{"id":372,"date":"2023-10-14T01:32:16","date_gmt":"2023-10-14T01:32:16","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=372"},"modified":"2023-10-14T01:32:16","modified_gmt":"2023-10-14T01:32:16","slug":"the-jacobson-density-theorem","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2023\/10\/14\/the-jacobson-density-theorem\/","title":{"rendered":"The Jacobson Density Theorem"},"content":{"rendered":"

I’ve never been at ease with the Jacobson density theorem, every proof that I’d seen felt somewhat weird and never had any stickiness in my mind. So I came up with my own proof, I hope you like it.<\/p>\n

The Theorem (Jacobson Density Theorem}<\/b>
\nLet \"k\" be a field and \"A\" a \"k\"-algebra (yes unital). Let \"S\" be a finite dimensional simple \"A\"-module and \"D=\operatorname{End}_A(S)\" (which is a division algebra by Schur’s Lemma). Then the canonical homomorphism from \"A\" to \"\operatorname{End}_D(S)\" is surjective.<\/p>\n

Proof<\/b>
\nFirst, without loss of generality (replacing \"A\" by a quotient), we can assume that \"S\" is faithful.<\/p>\n

Let us first deal with the case when \"A\" is a division algebra. Then \"S=A\". And \"D=A\" as well, acting by right multiplication. And we are done.<\/p>\n

Since \"A\" is not a division algebra it has a non-zero element \"a\" which is not invertible.<\/p>\n

If \"a\" is not nilpotent, then the minimal polynomial \"m(x)\in k[x]\" of \"a\" factors as \"x^np(x)\" where \"n,\deg(p(x))>0\". In particular it has two coprime factors, so by the Chinese remainder theorem \"k[x]/(m(x))\" has a non-trivial idempotent. In particular, \"A\" contains a non-trivial idempotent.<\/p>\n

If on the other hand \"a\" is nilpotent, then we can find \"v,w\in V\" with \"av=0\", \"aw=v\" and \"v\neq 0\". Since \"V\" is irreducible, there exists \"b\in A\" with \"bv=w\". Then \"(ba)v=0\" and \"(ba)w=w\". So \"ba\" is not nilpotent, and we can run the argument of the previous paragraph again to conclude that \"A\" contains a non-trivial idempotent.<\/p>\n

Let \"e\" be this non-trivial idempotent. We will use the fact that \"eS\" is an irreducible \"eAe\"-module. This is part of a general fact, that \"S\mapsto eS\" is a bijection betwen simple \"A\"-modules \"S\" with \"eS\neq 0\" and simple \"eAe\"-modules. Furthermore, from the nature of the constructions of both directions of this bijection, we can conclude that<\/p>\n

  <\/span>   <\/span>\"\[\operatorname{End}_A(S)\cong \operatorname{End}_{eAe}(eS).\]\"<\/p>\n

Now we can perform an induction on dimension. The inductive hypothesis tells us that \"eAe \to \operatorname{End}_{D}(eS)\" and \"(1-e)A(1-e)\to \operatorname{End}_{D}((1-e)S)\" are both surjective. To finish, given the symmetry between \"e\" and \"1-e\", it suffices to show that given any \"D\"-hyperplane \"H\" with \"eS\subset H\" and any \"D\"-line \"\ell\subset eS\", there is a nonzero element in \"A\" with kernel \"H\" and image \"\ell\". (I thought about this step myself in terms of matrices, but actually writing it up like that feels a little gauche).<\/p>\n

The inductive hypothesis applied to \"(1-e)A(1-e)\" allows us to find a nonzero element \"c\in A\" with \"ker(c)=H\". Pick nonzero \"v\in \im(c)\" and \"w\in \ell\". Since \"S\" is simple there exists \"d\in A\" with \"dv=w\". Then \"dc\" does the trick and our proof is complete.<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve never been at ease with the Jacobson density theorem, every proof that I’d seen felt somewhat weird and never had any stickiness in my mind. So I came up with my own proof, I hope you like it. The … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-372","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-60","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/372","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=372"}],"version-history":[{"count":5,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/372\/revisions"}],"predecessor-version":[{"id":383,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/372\/revisions\/383"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=372"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=372"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}