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K. Bongartz, "A generalization of a theorem of Auslander": Let R be a commutative ring and A an abelian R-linear category such that each morphism set in A has finite length as an R-module. Let …

The classification of $K[X,Y]$-modules of finite length is of wild type, meaning that a classification of these modules would contain within it a classification of all finite-dimensional $\Lambda$-modules … In fact, even the classification of those $K[X,Y]$-modules annihilated by $(X^2, XY^2,Y^3)$ is a wild problem, as shown by Drozd. …

See Fossum, Foxby, Griffith, and Reiten, "Minimal injective resolutions with applications to dualizing modules and Gorenstein modules" (Theorem 1.1) and also Roberts, "Two applications of dualizing complexes …

As soon as the dimension of $R$ is at least $1$ and $R$ is locally Gorenstein at the associated primes, then $K$ is isomorphic to an ideal of pure height $1$. In particular $K$ (as an ideal) contains …

$\mathrm{Ext} = \mathrm{Ext}^1$ does not classify the middle terms up to isomorphism. It classifies the short exact sequences up to equivalence, where the equivalence relation is generated by commuta …