# Some questions about Hartshorne chapter 2 proposition 2.6

In Hartshorne chapter 2 proposition 2.6, Hartshorne shows that there is a fully faithful functor $$t:\mathcal{Var}\rightarrow \mathcal{Sch}(k)$$ from the category of varieties over $$k$$ to the category of schemes over $$k$$. He proceeds as follows:

First, for any topological space $$X$$, define $$t(X)$$ to be the set of irreducible closed subsets of $$X$$. We can define a topology on $$t(X)$$ by taking as closed sets the subsets of the form $$t(Y)$$, where $$Y$$ is closed subset of $$X$$. If $$f:X_1\rightarrow X_2$$ is continuous, then define a map $$t(f):t(X_1)\rightarrow t(X_2)\\ Y\mapsto \overline{f(Y)},$$ where $$Y$$ is a irreducible closed subset of $$X_1$$. And, for any topological space $$X$$, define $$\alpha :X\rightarrow t(X)$$ by $$\alpha(p)=\overline{\{p\}}$$.

Question 1: Why is $$\overline{f(Y)}$$ a irreducible closed subset of $$X_2$$?

Next, let $$V$$ be an affine variety over $$k$$ with coordinate ring $$A$$, and $$O_V$$ its sheaf of regular functions, he then shows that $$(t(V), \alpha_*(O_V))$$ is isomorphic to the affine scheme $$(X,O_X)$$, where $$X=\operatorname{Spec}A$$.

Question 2: Why is $$(t(V),\alpha_*(O_V))$$ a locally ringed spaces, I mean why is $$(\alpha_*(O_V))_Y$$ a local ring for any $$Y\in t(V)$$?

Now define a morphism of locally ringed spaces $$\beta:(V,O_V)\rightarrow X=\operatorname{Spec}A,$$by $$\beta(p)=m_p$$. And for any open set $$U\subset X$$,define a homomorphism of rings $$O_X(U)\rightarrow \beta_*(O_V)(U)$$:given $$s\in O_X(U)$$,$$p\in \beta^{-1}(U)$$, we get a regular function $$g$$ on $$\beta^{-1}(U)$$ by $$g(p):=\overline{s_{\beta(p)}}\in A_{m_p}/m_p=k$$,where $$s_{\beta(p)}\in O_{X,\beta(p)}$$ and we identify the stalk $$O_{X,\beta(p)}$$ with the local ring $$A_{m_p}$$.

Question 3: Why is $$g$$ a regular function? It seems to me that $$g$$ is locally constant, is this true?

Then he claims that the above homomorphism $$O_X(U)\rightarrow \beta_*(O_V)(U)$$ is an isomorphism and uses the fact that there is a 1-1 correspondence $$t(V)\leftrightarrow \operatorname{Spec}A=X$$, then $$(X,O_X)\cong (t(V),\alpha_*(O_V))$$ as locally ringed spaces. I can't follow this part of proof.

Question 4: What is this $$\beta$$ used for? Shouldn't we construct a morphism $$(t(V),\alpha_*(O_V))\rightarrow (X,O_X)$$? And why is $$O_X(U)\rightarrow \beta_*(O_V)(U)$$ an isomorphism?

• question 1: because continuous maps preserve irreducibility and closure also preserves irreducibility.
– Seth
Commented Sep 7, 2014 at 16:00
• @Seth:You are right.This is easier than I thought.Thanks! Commented Sep 8, 2014 at 15:08

Question 1 is derived exactly as Seth's comment.

Question 2. We know that $$(\alpha_*(O_V))_Y=\varinjlim_{Y\in U}[\alpha_*(O_V)](U)=\varinjlim_{Y\in U}O_V(\alpha^{-1}(U)),$$ now, each open subset of $$t(V)$$ is of the form $$t(V)-t(W)$$ for some irreducible closed subset $$W$$ of $$V$$, let $$U=t(V)-t(W)$$ for some $$W$$, then $$Y\in U \Leftrightarrow Y\notin t(W)\Leftrightarrow \exists p\in Y, s.t. p\notin W\Leftrightarrow Y\cap (V-W)\neq \emptyset,$$ but $$V-W=\alpha^{-1}(U)$$, hence $$Y\in U \Leftrightarrow Y\cap \alpha^{-1}(U) \neq \emptyset$$. And $$\alpha$$ induce a 1-1 correspondence between open subsets of $$V$$ and $$t(V)$$, we have $$\varinjlim_{Y\in U}O_V(\alpha^{-1}(U))=\varinjlim_{Y\cap U\neq \emptyset}O_V(U),$$ where $$U$$ runs over all open subsets of $$V$$ s.t. $$Y\cap U\neq \emptyset.$$ Note that $$\varinjlim_{Y\cap U\neq \emptyset}O_V(U)$$ is just the local ring of a subvariety which has been shown to be a local ring(excercize 3.13 chapter 1).

