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I'm working on a problem to ultimately show that an open subset of $\text{Spec}\mathbb{Z}[t]$ is not an affine scheme. The way the problem suggests to do this is to find the neighborhoods $X_1 = D(2)$ and $X_2 = D(t)$, the domains of the two prime ideals $(2), (t)$, and compute the structure sheaf element $\mathcal{O}(X_1\cup X_2)$. My strategy is to compute $\mathcal{O}(X_1\cup X_2)$ as the limit of $\mathcal{O}(X_1)$ and $\mathcal{O}(X_2)$. Since the structure sheaf elements are the localizations around the specified prime ideals, I have

\begin{eqnarray*} \mathcal{O}(X_1) & = & \left (\mathbb{Z}[t]\right )_{(2)} \;\; =\;\; \left \{\left .\frac{p(t)}{q(t)} \right | \; 2 \; \text{doesn't divide} \;q(t) \right \} \\ \mathcal{O}(X_2) & = & \left (\mathbb{Z}[t]\right )_{(t)} \;\; =\;\; \left \{\left .\frac{p(t)}{q(t)} \right | \; t \; \text{doesn't divide} \;q(t) \right \} \\ \end{eqnarray*}

Given that we have restriction morphisms between these two sets, is it possible to conclude

$$ \mathcal{O}(X_1\cup X_2) \;\; =\;\; \left \{\left .\frac{p(t)}{q(t)} \right | \; \text{Neither} \; t \; \text{nor} \;2 \; \text{divide} \;q(t) \right \} $$

If this is true, then how do we know that $X_1\cup X_2$ isn't isomorphic to $\text{Spec}R$ for any ring $R$ and $\mathcal{O}(X_1\cup X_2)$ to corresponding affine scheme elements? Part of my issue here is that I'm not totally sure what the Zariski topology looks like in $\text{Spec}\mathbb{Z}[t]$.

Mnifldz
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    What do you mean by $\mathbb{Z}[t]$ as a scheme? It's a ring, and $Spec(\mathbb{Z}[t])$ is affine. Do you mean that some open subset of $Spec(\mathbb{Z}[t])$ is not affine? – David Lui Oct 05 '22 at 19:20
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    Your title is extremely confusing. $\mathbb{Z}[t]$ is a ring, so you presumably mean $\operatorname{Spec}(\mathbb{Z}[t])$. But this is clearly an affine scheme by definition. From the body of your post, it seems like you want to show that $X_1 \cup X_2$ (which is a subscheme of $\operatorname{Spec}(\mathbb{Z}[t])$) is not affine. – Viktor Vaughn Oct 05 '22 at 19:20
  • Hint: let $X = Spec(\mathbb{Z}[t])$. Show that $\frac{1}{2t}$ cannot be written as $f - g$, where $f \in \mathbb{Z}[t]_2$ and $g \in \mathbb{Z}[t]_t$, which shows that $H^1(X, O_X) \neq 0$. See this answer. – David Lui Oct 05 '22 at 19:25
  • Sorry all, my title was poorly written and I misused the terminology. I hope this is better. – Mnifldz Oct 05 '22 at 19:44
  • @DavidLui In the book I'm reading the author hasn't introduced cohomology that extensively (reserved for later chapters). Is there a clear brute-force approach that a novice like me could employ? Part of me is suspect that this book isn't perfectly axiomatic in its presentation... – Mnifldz Oct 05 '22 at 20:06
  • Usually, the (non-cohomological) method with these kinds of problems is to find a strictly larger affine open subset $X_1 \cup X_2 \subset U$ so that $\mathcal{O}(X_1 \cup X_2) = \mathcal{O}(U)$. If $X_1 \cup X_2$ was affine, this would have to be equality, contradicting the strict inclusion. – Daniel Oct 06 '22 at 02:13
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    Also, the structure sheaf over $D(2)$ should should be the localization $S^{-1}\mathbb{Z}[t]$ where $S$ is all the powers of $2$. The stalk of the structure sheaf at $(2)$ would be the localization $\mathbb{Z}[t]_{(2)}$. – Daniel Oct 06 '22 at 02:19

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