{"id":3554,"date":"2013-09-09T02:06:43","date_gmt":"2013-09-09T02:06:43","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3554"},"modified":"2013-09-09T02:06:43","modified_gmt":"2013-09-09T02:06:43","slug":"graded-idempotents","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3554","title":{"rendered":"Graded idempotents"},"content":{"rendered":"<p>Today, I was on and off wondering about idempotents in <b>Z<\/b>-graded associative algebras with a unit (which is assumed homogeneous). In my googling of this, I have found the terminology <i>graded idempotents<\/i> which refers to idempotents which are homogeneous of degree 1. This suggests that there exist others. And indeed, it is easy to make examples of non-homogeneous idempotents by conjugation with units. But we can ask for more.<\/p>\n<ol>\n<li>Is there an example of an idempotent which is not conjugate to a graded idempotent?<\/li>\n<li>Is there an example of a <b>Z<\/b>-graded associative algebra with a nontrivial idempotent but no nontrivial graded idempotents?\n<\/ol>\n<p>Hmm&#8230;?<\/p>\n<p>Some more searching and google finally turned up the paper <i>Idempotents in ring extensions<\/i> by Kanwar, Leroy, and Matczuk which provides the answer to 1. There&#8217;s probably tons of papers that make this observation. Namely, suppose that R is a (commutative) domain such that R[x, x^{-1}] and R don&#8217;t have the same Picard group. For example R = k[t^2, t^3] with k a field (details omitted). Let L be an invertible module over R[x, x^{-1}] which is not isomorphic to the pullback of an invertible module from R. Pick a surjection<\/p>\n<blockquote><p>R[x, x^{-1}]<sup>&oplus; n<\/sup> &#8212;> L<\/p><\/blockquote>\n<p>As L is a projective R[x, x^{-1}]-module we obtain an idempotent e in the <b>Z<\/b>-graded ring M_n(R[x, x^{-1}]) = M_n(R)[x, x^{-1}]. And this idempotent is not conjugate to an element of M_n(R) as that would mean L does come from R.<\/p>\n<p>So this answers 1. I do not know the answer to 2.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Today, I was on and off wondering about idempotents in Z-graded associative algebras with a unit (which is assumed homogeneous). In my googling of this, I have found the terminology graded idempotents which refers to idempotents which are homogeneous of &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3554\">Continue reading <span class=\"meta-nav\">&rarr;<\/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-3554","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\/3554","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=3554"}],"version-history":[{"count":16,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3554\/revisions"}],"predecessor-version":[{"id":3570,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3554\/revisions\/3570"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3554"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3554"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3554"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}