} and do it in such a way that they form a global holomorphic function. In other words, we would like to find a section of {\pi: \mathcal{O} \rightarrow \mathbb{C}} over {G}. We have to introduce a criterion to determine whether this section defines a holomorphic function on {G}. A section of {\pi: \mathcal{O} \rightarrow \mathbb{C}} over {G} simply means a choice of a convergent power series for every point of {G}, and these convergent power series may be completely independent of each other. It turns out the best way to introduce a criterion is to impose some topology on {\mathcal{O}} so that the sections we want are precisely the continuous sections. Why is the imposition of a topology on {\mathcal{O}} the best way to do it? To demand that a section {s} of {\pi: \mathcal{O} \rightarrow \mathbb{C}} over {G} is continuous with respect to a topology means that when {P} and {Q} are two points of {\mathbb{C}} that are sufficiently close together, we demand that {\pi(P)} and {\pi(Q)} are close together in a sense we are free to impose via the topology. We can achieve our goal by suitably interpreting the closeness of {\pi(P)} and {\pi(Q)} via the imposed topology, and the concept of continuity is one of the simplest in mathematics. So doing it through topology is the simplest way and gives us the most flexibility to develop a general theory. In our case, the closeness between {\pi(P)} and {\pi(Q)} is that the expansion at {Q} of the {P}-centered convergent power series {\pi(P)} is {\pi(Q)}. Closeness is determined by open sets of the topology. So we define our topology of {\mathcal{O}} as follows. We do it by specifying an open neighborhood basis of every point of {\mathcal{O}}. Take an element {f} of {\mathcal{O}}. Then {f} is represented by a convergent power series centered at some point {P} of {\mathbb{C}} with a domain of convergence {U}. For {Q \in U}, let {g_Q} be the expansion at {Q} of the power series {f}. Let {\mathcal{W}} be the set of all open neighborhoods {W} of {P} in {U}. For every {W \in \mathcal{W}}, let {\tilde{W} = \{g_Q : Q \in W\}}. Then an open neighborhood basis of {f} in our topology is {\{\tilde{W} : W \in \mathcal{W}\}}. It is straightforward to check that this gives a well-defined topology and it is the largest topology on {\mathcal{O}} so that every holomorphic function on an open subset {G} of {\mathbb{C}} defines a continuous section of {\pi: \mathcal{O} \rightarrow \mathbb{C}} over {G}. This way of defining a topology can be done in very general situations, because the use of convergent power series is purely of an illustrative nature and is unnecessary. One can say that the element {f} of {\mathcal{O}} is represented by a holomorphic function {f} on {U}, and {g_Q} is the element of {\mathcal{O}_Q} induced by the holomorphic function {f}.

This topology of {\mathcal{O}} makes {\pi: \mathcal{O} \rightarrow \mathbb{C}} a local homeomorphism, because clearly for every element {f} of {\mathcal{O}}, the restriction of {\pi} to {\tilde{W}} maps {\tilde{W}} homeomorphically onto {W}. We would like to point out that even though {\pi: \mathcal{O} \rightarrow \mathbb{C}} is a local homeomorphism, yet {\pi: \mathcal{O} \rightarrow \mathbb{C}} is not a topological covering map. The reason is that for a topological covering map {\sigma: X \rightarrow Y}, one must have the property that for every {y \in Y}, there is a connected open neighborhood {W} of {y} in {Y} so that the map {\sigma} maps every connected component of {\sigma^{-1}(W)} homeomorphically onto {W}. In particular, for every point {x \in \sigma^{-1}(y)}, the map {\sigma} maps some open neighborhood {W_x} of {x} in {X} homeomorphically onto {W}, and the image {W} of the homeomorphism {\sigma|W_x} is the same for all {x \in \sigma^{-1}(y)}. In the case of {\pi: \mathcal{O} \rightarrow \mathbb{C}}, the image {W} of the homeomorphism {\pi|\tilde{W}} must be at {P}, and we cannot have the same {W} for all elements of {\mathcal{O}_P}. So {\pi: \mathcal{O} \rightarrow \mathbb{C}} is not a topological covering map. For every element {f} of {\mathcal{O}}, the connected component of {\mathcal{O}} containing {f} corresponds precisely to the maximum analytic continuation of the germ {f}. The domain of definition of this maximum analytic in general of course is not the same as {\mathbb{C}}.