Question 3. In fact, the function $$g$$ is defined by the following map: let $$p\in \beta^{-1}(U)$$, $$O_X(U)\rightarrow O_{X,\beta(p)}\rightarrow A_{m_p}\rightarrow k\\s\mapsto s(\beta(p))\mapsto \frac{a}{b}\mapsto \frac{a(p)}{b(p)}=g(p),$$ where $$O_{X,\beta(p)}\cong A_{m_p}$$ and $$a,b\in A,b\notin m_p$$. Note that $$a,b$$ are polynomial functions on $$V$$ since $$A$$ is the coordinate ring of $$V$$, and by the definition of $$s\in O_X(U)$$,for any $$p\in \beta^{-1}(U)$$,$$s(\beta(p))$$ is locally quotient of elements in $$A$$,so $$g$$ is a regular function on $$\beta^{-1}(U)$$. Clearly, $$g$$ is a regular function in the usual sense and need not to be locally constant.

Question 4. Our original plan is to show $$(t(V),\alpha_*(O_V))\cong(X,O_X)$$. There is a natural homeomorphism $$\gamma:t(V)\rightarrow X$$, so it's left to show $$\gamma':O_X\rightarrow \gamma_*(\alpha_*(O_V))$$ is an isomorphism. But $$\beta=\gamma\circ\alpha$$, so it's enough to show $$O_X\cong \beta_*(O_V)$$.

We next show that for any open set $$U\subset X$$, $$\varphi:O_X(U)\rightarrow \beta_*(O_V)(U)$$ is an isomorphism.

$$\varphi$$ is injective: suppose $$g(p)=0$$ for all $$p\in\beta^{-1}(U)$$. For any $$p\in\beta^{-1}(U)$$, there exist an open set $$p\in W\subset U$$ and $$a,b\in A, b\notin m_p$$ s.t. for any $$q\in W$$ we have $$s(q)=\frac{a}{b}$$ and $$b\notin q$$. For any $$q\in\beta^{-1}(W), \beta(q)\in W$$, we have $$s(\beta(q))=\frac{a}{b}$$ and $$g(q)=\frac{a(q)}{b(q)}=0$$. It follows that $$a(q)=0$$ for all $$q\in\beta^{-1}(W)$$, since $$\beta^{-1}(W)$$ is open and dense in $$V$$,we see that $$a=0$$ in $$A$$. So $$s(\beta(q))=0$$ for all $$\beta(q)\in W$$. Since $$p$$ is a arbitrary point in $$U$$, these W can cover U, we see that $$s=0$$ in $$O_X(U)$$.

$$\varphi$$ is surjective: let $$g\in O_V(\beta^{-1}(U))$$, then for any $$p\in\beta^{-1}(U)$$, there exist $$W\subset \beta^{-1}(U)$$, $$p\in W$$ and $$a,b\in A, b\notin \beta(p)=m_p$$ such that $$g=\frac{a}{b}$$ on $$W$$. Let $$s(\beta(q)):=\frac{a}{b}$$ for all $$q\in W\subset\beta^{-1}(U)$$, since $$p$$ is a arbitrary point in $$\beta^{-1}(U)$$, these open sets $$\beta(W)$$ cover $$U$$, in this way we get $$s$$ as an element of $$O_X(U)$$. Then one can check that $$s$$ is well-defined and $$\varphi(s)=g$$.

Alternatively, one can show $$O_X\cong \beta_*(O_V)$$ by the following diagram: $$\begin{matrix} O_X(U) &\longrightarrow & \beta_*(O_V)(U)=O_V(\beta^{-1}(U)) \\\ \downarrow & & \downarrow \\\ O_{X,p}& \longrightarrow& [\beta_*(O_V)]_p=O_{V,\beta^{-1}(p)} \\\ \downarrow & & \downarrow \\\ A_p & \longrightarrow & A_p \\\ \end{matrix}$$