There’s a certain point where commutative algebra outgrows arguments that are phrased purely in terms of ideals (e.g. at some point in Matsumura the proofs stop being about ideals and elements and start being about long exact sequences and Ext, Tor). Once you get to that point, and even further to modern commutative algebra which is often about derived categories (I spent some years embedded in this community), I find that I’m essentially using a transplanted intuition from that “old world” but now phrased in terms of diagrams in derived categories.
E.g. a lot of Atiyah and Macdonald style arguments just reappear as e..g arguments about how to use the residue field to construct bounded complexes of finitely generated modules in the derived category of a local ring. Reconstructing that intuition in the derived category is part of making sense of the otherwise gun-metal machinery of homological algebra.
Ultimately I don’t see it as different, but the “externalised” view is the one that plugs into homological algebra and therefore, ultimately, wins.
(Edit: saw Simon’s reply after writing this, yeah agree!)
There’s a certain point where commutative algebra outgrows arguments that are phrased purely in terms of ideals (e.g. at some point in Matsumura the proofs stop being about ideals and elements and start being about long exact sequences and Ext, Tor). Once you get to that point, and even further to modern commutative algebra which is often about derived categories (I spent some years embedded in this community), I find that I’m essentially using a transplanted intuition from that “old world” but now phrased in terms of diagrams in derived categories.
E.g. a lot of Atiyah and Macdonald style arguments just reappear as e..g arguments about how to use the residue field to construct bounded complexes of finitely generated modules in the derived category of a local ring. Reconstructing that intuition in the derived category is part of making sense of the otherwise gun-metal machinery of homological algebra.
Ultimately I don’t see it as different, but the “externalised” view is the one that plugs into homological algebra and therefore, ultimately, wins.
(Edit: saw Simon’s reply after writing this, yeah agree!)