Difference Between Gluing of Manifolds and that of Schemes

I understand that both manifolds and schemes can be defined by gluing of Euclidean spaces and affine schemes respectively. For example, one may regard a manifold $$M$$ as a limit of a collection $$\{U_i\}_{i\in I}$$ of open subsets of $$\mathbb{R}^n$$, where $$n=\dim M$$, together with another open subsets $$U_{ij}$$ of $$\mathbb{R}^n$$ for each $$i\neq j\in I$$ and embeddings $$\phi^{ij}_i:U_{ij}\to U_i$$ and $$\phi^{ij}_j:U_{ij}\to U_j$$. This way we can glue two manifolds $$M$$, $$N$$ along another manifold $$L$$ of same dimension whenever embedding $$L\to M$$ and $$L\to N$$ is given.

I am not very good at algebraic geometry, so I am not sure of details, but I think similar things will be possible for schemes: defining schemes as a limit of affine schemes, and gluing two schemes along embeddings.

However, for me, they look very different. For me, it seems quite easier to patching manifolds up, but gluing schemes is very difficult and 'rigid.' For example, there are uncountably many ways to construct $$S^2$$ by patching up $$\mathbb{R}^2$$, but the complex projective line $$\mathbb{CP}^1$$, which is topologically homeomorphic to $$S^2$$, can only be constructed by glueing two affine lines $$\mathrm{Spec} \mathbb{C}[t]$$, $$\mathrm{Spec} \mathbb{C}[s]$$ along the relation $$\mathbb{C}[t]_t\to\mathbb{C}[s]_s, t\mapsto 1/s$$.

For another example, one can glue two real lines, each of them identified by open intervals $$(0,2)$$ and $$(1,3)$$, to make another real line $$(0,3)$$, but in the world of algebraic geometry, this looks impossible to me.

This is really strange and confusing to me, since gluing of two manifold is quite easy and straightforward, but the gluing of two schemes looks nearly impossible. Am I misunderstanding something? Or are they really different? Can someone give me a good example of gluing two schemes?

However, for me, they look very different. For me, it seems quite easier to patching manifolds up, but gluing schemes is very difficult and 'rigid.' p For another example, one can glue two real lines, each of them identified by open intervals $$(0,2)$$ and $$(1,3)$$, to make another real line $$(0,3)$$, but in the world of algebraic geometry, this looks impossible to me.

I think the difference is that you have a kind of "intuitive" understanding of manifolds, where schemes appear to be more abstract. To glue the both intervals formally, you need to construct a homeomorphism on the parts you want to glue. This is obvious in your case, because both contain the interval $$(1, 2)$$, and you can glue along the identity.

In the same sense, you can easily glue schemes which "already belong to each other".

This is really strange and confusing to me, since gluing of two manifold is quite easy and straightforward, but the gluing of two schemes looks nearly impossible. Am I misunderstanding something? Or are they really different? Can someone give me a good example of gluing two schemes?

For example take the schemes $$X = \mathbb{A}^1 = \text{Spec }k[x]$$, and the open set $$U = D(x) = \{ p \mid p \neq 0 \}$$ (that set-notation should not be taken literally). Then you can glue two copies of $$X$$ along $$U$$, and obtain a scheme $$Y$$ which has the zero-point twice. Think about this like a line with a double-point, a bit similar to glueing two copies of $$\mathbb{R}$$ along the set $$V = \{ p \mid p \neq 0 \}$$. Of course the latter is not a manifold, but still a topological space.

$$Y$$ is a famous example for a scheme that is integral and of finite type over a field $$k$$, but not separated, so not a variety.

but the complex projective line $$\mathbb{CP}^1$$, which is topologically homeomorphic to $$S^2$$, can only be constructed by glueing two affine lines $$\text{Spec }\mathbb{C}[t], \text{Spec }\mathbb{C}[s]$$ along the relation $$\mathbb{C}[t]_t \to \mathbb{C}[s]_s, t \mapsto 1/s$$

You could still apply linear transformations, glueing along $$\mathbb{C}[t]_{at + b} \to \mathbb{C}[s]_{cs + d}, t \mapsto \frac{1 - b(cs + d)}{a(cs + d)}$$ for $$a, b, c, d \in \mathbb{C}$$ and $$a, b \neq 0$$. This is just an isomorphic renaming.

• I see. I somewhat understand, but what I am still confused is what exactly is "scheme." For me, a manifold is a very natural concept, a topological space is a very general and strong concept, an affine variety is very important concept, but when I see schemes, I don't literally understand what they really are. A line with double point, for example, I don't understand why we should think of this kind of example. Why are they so important? What is a slogan to understand a scheme theory? – J1U Feb 23 '19 at 9:32
• I don't know where, but somewhere I read, that a scheme generalizes varieties in three different ways, which can be confusing: (1) schemes let you work over arbitrary rings, e.g. see this discussion, (2) schemes let you consider geometric object, which do not live in affine or projective space, obtained by glueing, and (3) schemes can have embedded components, which carry information, like intersection numbers, and have connections to nilpotent elements in rings. – red_trumpet Feb 23 '19 at 12:01
• I appreciate your detailed comment and answer. I will read them carefully. Thank you. – J1U Feb 23 '19 at 12:53