I am reading Geometry of Schemes By Eisenbud and Harris.
In section II.3.1 They talk about schemes that are affine varieties, except that they are nonreduced. They say that schemes with double points can arise as limits of reduced schemes. In the book an explanation of the meaning of the limit they are talking about is given, but I don't seem to have the right knowledge to understand what is ment. Below I will give the specefic example I was reading.
Let $a(t),b(t) \in k[t]$ by polynomials such that $a(0) = b(0) = 0$, and define the following ring and affine scheme:
$$ S_t \quad := \quad { k[x,y] \over (x,y)(x-a(t),y-b(t)) } $$ and $$ X_t \quad := \quad \text{Spec}(S_t) \ = \ \{(0,0),(a(t),b(t))\} \ \subseteq \ \mathbb{A}_k^2. $$ Now the limit is defined by defining the limit of $(x-a(t), y-b(t))$, but I don't know what this means. Should one think of a limit as in category theory? This is what the book says about it:
Of course, this only shifts the burden to describing what is the limit of a family of ideals! But this is easy: in the current case, for example, we can take their limit as codimension-2 subspaces of $K[x, y]$, viewed as a vector space over $K$. That this limit is again an ideal follows from the continuity of multiplication.
I don't know what the limit of vector spaces is. I could imagine how we can take the limit of this using category theory, but I don't know how to interpret it as an ideal again, it would just be $k^2$. I also could imagine that if $k$ is a metric space we could make some sort of metric on linear spaces of a fixed ambient space as well, but for many $k$ that wouldn't work.
Of course I also just googled to find out what it could mean, but I didn't find anything. I think the rest of II.3.1 just explains what this limit means intuitively, so that wouldn't really answer my question.
So what is this limit? please tell me.
The discussion of "limits" in section II.3.1 is, as far as I can tell, entirely intuitive and does not refer to any precise definition. It is an intuitive, geometric idea of limits as certain spaces "approaching" other spaces as a parameter tends to a value, and is not related to categorical limits. So you're not missing anything. A rigorous definition is given later in section II.3.4.
In particular, the line "we can take their limit as codimension-2 subspaces of $K[x, y]$" strikes me as particularly flippant and totally devoid of rigor. In the particular case that $K=\mathbb{C}$ you could make this precise using limits in the Euclidean topology, but for an arbitrary field $K$ there is no obvious sense of "limit" to be referring to and the authors are being very misleading in suggesting that there is one.