#### Approaching Scheme Theory as a Beginner

*Thank you to James Tao for explaining the following, geometrically intuitive, interpretation to me.*

A scheme is the result of patching affine schemes together, an affine scheme is any locally ringed space that is equivalent to Spec of a ring. To make an analogy, a manifold is the result of patching together copies of $\mathbb{R}^n$, in a way that preserves the smooth structure.

And the smooth structure of $\mathbb{R}^n$ is essentially the fact that it is a sheaf: over any open set, we have the space of smooth functions on that set, and these functions restrict and glue nicely, etc.

So when you go from $\mathbb{R}^n$ to “smooth manifolds,” you are creating spaces that locally look like $\mathbb{R}^n$, respect the same kind of smooth structure (as captured by the fact that smooth functions are defined on open sets in a way independent of the patching), yet have some global structure that might prevent them from just being the same as $\mathbb{R}^n$.

If you care about algebraic (polynomial / rational) functions defined on algebraic sets (zeroes of systems of polynomials), then you can do the same program! The “basic patch” is “affine scheme,” and you patch affine schemes together to get a scheme. Continue reading Spectrum of a Ring