{"id":910,"date":"2010-10-06T01:44:08","date_gmt":"2010-10-06T01:44:08","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=910"},"modified":"2010-10-07T20:14:04","modified_gmt":"2010-10-07T20:14:04","slug":"mittag-leffler-modules","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=910","title":{"rendered":"Mittag-Leffler modules"},"content":{"rendered":"

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>.<\/p>\n

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 \u220f_n k[[x]]\/(x^n) as k[[x]]-module.<\/p>\n

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<\/a>. 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.<\/p>\n

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<\/p>\n

M_1 —> M_2 —> M_3 —> M_4 —> …<\/p>\n

with each M_n finitely presented (see Lemma Tag 059W<\/a> and the proof of Lemma Tag 0597<\/a>). Another application of the Mittag-Leffler condition, using N = \u220f M_i and using that the system is countable, gives for each n an m \u2265 n and a map \u03c6 : 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 \u03c8 : 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 \u03c8 means that etale locally on R we have a direct sum decomposition M = M_unit \u2295 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 \u2295 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).<\/p>\n

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<\/em> 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.<\/p>\n

[Edit Oct 7, 2010: Some of the above is now in the stacks project, see Lemma Tag 05D2<\/a> for the existence of the maps ψ and see Lemma Tag 05D6<\/a> for the result on splitting M as a direct sum of finitely presented modules.]<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-910","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/910","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=910"}],"version-history":[{"count":27,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/910\/revisions"}],"predecessor-version":[{"id":954,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/910\/revisions\/954"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=910"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=910"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}