Last updates: 9 September 2012
Let be a topological space with topology . View as a category with morphisms inclusions of open sets .
Let be the category of commutative rings with identity.
Another way to state the overlap condition in the definition of a sheaf is that if is an open cover of and are such that then there is a unique such that for all .
Yet another way of stating the overlap condition in the definition of a sheaf is to say that the sequence is exact, where
Let be continuous.
The direct image
is given by ,
for an open set of .
The inverse image is the adjoint of ,
,
constructed by
, where .
The global sections functor is the direct image of ,
, where .
The constant sheaf functor is the inverse image of ,
, where .
The skyscraper sheaf functor is the direct image of ,
, where
The stalk functor is the inverse image of
where
HW 1: Show that the global sections of a sheaf on is the ring .
HW 2: Show that the constant sheaf on with values in is given by
HW 3: Show that the skyscraper sheaf at is given by
HW 4: Show that the stalk of at is given by
The inclusion, (or restriction, or forgetful) functor is
Sheafification is the left adjoint functor to inclusion given by
HW 5: Show that the sheafification of a presheaf is given by
see Macdonald.
HW 6: Show that
HW 7:
Let .
Define sheaves and on by
for open in . Thus
Show that
is an exact sequence of sheaves on which is not an exact sequence of presheaves on .
Show that
and in is generated by the function
which is a function in not in the image of in .
The section on sheaves and ringed spaces follows the presentations in [Go, §1.9], [Mac, ???] and [Bo, §AG4.2]. It is common to define a presheaf as a (contravariant) functor . Historically this was done because the language of categories was unfamiliar to most working research mathematicians, but this is no longer the case.
The section on direct and inverse images is based on the exposition of sheaves and presheaves in [Weibel]: the definitions of and are in [Weibel, Def. 1.6.5], HW 2 is [Weibel, Ex. 1.6.2], HW 7 is between [Weibel, Ex. 1.6.2] and [Weibel, Def. 1.6.6], the definition of sheafification is in [Weibel, Example 1.6.7], HW 3 and 4 are [Weibel, Example 2.3.2 and Exercise 2.3.6], the global sections functor is defined in [Weibel, Application 2.5.4] and HW 1 is in [Weibel, Exercise 2.6.3] and the direct image and inverse image are in [Weibel, Application 2.6.6].
The section on Presheaves and Sheafification is also based on the treatment in [Weibel].
[Bo] A. Borel, Linear Algebraic Groups, Section AG4.2, Graduate Texts in Mathematics 126, Springer-Verlag, Berlin, 1991, MR??????
[Go] R. Godement, Topologie algébrique et théorie des faisceaux, Section 1.9, Actualités scientifiques et industrielles 1252, Hermann, Paris, 1958. MR??????