1996/08/04 by Joseph Lipman, Lipman, Joseph, Pramathanath Sastry +1 · 1 citation
Mathematics · #13D99 #14B15 #14F99 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.alg-geom/9608005
openalex publication_date 1996/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a canonical pseudofunctor ^# on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor ^! to the full subcategory of the derived category having Cohen-Macaulay complexes as objects (a subcategory equivalent to the category of Cousin complexes, once a codimension function has been fixed). Specifically, for Cousin complexes M and any scheme map f:X -> Y as above, there is a functorial derived-category map γ: f^# M -> f^! M inducing a functorial isomorphism in the category of Cousin complexes f^# M \iso E(f^! M) (where E is the Cousin functor). γitself is an isomorphism if the complex f^! M is Cohen-Macaulay--which will be so whenever the map f is flat or whenever the complex M is injective. Also, f^# takes residual (resp. injective) complexes on Y to residual (resp. injective) complexes on X; and so the pseudofunctor ^# generalises--and makes canonical--the "variance theory" of residual complexes developed in Chapter VI of Hartshorne's "Residues and Duality." Moreover, we generalise the Residue Theorem of loc.cit., p.369 by defining a functorial Trace map of graded modules Trf(M): f_*f^# M -> M (a sum of local residues) such that whenever f is proper, Trf(M) is a map of complexes and the pair (f^# M, Trf(M)) represents the functor Hom(f_*C, M) of Cousin complexes C.