Integration of differential forms on schemes.

*(English)*Zbl 0712.14006The authors give an extensive treatment of the algebraic theory of residues. They consider separated morphisms \(f:X\to Y\) of noetherian schemes with f being equidimensional of dimension d and generically smooth. They are interested in the integration and calculation of residues of regular d-forms, resp. their local comohomology classes with supports in suitable finite subschemes \(Z/Y\) of \(X/Y\). Here “integral” means the “residue” at closed subschemes and “residue” the residue at a closed point of X.

In § 1-3 the affine theory is developed. Let R be an excellent noetherian ring with \(Ass(R)=Min(R)\) and \(S/R\) an algebra of finite type. It is assumed that \(Ass(S)=Min(S)\) and \(S/R\) equidimensional of dimension \(d\) and generically smooth. For any system \(t=\{t_ 1,...,t_ d\}\subseteq S\) such that \(S/(t)\) is a finite R-module and any regular differential \(\omega \in \omega^ d_{S/R}\) an integral \(\int_{S/R}\left[\begin{matrix} \omega \\ t\end{matrix} \right] \) of \(\omega\) with respect to t is constructed in § 1. Many technical details are necessary. The construction is based upon the theory of traces of differential forms for finite complete intersections. The connection to Lipman’s construction of residues is explained. The behaviour under flat base change is treated and a trace formula is proven.

§ 2 covers the definition of the residue of \(\omega\) at a maximal ideal \({\mathfrak P}\) of S similar to the definition of the integral. It is assumed that \(S^*_{{\mathfrak P}}/(t)S^*_{{\mathfrak P}}\) is a finite \(R^*_{{\mathfrak p}}\)-module, with \({\mathfrak p}={\mathfrak P}\cap R\) and \((.)^*\) denoting the completion with respect to the Jacobson radical. Contrary to the integral the residue \(Res_{{\mathfrak P}}\left[\begin{matrix} \omega \\ t \end{matrix} \right]\) is not an element of R but only of \(R^*_{{\mathfrak p}}\). After some technicalities about systems of parameters the authors give another description of the residue symbol via prime bases from which the transition formula about the behaviour of the residue with respect to the transition from one system t to another is derived. The close connection between integrals and residue symbols is shown in the following theorem 2.6 (\(\hat S\) denotes the (t)-adic completion): Let R be local with maximal ideal \({\mathfrak m}\) and let \({\mathfrak P}_ 1,...,{\mathfrak P}_ h\) be the maximal ideals of S with \((t)+{\mathfrak m}S\subseteq {\mathfrak P}_ i\). For each \(\omega\in {\tilde \omega}^ d_{\hat S/R}\) one has \(\int_{\hat S/R}{\omega \brack t} =\sum^{h}_{i=1}Res_{{\mathfrak P}_ i}{\omega \brack t}\). Together with the transition formula for the residue symbol one then gets a similar transition formula for the integral. Finally a trace formula for the residue symbol is proven.

§ 3 extends the theory to integration of local cohomology classes. Let J be an ideal with \(S/J\) a finite R-module, and assume that there exists a system \(t=\{t_ 1,...,t_ d\}\) with \(Rad(t)=Rad(J)\). The transition formula for the integral from § 2 shows that by passing to the direct limit the maps \(\int_{S/R}: \omega^ d_{S/R}/(t)\omega^ d_{S/R}\to R \) induce a canonical R-homomorphism \(\int_{S/R,J}H^ d_ J(\omega^ d_{S/R}) \to R\). A local duality theorem for \(\hat S\)- modules M is proven. Several facts on the local modules \(H^{d- i}_{J\hat S}(M)t\) are stated without proofs, and a new interpretation for the formula of theorem 2.6 is given.

§ 4 contains the globalization of the previous results by glueing affine pieces. The authors consider the situation \(f:X\to Y\), which was described above. Under additional assumptions on the closed subschemes Z/Y of X/Y they show the existence of the integral. To do this a generalized version of the local cohomology is developed. The authors show that their assumptions on Z are often satisfied especially in the important case of \(f:X\to Y\) being projective and \(Z\subseteq X\) a closed subscheme which is finite over Y.

