TITLE: Using schemes to prove things about rings
QUESTION [18 upvotes]: I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment.
The question is as follows:

What are examples of strict applications of the language of schemes/stacks/algebraic geometry to commutative rings?

Here a "strict" application means that the statement of the problem can be formulated without using any algebro-geometric language (stick to rings and modules and complexes, etc.) but a solution either requires or is very naturally obtained by using algebro-geometric language.
I don't know examples of this phenomenon off the top of my head, but here are two examples from algebraic topology:

Work on exotic spheres via homotopy theory (an example where this is the only known method to produce the results.)
(one of) Quillen's proof(s) of the Atiyah-Swan conjecture. While there is a purely algebraic, group-cohomological proof, it turns out to be very natural to prove this theorem using spaces with an action, as opposed to specializing to when the space is a point.

Motivation
Thanks to work of (insert all the usual suspects here), we now have a very strong theory of spectral algebraic geometry, i.e. algebraic geometry done with commutative ring spectra as opposed to commutative rings. While I don't know of any (hence this question), I am positive there exist strict applications of algebraic geometry to ring theory. It would be very neat if we could transplant these into strict applications of spectral algebraic geometry to the theory of ring spectra. Obviously I don't expect this to be straightforward, or literally possible, but I maintain that answers to this question would provide a useful insight in how to think about the relationship between non-affine and affine phenomena.

REPLY [2 votes]: We prove by elementary scheme-theoretic methods the following classical result:
CLAIM: Let $A$ be an artinian ring with maximal ideals $\mathsf{Spec} \, A=\lbrace \mathfrak m_1,\ldots,\mathfrak m_N\rbrace$. Then there is an isomorphism $$A\simeq \prod_{i=1}^NA_{\mathfrak m_i}.$$
PROOF: Let $X=\mathsf{Spec} \, A$. denote by $\mathcal{O}_X$ its structure sheaf. Since $X$ is discrete, by the gluing axiom, we have $$A\simeq \mathcal{O}_X(X)=\prod_{i=1}^N\mathcal{O}_X(\mathfrak m_i),$$ but $\mathcal{O}_X(\mathfrak{m}_i)\simeq A_{\mathfrak m_i}$.