An explicit construction of the Grothendieck residue complex. With an appendix by Pramathanath Sastry.

*(English)*Zbl 0788.14011
Astérisque. 208. Paris: Société Mathématique de France, 127 p., Appendix: 117-126 (1992).

Grothendieck’s original description (1958) of duality for coherent sheaves on a scheme \(X\) of finite type over a field \(k\) made use of the residue complex \(K^ 0_ X\), which is a direct sum of sheaves \(D(X/Y)\), with \(Y\subseteq X\) irreducible and closed. Later, Hartshorne (1966) reformulated duality theory in terms of derived categories, replacing \(K^ 0_ X\) by suitable functors \(f^ !:{\mathcal D}_ c^ +(Y)\to{\mathcal D}^ +_ c(X)\), assigned to morphisms \(f:X\to Y\). If \(X\) has finite type over \(k\), say with structure map \(\pi\), then the residue complex \(K^ 0_ X\) is the Cousin complex \(\pi^ \Delta k\) associated to \(\pi^ !k\in{\mathcal D}_ c^ +(X)\). In Hartshorne’s approach, the module structure of the residue complex is lost, however, essentially due to the fact that the summands of \(\pi^ \Delta k\) are local cohomologies of \(\pi^ !k\) and, as such, are not expressed in a concrete form.

The aim of this monograph is to present an explicit construction of the residue complex \(K^ 0_ X\), at least, when \(X\) is a reduced sheaf of finite type over a perfect field \(k\), i.e., based on a concrete realisation of this complex as an \({\mathcal O}_ X\)-module as well as a concrete description of the associated coboundary operator.

In order to realize this, the author first develops the theory of semitopological rings, needed in order to remedy the fact that the topologized rings one encounters in this set-up do not have adic topologies, hence may not be treated in the traditional way. In fact, these rings are not even topological rings: although left and right multiplication on such a ring \(A\) are continuous, the multiplication map \(A\times A\to A\) is not! A systematic study of semi-topological rings shows, however, that they behave nicely with respect to most of the traditional operations, including completions, and that they admit a differential calculus.

These techniques are then applied to topological local fields, i.e., local fields \(K\) containing a fixed perfect field \(k\) and which are semitopological \(k\)-algebras as such. For technical reasons, one also requires that there should be some parametrization \(k\simeq F((t_ n))\dots((t_ 1))\) with \(F\) discrete and rk\(_ F\Omega^ 1_{F/k}<\infty\). One of the main results here is a axiomatic treatment of the residue functor defined on the category of (reduced clusters of) topological local fields, which provides an improved version of the Parshin-Lomadze residue functor adapted to the present treatment.

In a subsequent chapter, the author introduces the Beilinson completion \({\mathcal M}_ \xi\) of a quasi coherent sheaf \({\mathcal M}\) on \(X\) with respect to a chain \(\xi\) in \(X\), i.e., a sequence of points \(\xi=(x_ 0,\dots,x_ l)\) with \(x_ i>x_{i+1}\) for all \(i\). This completion is a generalization of Beilinson’s sheaves of adèles, which occur as “local factors” in the present construction. It appears that the completion \({\mathcal O}_{X,\xi}\) is a commutative semitopological \(k\)- algebra for every chain \(\xi\) (which is semilocal if \(\xi\) is saturated) and that \({\mathcal M}_ \xi\) is a semitopological \({\mathcal O}_{X,\xi}\)- module for every quasicoherent sheaf \({\mathcal M}\), thus yielding an exact functor \((-)_ \xi\). In fact, if \(\xi=(x,\dots,y)\) is a saturated chain of length \(n\) then \(k(x)_ \xi=k(\xi)\) is an \(n\)-dimensional reduced cluster of topological local fields, whose spectrum is determined by repeated normalization. This (and other) results yield a link between the geometry of \(X\) and topological local fields. At this point the author is reado to describe explicitly the complex \((K^ 0_ X,\delta_ x)\). In fact, starting from the Parshin residue maps \[ \text{Res}_{\xi,\sigma}:\Omega^*_{k(x)/k}\to\Omega^{*,\text{sep} }_{k(\xi)/k} @>\text{Res}_{k(\xi)/k(y),\sigma}>>\Omega^*_{k(y)/k}, \] where \(\xi=(x,\dots,y)\) is a saturated chain in \(X\) and \(\sigma:k(y)\to{\mathcal O}_{X,(y)}=\widehat{\mathcal O}_{X,y}\) a coefficient field for \(y\), one may deduce for compatible coefficient fields \(\sigma/\tau\) for \(\xi\) a coboundary map \[ \sigma_{\xi,\sigma/\tau}:K(\sigma)\to K(\tau). \] Here, \(K(\tau)=\text{Hom}^{\text{cont}}_{k(y)}(\widehat{\mathcal O}_{X,y},\omega(y))\), where \(\omega(y)=\Omega^ d_{k(y)/k}\), with \(d=rk_{k(y)}\Omega^ 1_{k(y)/k}\). One may show (through base change) that this construction is independent of the choice of coefficient fields. The above complex is then just the sum of the local components over the \(x\in X\).

