Sheaves and Ringed Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 9 September 2012

Sheaves and ringed spaces

Let X be a topological space with topology 𝒯. View 𝒯 as a category with morphisms inclusions of open sets UV.

Let 𝒜 be the category of commutative rings with identity.

Another way to state the overlap condition in the definition of a sheaf is that if {Uα} is an open cover of U and fα 𝒪X(Uα) are such that fα| UαUβ = fβ| UαUβ ,for all α,β, then there is a unique f𝒪X (U) such that fα= f| Uα for all α.

Yet another way of stating the overlap condition in the definition of a sheaf is to say that the sequence 0𝒪X(U) i α 𝒪X(Uα) - k j α,β 𝒪X(Uα Uβ) is exact, where

  1. i is the map induced by the inclusions UαU,
  2. j is the map induced by the inclusions UαUβ Uα,
  3. k is the map induced by the inclusions UαUβ Uβ,
and exactness of the sequence means imi=ker( j-k).

Direct and inverse image

Let f:XY be continuous.

The direct image

f:Sheaves(X) Sheaves(Y) is given by (f)(V)= (f-1(V)) ,

for V an open set of Y.

The inverse image f-1:Sheaves(Y) Sheaves(X) is the adjoint of f,

HomSh(X) ( f-1𝒢, ) HomSh(Y) ( 𝒢,f ) ,

constructed by

f-1𝒢=sheafification() , where (U)= limV(U) 𝒢(V) .

The global sections functor Γ:Sheaves(X) AbelGps is the direct image of f:Xpt,

Γ=f, where f:Xpt.

The constant sheaf functor A:AbelGpsSheaves(X) is the inverse image of f:Xpt,

A=f-1, where f:Xpt.

The skyscraper sheaf functor (ιX): AbelGpsSheaves(X) is the direct image of ιX:ptX,

(ιX), where ιX:pt X X

The stalk functor ιX-1:Sheaves AbelGps is the inverse image of ιX:ptX

ιX-1() =X where ιX:pt X X

HW 1: Show that the global sections of a sheaf on X is the ring Γ()=(X).

HW 2: Show that the constant sheaf on X with values in A is given by

A(U)= { f:UA IfaAthen f-1(a) is open inU }

HW 3: Show that the skyscraper sheaf at x is given by

Ax(V)= { A , ifxU 0 , otherwise ,

HW 4: Show that the stalk of at x is given by

x= limV{x} (V)

Sheaves, presheaves and sheafification

The inclusion, (or restriction, or forgetful) functor is

Res:Sheaves(X) Presheaves(X) 𝒢 𝒢

Sheafification is the left adjoint functor to inclusion Sh:Presheaves(X) Sheaves(X) given by

HomSheaves(X) (Sh(),𝒢) HomPresheaves(X) (,Res(𝒢))

HW 5: Show that the sheafification of a presheaf is given by

see Macdonald.

HW 6: Show that

  1. Sheaves(X) is a subcategory of Presheaves(X)
  2. Sheaves(X) is an abelian category
  3. Presheaves(X) is an abelian category
  4. Sheaves(X) is not an abelian subcategory of Presheaves(X)

HW 7:

Let X=×.

Define sheaves 𝒪,𝒪× and on X by

𝒪(U) = { continuousf:U } 𝒪×(U) = { continuousf:U × } (U) = { continuousf:U } ,

for U open in ×. Thus

𝒪(X) = { continuousf: × } 𝒪×(X) = { continuousf: ×× } (X) = { continuousf: × } ,

Show that

02πi 𝒪exp𝒪×0

is an exact sequence of sheaves on × which is not an exact sequence of presheaves on ×.

Show that

coKer (𝒪exp𝒪×) = { 0 , in Sheaves(X) , =H (X,) , in Presheaves(X) ,

and coKer (𝒪exp𝒪×) in Presheaves(X) is generated by the function

f:× × z 1z

which is a function in 𝒪× not in the image of 𝒪 in Presheaves(X).

Notes and References

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 𝒯𝒜. 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 Presheaves(X) and Sheaves(X) 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??????

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