To piece together local functions to form global functions, we have to look at the discrepancies obtained by taking the differences of local functions. So we have to consider the algebraic process of addition. Every fiber {\mathcal{O}_P} of {\pi: \mathcal{O} \rightarrow \mathbb{C}} is an algebra over {\mathbb{C}}, i.e. the operations of addition and multiplication in it make it a ring, and at the same time, it is a vector space over {\mathbb{C}} compatible with its ring structure. In particular, {\mathcal{O}_P} is an additive abelian group. The algebraic structure of the fibers of {\pi: \mathcal{O} \rightarrow \mathbb{C}} is compatible with the topology of {\mathcal{O}} in the sense that the map from the fiber product

{\mathcal{O} \times_\pi \mathcal{O} = \{(f, g) \in \mathcal{O} \times_\pi \mathcal{O} :\pi(f) = \pi(g)\}}

to {\mathcal{O}} defined by addition or multiplication is continuous. Moreover, the map {\mathbb{C} \times \mathcal{O} \rightarrow \mathcal{O}} defined by scalar multiplication is continuous. Now we state the abstract definition of a sheaf.

Definition. Let {M} be a topological space. A sheaf of abelian groups over {M} is a topological space {\mathcal{S}} with a local homeomorphism {\pi: \mathcal{S} \rightarrow M} so that every fiber of {\pi: \mathcal{S} \rightarrow M} is an abelian group and the map from the fiber product {\mathcal{S} \times_\pi \mathcal{S}} to {\mathcal{S}} defined by the addition operation is continuous.

Similarly one can define a sheaf of rings, a sheaf of vector spaces, a sheaf of algebras, etc. so that every fiber carries the algebraic structure of a ring, a vector space, an algebraic, etc. and the algebraic operations are continuous. The set {\mathcal{O}} with the projection {\pi: \mathcal{O} \rightarrow \mathbb{C}} and its topology and the algebraic operations on the fibers of {\pi: \mathcal{O} \rightarrow \mathbb{C}} is a sheaf of {\mathbb{C}}-algebras over {\mathbb{C}}.

The fiber over {P} of a sheaf {\pi: \mathcal{S} \rightarrow M} over {M} is usually referred to as the stalk of {\mathcal{S}} at {P} and is denoted by {\mathcal{S}_P}. For an open subset {W} of {M} the restriction of {\mathcal{S}} to {W} denoted by {\mathcal{S}|W} is the sheaf {\pi^{-1}(W)} over {M} with the projection induced by {\pi} and the topology and algebraic structure induced from {\mathcal{S}}.

In the discussion of the construction of the sheaf {\mathcal{O}_\mathbb{C}} of germs of holomorphic functions on {\mathbb{C}}, for every open subset {G} of {\mathbb{C}}, one considers a class of functions on {G}, and this class of functions is the set of all holomorphic functions on {G}. Then one introduces the concept of a germ of a function in that class. For this step, one has to know how to restrict a function on {G} to a subset of {G}. Having a class of functions on {G} and knowing how to restrict a function on {G} to a subset of {G} are the only two ingredients needed to construct the sheaf {\mathcal{O}_\mathbb{C}}. We formulate these two ingredients in an abstract definition of a presheaf.

