More Examples

Some important examples and explications of Grothendieck duality which should have been included in Chapter 5 of [C1] were overlooked. Several such topics are treated here. One question which can be asked about the trace map H(X,ΩX/k) → k for a proper smooth scheme X of pure dimension n over a field k is how it relates to the trace map for a smooth divisor. Generalizing a little, one can ask whether there is an analogous statement if one replace “smooth” with “CM”. This matter is addressed in the relative case in §1, and the answer winds up involving a sign of (−1)n−1. In a different direction, one can ask how the Grothendieck trace map for a proper smooth variety over C is related to a “topological” trace map defined in terms of topological Poincaré duality, as in [C1, (2.3.4)]. They do not agree, but instead are off by a sign of (−1). In the projective case, one can analyze this problem quite pleasantly by using Bertini’s theorem, induction, and the divisor compatibility treated in §1, once one has also worked out the analytic analogue of §1. We work out such an analogue in §2, and the answer winds up involving a sign of (−1), a contrast with the sign of (−1)n−1 in the algebraic case, and it is this discrepancy of −1 that ultimately leads to the sign of (−1) when comparing the Grothendieck and topological traces in the projective algebraic case (since each of the n hyperplane slices contributes a sign of −1 in the comparison, but everything agrees in dimension 0). Despite naive impressions, our sign of (−1) in the analytic/algebraic trace comparison is consistent with the lack of such a sign in the analytic/algebraic comparison given in the appendix to [D]. The explanation for the consistency is given in the pp. 33, 36 remarks in [C2]. The analytic/algebraic comparison in the general proper case can be reduced to the projective case (and even just the case of projective spaces!) by using local cohomology, as we explain in §3. Finally, in §4 we work out the comparison between the residue map and the Chern class map for line bundles on curves (where the Chern class is viewed with values in the ground field rather than in Z). I am grateful to deJong, Ogus, and Serre for asking me questions about these topics, and to P. Sastry and Y. Tong for convincing me that one could use local cohomology to replace the use of alterations in the comparison of the analytic/algebraic traces.