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I'm working through some algebraic geometry exercises and I stumbled upon the following which I can't seem to get a satisfactory solution to: Prove that there exists a scheme which admits a covering by countably many closed subschemes each of which is isomorphic to $\mathbb{A}_1^k$ (over an algebraically closed field), indexed by $\mathbb{Z}$, such that the copies of $\mathbb{A}^1_k$ corrsponding to i and i+1 intersect in a single point, which is the point 0 when considered as a point in the ith copy, and the point 1 when considered as an element of the (i+1)th copy.

OK, so I have been trying to draw a picture of this and I can see somehow intersecting affine lines to get a geometric figure which satisfies the properties required. However, I feel a bit uneasy with this argument since my construction is very pictorial. So can anyone supply an example , or a hint, of how to construct a scheme that satisfies the requirements above?

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For any integer $i$, denote by $X_i$ the union of two copies of $\mathbb G_m=\mathbb A^{1}_k \setminus \{ 0\}$, denoted by $T_i^{-}$ and $T_i^{+}$, intersecting at a unique point $o_i$. Glue the $X_i$'s as following: $X_i$ and $X_{i+1}$ are glued along $$T_i^{+}\setminus \{ o_i\} \simeq T_{i+1}^{-}\setminus \{ o_{i+1} \};$$ and $X_i$ and $X_j$ dont have intersection if $|i-j|>1$.

Edit Some more details as required in the comments. Let us call $X$ the scheme obtained in the above construction. We view the $X_i$'s as open subschemes of $X$. Let $\Gamma_i=T_i^{+}\cup \{ o_{i+1}\}$. It is closed in the closed subset $X\setminus \cup_{j\ne i, i+1} X_j$, so it is closed in $X$. Clearly it is isomorphic to the affine line. It intersects $\Gamma_{i+1}$ at $o_{i+1}$ and $\Gamma_{i-1}$ at $o_i$. The points $o_i, o_{i+1}$ in $\Gamma_i$ can be considered as $0, 1$ in the affine line because any pair of distinct points in the affine line can be mapped to $0,1$ by an automorphism.

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Nice, thank you very much! :) –  Dedalus May 16 '12 at 20:23
    
Now that I have looked at this example more I am still a bit vague on how to express it as this union of closed subschemes. Would you mind elaborating a bit on your construction, and why they intersect in a point considered as 0 in one, and 1 in the other? –  Dedalus May 17 '12 at 16:15
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@Dedalus: hope it is clearer now. –  user18119 May 17 '12 at 21:35
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