{"id":3453,"date":"2013-08-01T17:27:13","date_gmt":"2013-08-01T17:27:13","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3453"},"modified":"2013-08-01T19:40:10","modified_gmt":"2013-08-01T19:40:10","slug":"simplicial-modules","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3453","title":{"rendered":"Simplicial modules"},"content":{"rendered":"

A simplicial ring A•<\/sub> is just a simplicial object in the category of rings. What is a simplicial module over A•<\/sub>? Well it is a simplicial object in the category of systems (A, M, +, *, +, *) where A is a ring and M is an A-module (so the + and * are multiplication and addition on A and M respectively) such that forgetful functor to the category of rings gives back A•<\/sub>.<\/p>\n

Of course this is annoying. Better: A simplicial ring A•<\/sub> is a sheaf on \u0394 (the category of finite ordered sets endowed with the chaotic topology). Then a simplicial module over A•<\/sub> is just a sheaf of modules.<\/p>\n

You can extend this to simplicial sheaves of rings over a site C. Namely, consider the category C x \u0394 together with the projection C x \u0394 —> C. This is a fibred category hence we get a topology on C x \u0394 inherited from C. Then a simplicial sheaf of rings A•<\/sub> is just a sheaf of rings on C x \u0394 and we define a simplicial module over A•<\/sub> as a sheaf of modules on C x \u0394 over this sheaf of rings. There is a derived category D(A*<\/sub>) and a derived lower shriek functor<\/p>\n

L\u03c0!<\/sub> : D(A•<\/sub>) ———-> D(C)<\/p>\n

as discussed in Tag 08RV<\/a>. Moreover, a map A•<\/sub> —> B•<\/sub> of simplicial rings on C gives rise to a morphism of ringed topoi, and hence a derived base change functor<\/p>\n

D(A•<\/sub>) ———-> D(B•<\/sub>)<\/p>\n

as well as a restriction functor the other way.<\/p>\n

Why am I pointing this out? The reason is to use it for the following. If A —> B is a map of sheaves of rings and M is a B-module, then a priori the Atiyah class “is” the extension of principal parts<\/p>\n

0 —> ΩP•<\/sub>\/A<\/sub> ⊗ M —> E —> M —> 0<\/p>\n

over the polynomial simplicial resolution P•<\/sub> of B over A. To get it in D(B) Illusie uses the base change along the map P•<\/sub> —> B. I was worried that we’d have to introduce lots of new stuff in the Stacks project to even define this, but all the nuts and bolts are already there. Cool!<\/p>\n

PS: Warning! The category D(A•<\/sub>) is not the same as the category D•<\/sub>(A•<\/sub>) defined in Illusie.<\/p>\n","protected":false},"excerpt":{"rendered":"

A simplicial ring A• is just a simplicial object in the category of rings. What is a simplicial module over A•? Well it is a simplicial object in the category of systems (A, M, +, *, +, *) where A … 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-3453","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\/3453","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=3453"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3453\/revisions"}],"predecessor-version":[{"id":3463,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3453\/revisions\/3463"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}