Connected scheme but not quasicompact

Question

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the $A_i`s$ all together, and construct a scheme $X$ which will be connected but not quasicompact. (Like the coordinate axis in 3 dimensional case). But can I do that ? I know that if I have two schemes with two open subsets in them then i can glue them. But can I glue the point $0$ of two affine schemes?

Answer 1

Here is an example without gluing:
Let $k$ be a field and put $\mathbb A^\infty_k=\operatorname{Spec}(k[x_0,x_1,x_2,\cdots])$.
Of course that scheme is quasi-compact, like all affine schemes, and irreducible since $k[x_0,x_1,x_2,\cdots]$ is a domain.
But if you delete the closed origin $O=\langle x_0,x_1,x_2,\cdots \rangle$ the resulting scheme $U=\mathbb A^\infty_k\setminus \{O\}$ is irreducible (like all non-empty open subsets of irrreducible spaces), hence connected.
However $U$ is not quasi-compact because the open covering $(U \setminus V(x_i))_{i\in \mathbb N}$ of $U$ has no finite subcovering.

Note
No example can be a subscheme of a noetherian scheme because any subset of a noetherian space is quasi-compact. Hence the necessity of taking infinitely many variables in the above example.