Using schemes to prove things about rings

Question

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:

1. Work on exotic spheres via homotopy theory (an example where this is the only known method to produce the results.)

2. (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.

Answer 1

Some examples:

A. A noetherian commutative ring has only finitely many minimal prime ideals. This is just a corollary of the easy observation that a noetherian space has only finitely many irreducible components.

B. The tensor product of two reduced (integral) $k$-algebras, where $k$ is an algebraically closed field, is again reduced (integral). After reducing to the finite type case, the argument of the proof is essentially geometric.

C. Diophantine equations, for example Fermat's Last Theorem (classify ring homomorphisms $\mathbb{Z}[x,y,z]/(x^n+y^n-z^n) \to \mathbb{Z}$), are (approximately) solved with the machinery of elliptic curves. The equation $x^2+y^3=z^7$ even needs algebraic stacks (see here)!

D. The classification of boolean rings. Or more generally rings whose elements satisfy a polynomial equation. As compared to the modern proof ($\underline{\mathbb{F}_2} \to \mathcal{O}_{\mathrm{Spec}(R)}$ is an isomorphism at stalks, hence globally), Stone's original one is quite clumsy. But of course, the historical importance of Stone's work cannot be overestimated. He can be seen as one of the innovators of the ideas of scheme theory.

E. The classification of integral domains generated by a single element. This comes down to the classification of prime ideals of $\mathbb{Z}[X]$, which is best done by looking at the fibers of $\mathrm{Spec}(\mathbb{Z}[X]) \to \mathrm{Spec}(\mathbb{Z})$. As in the examples above, this can also be done purely algebraically, but then it gets clumsy.

F. Define an $R$-module $M$ to be locally free of finite rank if there are elements $\{f_i\}$ of $R$ generating the unit ideal such that $M_{f_i}$ is free of finite rank over $R_{f_i}$. There is a purely algebraic proof that $M$ is locally free of finite rank if and only if $M$ is flat and of finite presentation, if and only if $M$ is finitely generated projective. However, at least for me as a beginner, it was hard to really grasp what is going on in that proof. But when you view $M_{f_i}$ as the restriction of (the quasi-coherent sheaf associated to) $M$ to (the open subscheme defined by) $R_{f_i}$, every step is clear as crystal. More generally, there are many theorems about modules over commutative rings which are best formulated, understood and proven more generally for quasi-coherent sheaves on a scheme. For another example, see David Lehavi's comment to Emerton's slick proof of the structure theorem for finitely generated modules over a PID.

G. There are non-isomorphic commutative rings $R,S$ such that $R[x]$ and $S[x]$ are isomorphic. The first example was found by Mel Hochster with geometric ideas (see here).

H. Affine algebraic geometry is full of problems, which can be formulated in terms of ring and module theory, but are attacked with algebraic-geometric methods. For a survey, see here. Perhaps I should stop here, because the list will never end ...

I. There are various algebraic constructions and invariants for rings which are best understood in geometric terms, such as the Krull dimension. The associated graded ring $\bigoplus_n I^n / I^{n+1}$ of an ideal $I \subseteq R$ roughly contains the infinitesimal information of $\mathrm{Spec}(R)$ at the closed subscheme $V(I)$. Modules of differentials provide another infinitesimal invariant. For function fields over perfect fields we have the geometric genus (of the corresponding proper normal curve). As always, such invariants are useful for example when one wants to prove that two rings are not isomorphic, replacing painful direct computations (see for example math.SE / 128918, 151714, 296737), but also to provide parameters for a possible classification.

J. Even projective schemes are useful in general ring theory, in particular in the context of homogeneous polynomials, see for example Will Sawin's answer in MO/110250, François Brunault's answer in MO/98043 and Qiaochu Yuan's answer at MO/14076.

K. (recreational) In the game on noetherian rings, a move consists of replacing a ring $R$ by $R/(a)$ for some $0 \neq a \in R$. You win when your oponent gives you the trivial ring. A complete analysis of this game is still out of reach, but the first attempts by Will Sawin and Kevin Buzzard illustrate the usage of algebraic geometry. Actually it is a game on (affine) schemes, where each move replaces $X$ by a closed subscheme $X' \subseteq X$ cut out by a single nontrivial equation.

L. Let $k$ be a field and $A \to C \leftarrow B$ homomorphisms of finitely generated $k$-algebras. Is the fiber product $A \times_C B$ again finitely generated? At first sight this seems to be elementary and should be well-known for decades, but it seems to be an open problem(?). Jakob Scholbach has proven it when $A \to C$ and $B \to C$ are regular (i.e. the ideal is generated by codim many elements), using quite a bit of projective algebraic geometry.

Answer 2

Primes in a (commutative) Jacobson ring

This question was phrased purely algebraically, but I only arrived at the solution by geometric arguments.

Answer 3

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 $\mathcal{O}_X(\mathfrak{m}_i)\simeq A_{\mathfrak m_i}$.