Let x* A be a skyscraper sheaf, as in Example 5.72. (i)...

Let $x_{*} A$ be a skyscraper sheaf, as in Example $5.72$. (i) Prove, for every sheaf $\mathcal{G}$, that there is an isomorphism $$ \operatorname{Hom}_{\mathbb{Z}}\left(\mathcal{G}_{x}, A\right) \cong \operatorname{Hom}_{S h}\left(\mathcal{G}, x_{*} A\right) $$ that is natural in $\mathcal{G}$. (ii) Every sheaf map $\varphi: \mathcal{F} \rightarrow \mathcal{G}$ induces homomorphisms of stalks $\varphi_{y}: \mathcal{F}_{y} \rightarrow \mathcal{G}_{y}$ for all $y \in X .$ Choose $x \in X .$ If $\mathcal{F}$ is a sheaf over $X$ with stalk $\mathcal{F}_{x}=A$, prove that there is a sheaf map $\varphi: \mathcal{F} \rightarrow x_{*} A$ with $\varphi_{x}=1_{A}$.

Let $x_{*} A$ be a skyscraper sheaf, as in Example $5.72$.
(i) Prove, for every sheaf $\mathcal{G}$, that there is an isomorphism
$$
\operatorname{Hom}_{\mathbb{Z}}\left(\mathcal{G}_{x}, A\right) \cong \operatorname{Hom}_{S h}\left(\mathcal{G}, x_{*} A\right)
$$
that is natural in $\mathcal{G}$.
(ii) Every sheaf map $\varphi: \mathcal{F} \rightarrow \mathcal{G}$ induces homomorphisms of stalks $\varphi_{y}: \mathcal{F}_{y} \rightarrow \mathcal{G}_{y}$ for all $y \in X .$ Choose $x \in X .$ If $\mathcal{F}$ is a sheaf over $X$ with stalk $\mathcal{F}_{x}=A$, prove that there is a sheaf map $\varphi: \mathcal{F} \rightarrow x_{*} A$ with $\varphi_{x}=1_{A}$.

Key Concepts

Sheaf Map and Induced Map on Stalks

A sheaf map is a morphism between sheaves that commutes with the restriction maps. Crucially, every sheaf map induces corresponding maps between the stalks at each point. This concept is important when constructing morphisms that realize specified behavior at the level of stalks, as seen in the requirement that the induced map at x is the identity.

Adjunction

Adjunction in sheaf theory refers to a pair of functors being adjoint to each other, meaning that there is a natural isomorphism between certain Hom sets. In the context of the problem, the stalk functor and the skyscraper sheaf functor exhibit an adjunction, allowing one to identify Hom sets involving stalks with Hom sets involving skyscraper sheaves.

Natural Isomorphism

A natural isomorphism is an isomorphism between two functors that behaves 'naturally' with respect to the objects and morphisms in the category. In this problem, the isomorphism between Hom_Z(?_x, A) and Hom_Sh(?, x_*A) is natural in ?, meaning it is coherent with the morphisms between sheaves and respects the sheaf structure.

Skyscraper Sheaf

A skyscraper sheaf is a sheaf that is concentrated at a single point of the topological space, meaning it assigns a specified group (or set, module, etc.) at that point and trivial sections (often a zero or trivial object) at all other points. This concept is useful for isolating local behavior and for constructing examples or counterexamples in sheaf theory.

Stalk of a Sheaf

The stalk of a sheaf at a point is the colimit of the sections over all open neighborhoods containing that point. It captures the local behavior of the sheaf around that point by collecting all germs of sections. This is essential in understanding how global properties of sheaves relate to their local data.

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