Residual intersections.

*(English)*Zbl 0732.13004The paper is mainly concerned with when a residual intersection is Cohen- Macaulay. The concept of residual intersection was introduced by M. Artin and M. Nagata [J. Math. Kyoto Univ. 12, 307-323 (1972; Zbl 0263.14019)], and has been investigated since then by several authors. Let R be a local Cohen-Macaulay ring and let I be an ideal of R and \(A=(a_{1},...,a_{s})\subset I\), but with \(A\neq I\). \(J=A:I\) is said to be an s-residual intersection of A, or simply a residual intersection, if ht(J)\(\geq s\geq ht(I)\). If furthermore \(I_{p}=A_{p}\) for all \(p\in V(I)\) with ht(p)\(\leq s\), then J is said to be a geometric residual intersection. This notion generalizes the notion of linkage and geometric linkage. I is said to satisfy \(G_{s}\) if \(\mu (I_{p})\leq ht(p)\) for all \(p\in V(I)\) such that ht(p)\(\leq s-1\), and I is said to satisfy \(G_{\infty}\) if it satisfies \(G_{s}\) for all s. Finally, to explain the results we need to recall that I is said to be strongly Cohen- Macaulay (SCM) if the Koszul homology of I is Cohen-Macaulay.

Section 2 contains an explicit description of the canonical module of a geometric residual intersection. The precise statement is this: Let R be a local Gorenstein ring and let I be an ideal of R of height g which is SCM and \(G_{s}\). Suppose that J is a geometric s-residual intersection of I, and let * denote the homomorphism from R to R/J. Then \(\omega_{R}\simeq (I^{*})^{s-g+1}\). Section 2 also contains a key lemma of this paper which says that the property \(G_{s}\) is preserved by generic linkage.

Using these results and facts concerning linkage proved by the authors in another paper [Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003)] as well as deformation arguments which allow one, under certain conditions, to specialize residual intersections, the authors are able to prove the following very general theorem: Let (R,m) be a local Gorenstein ring and suppose that I is an ideal of height g which is evenly linked to an ideal K which is SCM and satisfies \(G_{\infty}\). Let \(J=A:I\) be any s-residual intersection of I. Then R/J is CM, depth\((R/A)=\dim (R)-s\) and \(ht(J)=s\). Furthermore the canonical module of R/J is isomorphic to \(Sym_{s-g+1}(I/A)\). It is also isomorphic to \((I^{s-g+1}+J)/J\) in the case that J is a geometric residual intersection.

Several applications of this theorem are given. One of them answers a question of Pellikaan on the germ of an isolated line singularity.

The key lemma is used again in the final section to prove nice results on rigid licci algebras (”licci” means ”linked to a complete intersection”). Let S be a formal power series ring over a field k and let J be an ideal of S such that S/J is licci and rigid. It is shown that such an ideal is generated by a d-sequence. Since J is SCM by an earlier result of C. Huneke [Am. J. Math. 104, 1043-1062 (1982; Zbl 0505.13003)] it follows in particular that Sym(J) is isomorphic to the Rees-algebra S[Jt], that this algebra is CM, and that the associated graded ring \(gr_{J}(S)=\oplus_{n\geq 0}J^{n}/J^{n+1} \) is a Gorenstein algebra. In fact it is shown that \(gr_{J}(S)\) is a normal domain as well.

Section 2 contains an explicit description of the canonical module of a geometric residual intersection. The precise statement is this: Let R be a local Gorenstein ring and let I be an ideal of R of height g which is SCM and \(G_{s}\). Suppose that J is a geometric s-residual intersection of I, and let * denote the homomorphism from R to R/J. Then \(\omega_{R}\simeq (I^{*})^{s-g+1}\). Section 2 also contains a key lemma of this paper which says that the property \(G_{s}\) is preserved by generic linkage.

Using these results and facts concerning linkage proved by the authors in another paper [Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003)] as well as deformation arguments which allow one, under certain conditions, to specialize residual intersections, the authors are able to prove the following very general theorem: Let (R,m) be a local Gorenstein ring and suppose that I is an ideal of height g which is evenly linked to an ideal K which is SCM and satisfies \(G_{\infty}\). Let \(J=A:I\) be any s-residual intersection of I. Then R/J is CM, depth\((R/A)=\dim (R)-s\) and \(ht(J)=s\). Furthermore the canonical module of R/J is isomorphic to \(Sym_{s-g+1}(I/A)\). It is also isomorphic to \((I^{s-g+1}+J)/J\) in the case that J is a geometric residual intersection.

Several applications of this theorem are given. One of them answers a question of Pellikaan on the germ of an isolated line singularity.

The key lemma is used again in the final section to prove nice results on rigid licci algebras (”licci” means ”linked to a complete intersection”). Let S be a formal power series ring over a field k and let J be an ideal of S such that S/J is licci and rigid. It is shown that such an ideal is generated by a d-sequence. Since J is SCM by an earlier result of C. Huneke [Am. J. Math. 104, 1043-1062 (1982; Zbl 0505.13003)] it follows in particular that Sym(J) is isomorphic to the Rees-algebra S[Jt], that this algebra is CM, and that the associated graded ring \(gr_{J}(S)=\oplus_{n\geq 0}J^{n}/J^{n+1} \) is a Gorenstein algebra. In fact it is shown that \(gr_{J}(S)\) is a normal domain as well.

##### MSC:

13C40 | Linkage, complete intersections and determinantal ideals |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

14M06 | Linkage |

14M10 | Complete intersections |