To revisit Bruce's earlier question, it might be useful to suggest a more sceptical alternative to Ben's answer and the related comments. I doubt that there will be a "correct" version of the CDE-triangle that has enough breadth to take in the variety of analogues that have emerged in representation theory, though it's obviously important to make the assumptions precise about underlying fields, local rings, etc. See also the conference paper: MR2184010 (2006g:17027) 17B67 (17B10)
Ekedahl, Torsten (S-STOC),
Kac-Moody algebras and the cde-triangle.
Noncommutative geometry and representation theory in mathematical physics, 49–58, Contemp.
Math., 391, Amer. Math. Soc., Providence, RI, 2005.

For me the essential ingredients to start with are suitable module categories (or more general analogues) in which three distinctive types of objects play a leading role: (1) simple modules, which are "small" and basic but often hard to get at directly; (2) indecomposable projectives (or injectives ...) which are also naturally present in suitable categories and usually have just finitely many composition factors, but are typically "large" and messy to study; (3) *intermediate* objects, easier to construct and often having known dimensions (if finite) or "formal" characters. The idea then is to express (say) a projective cover of a simple module formally, perhaps via a filtration, in terms of the intermediate objects. To make this interesting, when all composition factor multiplicities are finite or otherwise controllable, one wants a kind of "reciprocity" between multiplicities of simples in intermediate objects and multiplicities of the latter in the big modules. Here is some brief history followed by a sample of more recent instances. (It does get a bit long...)

(1) Elie Cartan actually studied what we now call finite dimensional associative algebras and emphasized the importance of knowing the composition factor multiplicities of indecomposable projectives (now dubbed *Cartan invariants*). For a finite group with $r$ classes of elements having orders not divisible by a given prime $p$, you get an $r \times r$ matrix $C$ (over a large enough field).

(2) Work by Richard Brauer and his Toronto student Cecil Nesbitt after 1937 introduced in the finite group setting (for a prime $p$ dividing the group order and large enough fields) an $s \times r$ *decomposition matrix* $D$ showing how to express the $s$ ordinary irreducible characters (= number of classes) as formal sums of $p$-modular irreducible characters (counting composition factor multiplicities in reduction mod $p$ for any suitable lattice in a module). Then $C = D^{t} D$ (so $C$ is symmetric), where the transpose of $D$ shows how to express a projective (lifted to characteristic 0) as a combination of ordinary characters.
These ideas were exposed by Curtis-Reiner (1962) in Section 83, etc. Brauer was studying $p$-*blocks*, which show up in the block decompositions of the matrices.

(3) Following Swan's formalism, Serre (1971) formulated the more abstract cde-triangle in his part III, using homomorphisms between various Grothendieck groups. This was further codified by Curtis-Reiner in their later 1981 book, Section 18, with a lot of attention to the rings and fields involved.

(4) Having studied the old CR book in a 1963-64 course at Yale taught by Jacobson (!), I later tried to adapt $C = D^{t} D$ to modular representations of Lie algebras of simple algebraic groups (working just in prime characteristic). The f.d. restricted enveloping algebra imitates a finite group algebra in some ways. Here the "intermediate" objects were f.d. analogues of Verma modules. At first I studied blocks and multiplicities but not filtrations. This rough version appeared in J. Algebra (1971) and inspired Verma's introduction of affine Weyl groups relative to $p$, as well as much more sophisticated work by Jantzen treating filtrations of projectives, plus action of a maximal torus in the group.

(5) Work by Bernstein-Gelfand-Gelfand in the early 1970s was partly inspired by Jantzen's work and by my 1971 paper, leading to their "BGG category" and "BGG reciprocity" in 1976. Then Kazhdan-Lusztig theory for finite and affine Weyl groups came into play, etc.

(6) Eventually some but not all of the ideas spread elsewhere in representation theory, including the work by Alvany Rocha and her thesis advisor Nolan Wallach
and much other work on Kac-Moody algebras (recently by Arakawa-Fiebig for the mysterious critical level in the affine case). Plus early work by Dan Nakano on other modular Lie algebra settings, recent work on rational Cherednik algebras, and so on.

(7) The most fruitful general formulation for some purposes was given in a 1988 Crelle paper by Cline-Parshall-Scott on highest weight categories and quasi-hereditary algebras. Other general settings were proposed by Ron Irving and by Apoorva Khare. It's hard though to find just one common framework.