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 and point of . We would like to produce a global meromorphic function on on with a given principal part at . Locally we can always do it. We cover by open coordinate charts so that belongs to only one chart . We can choose meromorphic functions on so that the principal part of at is if belongs to . Of course, in general, these local meromorphic functions cannot be patched together to give a global meromorphic function with principal part at . The discrepancies are given by
on , which is a holomorphic function on . Observe that is skew-symmetric in and and
on . If we can modify on by a holomorphic function on to make the discrepancies vanish, then we can piece the local meromorphic functions together to form a global meromorphic function. In other words, we want to find a holomorphic function on so that
Then the meromorphic function which is equal to is a global meromorphic function whose principal part at is .
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 of a topological space . Suppose we have two functions and defined respectively on open neighborhoods and of . We introduce the following equivalence relation. The two functions and are equivalent if there exists some open neighborhood of in so that . By the germ of a function at , we mean an equivalence class of functions defined on open neighborhoods of 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 of . Consider the set of germs at of all holomorphic functions of defined locally near . This set is simply the set of all convergent power series centered at . We denote this set by or . We denote the union of for all by or . There is a natural projection so that maps the element of to . 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 is an open subset of and we want to find a global holomorphic function on . We have to find for every in an element of 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 over . We have to introduce a criterion to determine whether this section defines a holomorphic function on . A section of over simply means a choice of a convergent power series for every point of , 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 so that the sections we want are precisely the continuous sections. Why is the imposition of a topology on the best way to do it? To demand that a section of over is continuous with respect to a topology means that when and are two points of that are sufficiently close together, we demand that and 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 and 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 and is that the expansion at of the -centered convergent power series is . Closeness is determined by open sets of the topology. So we define our topology of as follows. We do it by specifying an open neighborhood basis of every point of . Take an element of . Then is represented by a convergent power series centered at some point of with a domain of convergence . For , let be the expansion at of the power series . Let be the set of all open neighborhoods of in . For every , let . Then an open neighborhood basis of in our topology is . It is straightforward to check that this gives a well-defined topology and it is the largest topology on so that every holomorphic function on an open subset of defines a continuous section of over . 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 of is represented by a holomorphic function on , and is the element of induced by the holomorphic function .
This topology of makes a local homeomorphism, because clearly for every element of , the restriction of to maps homeomorphically onto . We would like to point out that even though is a local homeomorphism, yet is not a topological covering map. The reason is that for a topological covering map , one must have the property that for every , there is a connected open neighborhood of in so that the map maps every connected component of homeomorphically onto . In particular, for every point , the map maps some open neighborhood of in homeomorphically onto , and the image of the homeomorphism is the same for all . In the case of , the image of the homeomorphism must be at , and we cannot have the same for all elements of . So is not a topological covering map. For every element of , the connected component of containing corresponds precisely to the maximum analytic continuation of the germ . The domain of definition of this maximum analytic in general of course is not the same as .
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 of is an algebra over , 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 compatible with its ring structure. In particular, is an additive abelian group. The algebraic structure of the fibers of is compatible with the topology of in the sense that the map from the fiber product
to defined by addition or multiplication is continuous. Moreover, the map defined by scalar multiplication is continuous. Now we state the abstract definition of a sheaf.
Definition. Let be a topological space. A sheaf of abelian groups over is a topological space with a local homeomorphism so that every fiber of is an abelian group and the map from the fiber product to 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 with the projection and its topology and the algebraic operations on the fibers of is a sheaf of -algebras over .
The fiber over of a sheaf over is usually referred to as the stalk of at and is denoted by . For an open subset of the restriction of to denoted by is the sheaf over with the projection induced by and the topology and algebraic structure induced from .
In the discussion of the construction of the sheaf of germs of holomorphic functions on , for every open subset of , one considers a class of functions on , and this class of functions is the set of all holomorphic functions on . 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 to a subset of . Having a class of functions on and knowing how to restrict a function on to a subset of are the only two ingredients needed to construct the sheaf . We formulate these two ingredients in an abstract definition of a presheaf.
Definition. Let be a topological space. A presheaf of abelian groups over is an assignment to every open subset of an abelian group and an assignment to every inclusion map a homomorphism of abelian groups so that the composite of and is when one has inclusion maps . In other words, is a functor from the category of open subsets of and inclusions to the category of abelian groups and homomorphisms.
In most of the applications in complex analytic geometry, the topological space is, not surprisingly, a complex manifold. In order to avoid spending time to handle exceptional pathological cases, we assume that the base topological space is always a locally compact metrizable topological space.
An example of a presheaf is and all holomorphic functions on . In general, from a presheaf , one can construct an associated sheaf in the following way. The fiber over the point of is the direct limit of for in the directed set of all open neighborhoods of in . An open neighborhood basis of the point of coming from an element of consists of all sets of the form , where is an open neighborhood of in . Conversely, when we are given a sheaf , we can construct a presheaf whose associated sheaf is by setting equal to the set of all continuous sections of over .
In general, the topology of a sheaf may not be Hausdorff. Consider the sheaf of germs of functions on . It comes from the presheaf which assigns to every open subset of the set of all functions on . Let be the function defined by for and otherwise. Let be the germ of at , and let be the germ at of the function that is identically zero. Then the elements and of the stalk cannot be separated by open subsets, because an open neighborhood of contains the set
for some open neighborhood of in , and an open neighborhood of contains the set
for some open neighborhood of in . The germs of and the zero function coincide at any point in with and this common germ belong to both and . However, the topology of the sheaf 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 and over a topological space . A sheaf-homomorphism is a continuous map so that the map between the stalks at every point of induced by is a group homomorphism. Since both and are locally homeomorphic to under their respective projections, it follows a sheaf-homomorphism between and 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.