The appendix (written by P. Sastry) describes the canonical isomorphism \(K^ 0_ X\equiv\pi^ !k\).

The aim of this monograph is to present an explicit construction of the residue complex \(K^ 0_ X\), at least, when \(X\) is a reduced sheaf of finite type over a perfect field \(k\), i.e., based on a concrete realisation of this complex as an \({\mathcal O}_ X\)-module as well as a concrete description of the associated coboundary operator.

In order to realize this, the author first develops the theory of semitopological rings, needed in order to remedy the fact that the topologized rings one encounters in this set-up do not have adic topologies, hence may not be treated in the traditional way. In fact, these rings are not even topological rings: although left and right multiplication on such a ring \(A\) are continuous, the multiplication map \(A\times A\to A\) is not! A systematic study of semi-topological rings shows, however, that they behave nicely with respect to most of the traditional operations, including completions, and that they admit a differential calculus.

These techniques are then applied to topological local fields, i.e., local fields \(K\) containing a fixed perfect field \(k\) and which are semitopological \(k\)-algebras as such. For technical reasons, one also requires that there should be some parametrization \(k\simeq F((t_ n))\dots((t_ 1))\) with \(F\) discrete and rk\(_ F\Omega^ 1_{F/k}<\infty\). One of the main results here is a axiomatic treatment of the residue functor defined on the category of (reduced clusters of) topological local fields, which provides an improved version of the Parshin-Lomadze residue functor adapted to the present treatment.

In a subsequent chapter, the author introduces the Beilinson completion \({\mathcal M}_ \xi\) of a quasi coherent sheaf \({\mathcal M}\) on \(X\) with respect to a chain \(\xi\) in \(X\), i.e., a sequence of points \(\xi=(x_ 0,\dots,x_ l)\) with \(x_ i>x_{i+1}\) for all \(i\). This completion is a generalization of Beilinson’s sheaves of adèles, which occur as “local factors” in the present construction. It appears that the completion \({\mathcal O}_{X,\xi}\) is a commutative semitopological \(k\)- algebra for every chain \(\xi\) (which is semilocal if \(\xi\) is saturated) and that \({\mathcal M}_ \xi\) is a semitopological \({\mathcal O}_{X,\xi}\)- module for every quasicoherent sheaf \({\mathcal M}\), thus yielding an exact functor \((-)_ \xi\). In fact, if \(\xi=(x,\dots,y)\) is a saturated chain of length \(n\) then \(k(x)_ \xi=k(\xi)\) is an \(n\)-dimensional reduced cluster of topological local fields, whose spectrum is determined by repeated normalization. This (and other) results yield a link between the geometry of \(X\) and topological local fields. At this point the author is reado to describe explicitly the complex \((K^ 0_ X,\delta_ x)\). In fact, starting from the Parshin residue maps \[ \text{Res}_{\xi,\sigma}:\Omega^*_{k(x)/k}\to\Omega^{*,\text{sep} }_{k(\xi)/k} @>\text{Res}_{k(\xi)/k(y),\sigma}>>\Omega^*_{k(y)/k}, \] where \(\xi=(x,\dots,y)\) is a saturated chain in \(X\) and \(\sigma:k(y)\to{\mathcal O}_{X,(y)}=\widehat{\mathcal O}_{X,y}\) a coefficient field for \(y\), one may deduce for compatible coefficient fields \(\sigma/\tau\) for \(\xi\) a coboundary map \[ \sigma_{\xi,\sigma/\tau}:K(\sigma)\to K(\tau). \] Here, \(K(\tau)=\text{Hom}^{\text{cont}}_{k(y)}(\widehat{\mathcal O}_{X,y},\omega(y))\), where \(\omega(y)=\Omega^ d_{k(y)/k}\), with \(d=rk_{k(y)}\Omega^ 1_{k(y)/k}\). One may show (through base change) that this construction is independent of the choice of coefficient fields. The above complex is then just the sum of the local components over the \(x\in X\).

The appendix (written by P. Sastry) describes the canonical isomorphism \(K^ 0_ X\equiv\pi^ !k\).

Reviewer: A.Verschoren (Antwerp)

##### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

13J10 | Complete rings, completion |

14B15 | Local cohomology and algebraic geometry |