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One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, so that this description of commutative algebra is misleading. It feels to me like in most commutative algebra circles, the main approaches taken are geometric, rather than, say, multiplicative ideal theoretic. For example, in Eisenbud's commutative algebra book, less than one page is devoted to Dedekind domains, except for some scattered exercises.

If I were to make a Venn diagram, I would depict commutative algebra, algebraic geometry, and non-commutative algebra as all having nontrivial intersections with each other (rather than commutative algebra being subsumed by algebraic geometry).

My question is threefold: is there a good example of an important problem in commutative algebra that requires a geometric approach? is there a good example of an important problem in commutative algebra that rather requires methods from non-commutative algebra? And is there a good example of a problem in commutative algebra that requires both (e.g., that requires some results in "non-commutative geometry", a term that has multiple meanings)?

To answer a comment: I am not looking for a list of examples. I'm looking for a single important and good example of each, as I don't know any good examples of the three. I will accept an answer that provides three important examples, as requested. I just want to know if there are any important examples of all three at all.

ADDED: As I explained in the (now deleted) comments, I don't think this question should be or should have been moved to community wiki. If I ask for a proof of Theorem X, then there will likely not be a unique answer, and the OP can accept whichever proof that was offered as an answer that they believe was the best among them. The situation is the same here. I'm asking whether or not there exist important examples, which are not unique, but there are criteria for whether an example is an example or not, just as there are criteria for whether a proof is a proof or not. If there are no such examples, then I would like to know that. If there are such examples, then I would like to know that. It doesn't matter what the examples are, as long as there are indeed examples.

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  • $\begingroup$ A thing I think its worth noticing is that some important parts of noncommutative algebra become trivial for commutative rings. For instance, a fundamental result for noncommutative rings is the Structure Theorem for (left) Primitive Rings. But in commutative álgebra this result says nothing, for a left (or right) commutative primitive ring is just a field. $\endgroup$
    – jg1896
    Commented Nov 20, 2023 at 2:43
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    $\begingroup$ I think this is too broad a question to admit a uniform answer. But here is one example that might be worth pondering: consider a finite subgroup $G<SL(2,{\mathbb C})$; its action on ${\mathbb C}[x,y]$ gives rise to the twisted group ring ${\mathbb C}G\star {\mathbb C}[x,y]$, as well as the Morita-equivalent, "multiplicity-free" version $\Pi_G$, the associated preprojective algebra. These are mindly noncommutative (finite over their centre); they have been studied from all three points of view: CA (max CM modules), non-CA (quiver techniques) and geometry (Nakajima quiver varieties)... $\endgroup$
    – Balazs
    Commented Nov 21, 2023 at 8:34
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    $\begingroup$ ...and then as one deforms $\Pi_G$, it becomes ''more non-commutative'', and so non-commutative and geometric techniques become more important. $\endgroup$
    – Balazs
    Commented Nov 21, 2023 at 8:36
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    $\begingroup$ For problems in commutative algebra requiring algebraic geometry: I think the direct summand conjecture is an obvious candidate. The statement is clearly one in commutative algebra (and for the cases that classically come up in algebraic geometry, namely finite type $k$-algebras, the result was already known). But André's proof (and later simplifications, e.g. by Bhatt or Ma) uses perfectoid spaces, which probably falls under (a sufficiently broad definition of) algebraic geometry. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 26, 2023 at 17:36
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    $\begingroup$ @JesseElliott, if you consider field theory as a part of commutative algebra (this depends on the point of view), the positive solution of Lüroth problem for $n=2$, which is a purely field-theoretic question, follows from a geometric fact: Castelnuovo's rationality criterion for surfaces. $\endgroup$
    – jg1896
    Commented Nov 29, 2023 at 12:58

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Here a maybe not too important problem for a commutative algebra ,which is solved using non-commutative algebra and the solution is quite non-trivial:

Problem: Let $K$ be a field, classify the finite dimensional $A$-modules of the finite dimensional commutative $K$-algebra $A=K[x,y]/(x^2,y^2)$.

I do not know a "commutative algebra" solution. One way is to reduce it to the problem to classify the modules over the Kroenecker algebra, which is a non-commutative algebra, see for example chapter 4.3 in the book "Representation theory and Cohomology: Volume 1" by Benson.

Of course one can argue that this is not a typical commutative algebra problem, but it is a problem about a commutative algebra! And it sounds like a natural problem to look at.

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    $\begingroup$ Maybe a similar class of examples: cluster algebras are by definition commutative algebras; but it is mostly techniques from noncommutative algebra (e.g., representation theory of associative algebras) which is used to study cluster algebras. $\endgroup$
    – Sam Hopkins
    Commented Nov 29, 2023 at 15:27

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