What is a Mittag-Leffler module? Let R be a ring and let M be an R-module. Write M = colim_i M_i as a directed colimit of finitely presented R-modules. (This is always possible.) Pick any R-module N. Then consider the inverse system (Hom_R(M_i, N))_i. We say M is Mittag-Leffler if this inverse system is a Mittag-Leffler system for any N. It turns out that this condition is independent of the choices made, see Proposition Tag 059E.

A prototypical example of a Mittag-Leffler module is an arbitrary direct sum of finitely presented modules. Some examples of non-Mittag-Leffler modules are: Q as Z-module, k[x, 1/x] as k[x]-module, k[x, y, t]/(xt – y) as k[x,y]-module, and ∏_n k[[x]]/(x^n) as k[[x]]-module.

Why is this notion important? It turns out that an R-module P is projective if and only if P is (a) flat, (b) a direct sum of countably generated modules, and (c) Mittag-Leffler, see Theorem Tag 059Z. This characterization is a key step in the proof of descent of projectivity. For us this characterization is also important because it turns out that if R —> S is a finitely presented ring map, which is flat and “pure” (I hope to discuss this notion in a future post), then S is Mittag-Leffler as an R-module and hence projective as an R-module. This result is a key lemma in Raynaud-Gruson.

Let me say a bit about the structure of countably generated Mittag-Leffler R-module M. First, you can write M as the colimit of a system

M_1 —> M_2 —> M_3 —> M_4 —> …

with each M_n finitely presented (see Lemma Tag 059W and the proof of Lemma Tag 0597). Another application of the Mittag-Leffler condition, using N = ∏ M_i and using that the system is countable, gives for each n an m ≥ n and a map φ : M —> M_m such that M_n —> M —> M_m is the transition map M_n —> M_m. In other words, there exists a self map ψ : M —> M which factors through a finitely presented R-module and which equals 1 on the image of M_n in M. Loosely speaking M has a lot of “compact” endomorphisms. Continuing, I think the existence of ψ means that etale locally on R we have a direct sum decomposition M = M_unit ⊕ M_rest with M_unit finitely presented and such that M_n maps into M_unit. Formulated a bit more canonically we get: (*) Given any map F —> M from a finitely presented module F into M there exists etale locally on R a direct sum decomposition M = A ⊕ B with A a finitely presented module such that F —> M factors through A. It seems likely that (*) also implies that M is Mittag-Leffler (but I haven’t checked this).

In the last couple of weeks I have tried, without any success, to understand what it means for a finitely presented R-algebra S to be Mittag-Leffler as an R-module, without assuming S is flat over R. If you know a nice characterization, or if you think there is no nice characterization please email or leave a comment.

[Edit Oct 7, 2010: Some of the above is now in the stacks project, see Lemma Tag 05D2 for the existence of the maps ψ and see Lemma Tag 05D6 for the result on splitting M as a direct sum of finitely presented modules.]