{"id":4912,"date":"2023-09-01T21:33:59","date_gmt":"2023-09-01T21:33:59","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4912"},"modified":"2023-09-01T21:33:59","modified_gmt":"2023-09-01T21:33:59","slug":"endomorphisms-of-the-koszul-complex","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4912","title":{"rendered":"Endomorphisms of the Koszul complex"},"content":{"rendered":"

Let k be a ring, for example a field. Let R be a k-algebra. Let f_1, …, f_r be a regular sequence in R such that k → R\/(f_1, …, f_r) is an isomorphism. Let K be the Koszul complex over R on f_1, …, f_r viewed as a cochain complex sitting in degrees -r, …, 0. See Tag 0621<\/a>. Then we are interested in the hom-complex<\/p>\n

E = Hom_R(K, K)<\/p><\/blockquote>\n

constructed in Tag 0A8H<\/a> which we may and do view as a differential graded R-algebra, see for example Tag 0FQ2<\/a>. This algebra is interesting for many reasons; for example because there is an equivalence<\/p>\n

D(E) = D_{QCoh, V(f_1, …, f_r)}(Spec(R))<\/p><\/blockquote>\n

Here the LHS is the derived category of dg E-modules and the RHS is the derived category of complexes of quasi-coherent modules on Spec(R) supported set theoretically on f_1 = … = f_r = 0. To prove this equivalence, use Tag 09IR<\/a> to see that K gives a generator of the RHS and argue as in the proof of Tag 09M5<\/a>.<\/p>\n

Recall that the underlying R-module of K is the exterior algebra on the free module of rank r, say with basis e_1, …, e_r. For i = 1, .., r let v_i : K → K be the operator given by contraction by the dual basis element to e_i. Of course v_i has degree 1 and a computation shows that v_i : K → K[1] is a map of complexes. Similarly, the reader shows that v_i ∘ v_j = – v_j ∘ v_i and v_i ∘ v_i = 0. Thus we obtain a map of differential graded k-algebras<\/p>\n

k ⟨ v_1, …, v_r ⟩ → E<\/p><\/blockquote>\n

with target E and source the exterior algebra over k on v_1, …, v_r in degree 1 and vanishing differential. A bit more work shows that this map is a quasi-isomorphism of differential graded k-algebras; this is where we use the assumption that f_1, …, f_r is a regular sequence so the koszul complex is a resolution of R\/(f_1, …, f_r) by Tag 062F<\/a>.<\/p>\n

Applying Tag 09S6<\/a> we conclude that<\/p>\n

D(k ⟨ v_1, …, v_r ⟩) = D_{QCoh, V(f_1, …, f_r)}(Spec(R))<\/p><\/blockquote>\n

Cheers!<\/p>\n","protected":false},"excerpt":{"rendered":"

Let k be a ring, for example a field. Let R be a k-algebra. Let f_1, …, f_r be a regular sequence in R such that k → R\/(f_1, …, f_r) is an isomorphism. Let K be the Koszul complex … 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-4912","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\/4912","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=4912"}],"version-history":[{"count":17,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4912\/revisions"}],"predecessor-version":[{"id":4929,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4912\/revisions\/4929"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4912"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4912"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4912"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}