{"id":2253,"date":"2012-03-27T01:46:39","date_gmt":"2012-03-27T01:46:39","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2253"},"modified":"2012-03-27T01:46:39","modified_gmt":"2012-03-27T01:46:39","slug":"compact-and-perfect-objects","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2253","title":{"rendered":"Compact and perfect objects"},"content":{"rendered":"<p>Let R be a ring. Let D(R) be the derived category of R-modules. An object K of D(R) is <em>perfect<\/em> if it is quasi-isomorphic to a finite complex of finite projective R-modules. An object K of D(R) is called <em>compact<\/em> if and only of the functor Hom_{D(R)}(K, &#8211; ) commutes with arbitrary direct sums. In the <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2227\">previous post<\/a> I mentioned two results on perfect complexes which I added to the stacks project today. Both are currently in the <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/more-algebra.pdf\">second chapter on algebra of the stacks project<\/a>. Here are the statements with corresponding tags:<\/p>\n<ol>\n<li>An object K of D(R) is perfect if and only if it is compact. This is <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=07LT\">Proposition Tag 07LT<\/a>.<\/li>\n<li>If I &subset; R is an ideal of square zero and K &otimes;^L R\/I is a perfect object of D(R\/I), then K is a perfect object of D(R). This is <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=07LU\">Lemma Tag 07LU<\/a>.<\/li>\n<\/ol>\n<p>Enjoy! If anybody knows a reference for the first result which predates the paper &#8220;Morita theory for derived categories&#8221; by Rickard I&#8217;d love to hear about it. Thanks.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let R be a ring. Let D(R) be the derived category of R-modules. An object K of D(R) is perfect if it is quasi-isomorphic to a finite complex of finite projective R-modules. An object K of D(R) is called compact &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2253\">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-2253","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\/2253","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=2253"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2253\/revisions"}],"predecessor-version":[{"id":2258,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2253\/revisions\/2258"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2253"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2253"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}