In § 1-3 the affine theory is developed. Let R be an excellent noetherian ring with \(Ass(R)=Min(R)\) and \(S/R\) an algebra of finite type. It is assumed that \(Ass(S)=Min(S)\) and \(S/R\) equidimensional of dimension \(d\) and generically smooth. For any system \(t=\{t_ 1,...,t_ d\}\subseteq S\) such that \(S/(t)\) is a finite R-module and any regular differential \(\omega \in \omega^ d_{S/R}\) an integral \(\int_{S/R}\left[\begin{matrix} \omega \\ t\end{matrix} \right] \) of \(\omega\) with respect to t is constructed in § 1. Many technical details are necessary. The construction is based upon the theory of traces of differential forms for finite complete intersections. The connection to Lipman’s construction of residues is explained. The behaviour under flat base change is treated and a trace formula is proven.

§ 2 covers the definition of the residue of \(\omega\) at a maximal ideal \({\mathfrak P}\) of S similar to the definition of the integral. It is assumed that \(S^*_{{\mathfrak P}}/(t)S^*_{{\mathfrak P}}\) is a finite \(R^*_{{\mathfrak p}}\)-module, with \({\mathfrak p}={\mathfrak P}\cap R\) and \((.)^*\) denoting the completion with respect to the Jacobson radical. Contrary to the integral the residue \(Res_{{\mathfrak P}}\left[\begin{matrix} \omega \\ t \end{matrix} \right]\) is not an element of R but only of \(R^*_{{\mathfrak p}}\). After some technicalities about systems of parameters the authors give another description of the residue symbol via prime bases from which the transition formula about the behaviour of the residue with respect to the transition from one system t to another is derived. The close connection between integrals and residue symbols is shown in the following theorem 2.6 (\(\hat S\) denotes the (t)-adic completion): Let R be local with maximal ideal \({\mathfrak m}\) and let \({\mathfrak P}_ 1,...,{\mathfrak P}_ h\) be the maximal ideals of S with \((t)+{\mathfrak m}S\subseteq {\mathfrak P}_ i\). For each \(\omega\in {\tilde \omega}^ d_{\hat S/R}\) one has \(\int_{\hat S/R}{\omega \brack t} =\sum^{h}_{i=1}Res_{{\mathfrak P}_ i}{\omega \brack t}\). Together with the transition formula for the residue symbol one then gets a similar transition formula for the integral. Finally a trace formula for the residue symbol is proven.

§ 3 extends the theory to integration of local cohomology classes. Let J be an ideal with \(S/J\) a finite R-module, and assume that there exists a system \(t=\{t_ 1,...,t_ d\}\) with \(Rad(t)=Rad(J)\). The transition formula for the integral from § 2 shows that by passing to the direct limit the maps \(\int_{S/R}: \omega^ d_{S/R}/(t)\omega^ d_{S/R}\to R \) induce a canonical R-homomorphism \(\int_{S/R,J}H^ d_ J(\omega^ d_{S/R}) \to R\). A local duality theorem for \(\hat S\)- modules M is proven. Several facts on the local modules \(H^{d- i}_{J\hat S}(M)t\) are stated without proofs, and a new interpretation for the formula of theorem 2.6 is given.

§ 4 contains the globalization of the previous results by glueing affine pieces. The authors consider the situation \(f:X\to Y\), which was described above. Under additional assumptions on the closed subschemes Z/Y of X/Y they show the existence of the integral. To do this a generalized version of the local cohomology is developed. The authors show that their assumptions on Z are often satisfied especially in the important case of \(f:X\to Y\) being projective and \(Z\subseteq X\) a closed subscheme which is finite over Y.

Reviewer: R.Berger

##### MSC:

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

14B15 | Local cohomology and algebraic geometry |

13N05 | Modules of differentials |