Definition. Let {M} be a topological space. A presheaf of abelian groups over {M} is an assignment to every open subset {U} of {M} an abelian group {\mathcal{P}_U} and an assignment to every inclusion map {V \rightarrow U} a homomorphism of abelian groups {\mathcal{P}_V \rightarrow \mathcal{P}_U} so that the composite of {\mathcal{P}_W \rightarrow \mathcal{P}_V} and {\mathcal{P}_V \rightarrow \mathcal{P}_U} is {\mathcal{P}_W \rightarrow \mathcal{U}} when one has inclusion maps {W \rightarrow V \rightarrow U}. In other words, {\mathcal{P}} is a functor from the category of open subsets of {M} and inclusions to the category of abelian groups and homomorphisms.

In most of the applications in complex analytic geometry, the topological space {M} is, not surprisingly, a complex manifold. In order to avoid spending time to handle exceptional pathological cases, we assume that the base topological space {M} is always a locally compact metrizable topological space.

An example of a presheaf is {M = \mathbb{C}} and {\mathcal{P}_U =} all holomorphic functions on {U}. In general, from a presheaf {\mathcal{P}}, one can construct an associated sheaf {\mathcal{S}} in the following way. The fiber {\mathcal{S}_P} over the point {P} of {M} is the direct limit of {\mathcal{P}_U} for {U} in the directed set of all open neighborhoods of {P} in {M}. An open neighborhood basis of the point {s} of {\mathcal{S}_P} coming from an element {s} of {\mathcal{P}_U} consists of all sets of the form {\{\text{image of }s \in \mathcal{S}_Q : Q \in W\}}, where {W} is an open neighborhood of {P} in {U}. Conversely, when we are given a sheaf {\mathcal{S}}, we can construct a presheaf {\mathcal{P}} whose associated sheaf is {\mathcal{S}} by setting {\mathcal{P}_U} equal to the set of all continuous sections of {\mathcal{S}} over {U}.

In general, the topology of a sheaf may not be Hausdorff. Consider the sheaf {\mathcal{E}} of germs of {C^\infty} functions on {\mathbb{R}}. It comes from the presheaf which assigns to every open subset {G} of {\mathbb{R}} the set of all {C^\infty} functions on {G}. Let {f} be the function defined by {f(x) = \exp(-{1/x^2})} for {x > 0} and {f(x) = 0} otherwise. Let {f} be the germ of {f} at {x = 0}, and let {\boldsymbol{0}} be the germ at {x=0} of the function that is identically zero. Then the elements {f} and {\boldsymbol{0}} of the stalk {\mathcal{E}_0} cannot be separated by open subsets, because an open neighborhood of {f} contains the set

\displaystyle \tilde{G} = \{\text{the germ of }f\text{ at }x\text{ }:\text{ }x \in G\},

for some open neighborhood {G} of {0} in {\mathbb{R}}, and an open neighborhood of {\boldsymbol{0}} contains the set

\displaystyle \tilde{H} = \{\text{the germ of the zero function at }x\text{ }:\text{ }x \in H\},

for some open neighborhood {H} of {0} in {\mathbb{R}}. The germs of {f} and the zero function coincide at any point {x} in {G \cap H} with {x < 0} and this common germ belong to both {\tilde{G}} and {\tilde{H}}. However, the topology of the sheaf {\mathcal{O}_\mathbb{C}} is Hausdorff because a holomorphic function on a domain is identically zero everywhere if it is identically zero on a nonempty open set.

Suppose we have two sheaves of abelian groups {\mathcal{S}} and {\mathcal{T}} over a topological space {M}. A sheaf-homomorphism {\varphi: \mathcal{S} \rightarrow \mathcal{T}} is a continuous map so that the map {\varphi_P: \mathcal{S}_P \rightarrow \mathcal{T}_P} between the stalks at every point {P} of {M} induced by {\varphi} is a group homomorphism. Since both {\mathcal{S}} and {\mathcal{T}} are locally homeomorphic to {M} under their respective projections, it follows a sheaf-homomorphism between {\mathcal{S}} and {\mathcal{T}} is a local homeomorphism. It is clear that the kernel, the image, and the cokernel of a sheaf-homomorphism with the induced topology and algebraic structure are sheaves over the same base space.

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