{"id":4093,"date":"2016-02-17T02:47:40","date_gmt":"2016-02-17T02:47:40","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=4093"},"modified":"2016-03-01T17:30:36","modified_gmt":"2016-03-01T17:30:36","slug":"canonical-divisor","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4093","title":{"rendered":"Canonical divisor"},"content":{"rendered":"

Let k be a field and let X be a finite type scheme over k. Let F be a coherent O_X-module which is generically invertible<\/em>. This means there exists a an open dense subscheme such that F is an invertible module when restricted to that open.<\/p>\n

Lemma:<\/b> There exists an open subscheme U containing all codimension 1 points, an invertible O_U-module L, and a map a : L → F|_U which is generically an isomorphism, i.e., there exists an open dense subscheme of U such that a restricted to that open is an isomorphism.<\/p>\n

Proof.<\/em> We already have a triple (U, L, a) for some dense open U in X. To prove the lemma we can proceed by adding 1 codimension 1 point \u03be at a time. To do this we may work over the 1-dimensional local ring at \u03be, where the existence of the extension is more or less clear.<\/p>\n

Now assume that X is equidimensional of dimension d. Then we have a Chow group A_{d-1}(X) of codimension 1 cycles. If X is integral this is called the Weil divisor class group. For F as above we pick (U, L, a) as in the lemma. Observe that A_{d-1}(U) = A_{d-1}(X).<\/p>\n

Def:<\/b> The divisor associated to<\/em> F is c_1(L) \u2229 [U]_d + [Coker(a)]_{d-1} – [Ker(a)]_{d-1}<\/p>\n

The notation here is as in the chapter Chow Homology of the Stacks project. The first term c_1(L) \u2229 [U]_d is the first chern class of L on U and the other two terms involve taking lengths at codimension 1 points. Using the lemma to compare different triples for F it is easy to verify this is well defined as an element of A_{d-1}(X).<\/p>\n

Def:<\/b> Assume in addition X is generically Gorenstein, i.e., there exists a dense open which is Gorenstein. Let \u03c9 and \u03c9’ be the cohomology sheaves of the dualizing complex of X in degrees -d and -d+1. The canonical divisor<\/em> K_X is the divisor associated to \u03c9 minus [\u03c9’]_{d-1}.<\/p>\n

There you go; you’re welcome!<\/p>\n

Rmks:<\/b>
\n1. Fulton’s “Intersection Theory” defines the todd class of X in complete generality.
\n2. If X is generically reduced, then X is generically regular, hence generically Gorenstein and our definition applies.
\n3. The term [ω’]_{d-1} is zero if X is Cohen-Macaulay in codim 1.
\n4. If X is Gorenstein in codimension 1, then our canonical divisor agrees with the canonical divisor you find in many papers.
\n5. A canonical divisor of an equidimensional X can always be defined: either by Fulton or by generalizing the definition of the divisor associated to F to the case where F and O_X define the same class in K_0(Coh(U)) for some dense open U. This will always be true for ω. Just takes a bit more work.
\n6. If X is proper and equidimensional of dimension 1, then χ(F) = deg(divisor asssociated to F) + χ(O_X) whenever F is generically invertible.
\n7. If X is proper and equidimensional of dimension 1, then deg(K_X) = – 2χ(O_X).
\n8. If X is a curve and f : Y → X is the normalization, then K_X = f_*(K_Y) + 2 ∑ \u03b4_P P where \u03b4_P is the delta invariant at the point P (Fulton, Example 18.3.4).
\n9. If X is equidimensional of dimension 1 and Z ⊂ X is the largest CM subscheme agreeing with X generically, then K_X = K_Z – 2 ∑ t_P P where t_P is the length of the torsion submodule in O_{X,P}.<\/p>\n

Edit 3\/1\/2016:<\/b> Jason Starr commented below that there is a refinement which is sometimes useful, namely, one can ask for a Todd class and Riemann-Roch in K-theory and he just added by email: “In our joint work on rational simple connectedness of low degree complete intersections, we need to know that certain (integral) Cartier divisor classes on moduli spaces are Q-linearly equivalent. It is not enough to know that the pushforward cycles classes to the (induced reduced) coarse moduli scheme are rationally equivalent. So we need the Riemann-Roch that works on K-theory. In fact, the relevant computations are in our earlier manuscript about “Virtual canonical bundle …”, and we slightly circumvent Riemann-Roch in the computation. But, morally, we are using a Todd class that lives in K-theory, not just in CH_*.”<\/p>\n","protected":false},"excerpt":{"rendered":"

Let k be a field and let X be a finite type scheme over k. Let F be a coherent O_X-module which is generically invertible. This means there exists a an open dense subscheme such that F is an invertible … 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-4093","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\/4093","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=4093"}],"version-history":[{"count":20,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4093\/revisions"}],"predecessor-version":[{"id":4114,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4093\/revisions\/4114"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4093"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4093"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4093"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}