Sheafs of abelian groups are the same as $\underline{\mathbb{Z}}$-modules

I worked things out and have the following conclusion.

Abelian groups are the same as $\mathbb{Z}$-modules, therefore sheafs of abelian groups are the same as $\underline{\mathbb{Z}}^{pre}$-modules. However, due to the fact that sheafs have the identity and gluability property, this $\underline{\mathbb{Z}}^{pre}$ action can be canonically extended to a $\underline{\mathbb{Z}}$ action, the way @hoot described, where $\underline{\mathbb{Z}}^{pre}$ is identified as the constant functions inside $\underline{\mathbb{Z}}$.

This can also be explained from the adjunction of embedding and sheafification as @Zhen Lin described. The fact that the sheafification of the constant presheaf is the constant sheaf implies that a $\underline{\mathbb{Z}}^{pre}$-module extends uniquely to a $\underline{\mathbb{Z}}$-module.

Tags:

Sheaf Theory

Algebraic Geometry

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