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kernels, cokernels, etc. on sheaves of modules

In this subsection we restrict ourselves to deal with sheaves of modules.

To shorten our statements, we call a presheaf which satisfies (only) the sheaf axiom (1) (locality) a ``(1)-presheaf''.

LEMMA 06.18   Let $ \varphi: \mathcal F\to \mathcal G$ be a homomorphism between sheaves of modules. Then we have

  1. The presheaf kernel of $ \varphi$ is a sheaf. We call it the sheaf kernel $ \operatorname{Ker}(\varphi)$ of $ \varphi$ .
  2. The presheaf image of $ \varphi$ is not necessarily a sheaf, but it is a (1)-presheaf. We call the sheafification of the presheaf image as the sheaf image $ \operatorname{Image}(\varphi)$ of $ \varphi$ .
  3. The presheaf cokernel of $ \varphi$ is not necessarily a sheaf. We call the sheafification of the cokernel as the sheaf cokernel $ \operatorname{Coker}(\varphi)$ of $ \varphi$ .

DEFINITION 06.19   A sequence of homomorphisms of sheaves of modules

$\displaystyle \mathcal F_1
\overset{f_1}{\to}
\mathcal F_2
\overset{f_2}{\to}
\mathcal F_3
$

is said to be exact if $ \operatorname{Image}(f_1)=\operatorname{Ker}(f_2)$ holds.

LEMMA 06.20   A sequence of homomorphisms of sheaves of modules

$\displaystyle \mathcal F_1
\overset{f_1}{\to}
\mathcal F_2
\overset{f_2}{\to}
\mathcal F_3
$

is exact if and only if it is exact stalk wise, that means, if and only if the sequence

$\displaystyle (\mathcal F_1)_P
\overset{f_1}{\to}
(\mathcal F_2)_P
\overset{f_2}{\to}
(\mathcal F_3)_P
$

is exact for all point $ P$ .


next up previous
Next: About this document ... Up: Resolutions of singularities. Previous: stalk of a presheaf
2014-07-03