Related question: How do you use this resource?
[edit] To make it clear: It's a wonderful thing that this exists.
The definitions and machinery make sense if you have enough background, but the why and how we got here is often very unclear.
If you don't care about schemes, stacks and current modern viewpoint of Algebraic Geometry, it's not hard to get a decent understanding of algebraic varieties which are the original motivation in the field. An undergrad book on the subject Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea does a great job of introducing the subject. And has a really cool project on calculating Groebner bases for polynomial equations in sin and cosine to define the configuration space of different types of robotic arms.
I get that affine schemes are somehow the "geometric" dual of a commutative ring. A motivating example is Spec(R[X,Y]/(X^2 + Y^2 - 1)), which is simple enough as a ring (if you remove the "Spec"), but as an affine scheme it is a circle. The slash is almost acting like a subset formation operation. I know enough category theory to see that the reason why the slash is acting that way is because equalisers are the dual construction to coequalisers; the slash (ring quotienting) is a coequaliser, and in the category of affine schemes it becomes an equaliser, and equalisers on "spaces" are supposed to form subsets somehow. Another example is the ring R[X]/(X^2), sometimes called the dual numbers, whose affine scheme (or Spec) is a lot like an infinitely small line segment. The fact that the affine scheme behaves like an infinitely small region of space is dual to the algebraic fact that the dual numbers are a local ring.
Finally, I have a vague understanding that a scheme is the result of gluing some affine schemes together. Sheaf stuff is involved.
Anyway, the above summarises my understanding of schemes. I don't know any differential geometry as such. I rely a lot on naive, and sometimes not wholly rigorous intuition. I have no idea how you compute with this, especially given how elaborate the definitions are.
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I'm hoping this might present a shortcut for someone like me: https://www.ingo-blechschmidt.eu/research.html It's especially promising because the computations look more familiar to me.
Try these two: https://faculty.math.illinois.edu/Macaulay2/Book/Computation... https://faculty.math.illinois.edu/Macaulay2/Book/Computation...
There's more in the book "Computations in algebraic geometry with Macaulay 2," which is free online. See also Schenck's "Computational Algebraic Geometry."
edit: Also, topos theory is definitely not the way to go here. Getting your hands dirty with algebraic curves is. See e.g. Griffiths' Introduction to Algebraic Curves.
Also the original motivation for sheaves was about creating a way to deal with multi-valued complex function. The complex log function is multi-valued so in intro complex analysis it’s studied locally by choosing a branch of the range where it’s singular valued. Thus it’s impossible to “do differential geometry” by talking about a global ring of analytic functions. But you can talk about the “local ring of analytic functions” at a point and specific branch and glue these locally ringed spaces together to get global insight.
> Finally, I have a vague understanding that a scheme is the result of gluing some affine schemes together. Sheaf stuff is involved.
For most mathematicians, this step is actually rather clear, because the intuition here is that "you glue schemes from affine schemes same way you glue (topological/differential) manifolds from pieces that look like R^n", and people studying mathematics typically have extensive experience with manifolds before they encounter schemes.
> and in the category of affine schemes it becomes an equaliser, and equalisers on "spaces" are supposed to form subsets somehow.
The intuition here is something like this: let's assume that R is algebraically closed, for example let R = C, ring of complex numbers. Then C[x, y] is a ring of polynomial functions defined on complex space C^2, and C[x, y]/(x^2 + y^2 - 1) is a ring of polynomial functions defined on the subspace V = { x^2 + y^2 - 1 = 0 }: if you have two functions f, g \in C[x, y], such that f(z) = g(z) for all z in V, then function h = f - g must be zero on the entire V, so (by Hilbert's Nullstellensatz), h must be in the ideal (x^2 + y^2 - 1). So, two elements f, g of C[x, y] restrict to the same function on V = { x^2 + y^2 - 1 = 0 } precisely when their difference is in the ideal I = (x^2 + y^2 - 1), so the ring of functions on V is exactly the quotient ring C[x, y]/I (or, as you call it, equalizer, which is correct, but it's never called this way by geometers).
> I have no idea how you compute with this, especially given how elaborate the definitions are.
Let me give you an example that I found very illuminating, which just so happen to talk about the scheme Spec(R[X,Y]/(X^2 + Y^2 - 1)) you mentioned: see section "6.5.8. More examples of rational maps." in http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pd...