Lecture 16, Jul 19, 2021

Sheaves, germs, Cech cohomology

In this concluding lecture we return to the basic notion of the “structural algebra” of a manifold which allows to introduce in the coordinate-free way all other geometric objects. Actually, the structural algebra is too large (and may be too rigid an object) to manipulate with. For instance, the property of a function to be invertible (a unit in the algebra) is rather rare in (the function must be nonvanishing), but may well hold on very large parts of where the function has no zeros.

Do deal with this, an idea of a sheaf was suggested. Morally sheafs are “functions with unspecified domains of definition”. Formally this means the following (we discuss only the sheaf of -smooth functions).
We consider various coverings of the manifold , and instead of a single algebra latex \mathscr C(M)$ we consider collection of algebras , each of which should be thought of as the algebra of -smooth functions defined in the open set . Clearly, these algebras cannot be totally independent. One obvious assumption is that if , then any element from can be “restricted” on , i.e., there exists a natural morphism of algebras with obvious composition rule for three morphisms associated with any triple .
Given this “rectriction” morphisms, for any with a nonempty intersection , we may compare two elements and by looking at their restrictions on . We say that these two elements agree on , if their restrictions coincide. Given a family of elements, we can generalize this notion for unions of several open sets.

Definition. Collection of algebras as above is called a sheaf, if any collection of elements which agree on the intersections, always corresponds to a unique element , which after restriction on each coincides with .

The flexibility contained in this construction allows to simplify formulations of many statements.

Definition. Let be a point on the manifold and a set of functions , each defined (at least) in an open neighborhood of (each for each function) and -smooth in this neighborhood. Two functions are called equivalent, if there exists an open neighborhood , such that on . The equivalence classes are called germ of smooth functions, and the set of all germs is usually denoted by .

Theorem. Germs at a given point form a commutative -algebra with the natural operations. This algebra is local: the set is the unique maximal ideal in it.

The notion of a germ is the realization of the “maximal localization” of objects (functions, vector fields, differential forms) to a smallest, “indivisible” domain of definition which still follows to work with analytic tools (computing derivatives of any order, study convergence of the Taylor series etc.).

Scribbled notes: https://drive.google.com/file/d/18CNcbdFfhd5ctR7K8tPiQkQETZu7T9qR/view?usp=sharing
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