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**The structure of linkage.**
*(English)*
Zbl 0638.13003

Two unmixed ideals I, J in a Cohen-Macaulay local ring R are said to be linked (written \(I\sim J)\) if there is a regular sequence \(x=x_ 1,...,x_ g\) in \(I\cap J\) such that \((x):I=J\) and \((x):J=I\). Two ideals are in the same linkage class if there is a finite chain of links from one to the other; ideals in the linkage class of a complete intersection are called licci. While many properties are invariants of a linkage class (or at least of the even linkage class of an ideal), better results seem to require more control over the intermediate steps of the linking chain. The authors prove a remarkable theorem asserting that given I, there is a universal chain of links \(IS\sim L^ 1(I)\sim...\sim L^ n(I)\) in a suitable extension \(S=R[X]_{{\mathfrak m}_ R[X]}\). Universality means that if \(I\sim I_ 1\sim...\sim I_ n\) is any chain of links in R, then \((S,L^ i(I))\) is essentially a deformation of \((R,I_ i)\) for every i, that is, obtained by deformation, localization, and automorphism. In particular, since I is linked to itself in two steps each \(L^{2k}(I)\) is essentially a deformation of I. The result is an explicit construction based on the idea of generic linkage. If \(I=(f_ 1,...,f_ n)=(f)\) then \(\alpha =fX\) is a regular sequence in \(R'=R[X]\) where X is an \(n\times g\) matrix of indeterminates. Then \(L_ 1(f)=(\alpha):IR'\) is called the first generic link, and \(L^ 1(f)=(\alpha):IS\) is the first universal link. By definition \(L^ i(I)=L^ 1(L^{i-1}(I))\) for \(i>0.\)

The utility of the authors’ result has two main aspects. First, since it is an explicit construction, they are able to actually compute the non- Gorenstein and non-complete intersection loci of the second universal link (this is the critical step) and to prove that the defining ideals of these loci have prime radicals. This enables them to establish sharp smoothability results. Secondly, there is the philosophy, amply demonstrated here, that if (S,J) is essentially a deformation of (R,I) then S/J is “at least as good” as R/I. The authors prove that if R/I is Cohen-Macaulay, Gorenstein, complete intersection, \((R_ k)\), respectively, then so is S/J. The multiplicity, Cohen-Macaulay type, embedding codimension of S/J, and deviation of J are all less than or equal to the corresponding values for R/I and I.

Applications are to ideals in a regular local ring, which for definiteness we shall take to be power series over a field. The authors demonstrate that with respect to smoothability all licci ideals, independent of codimension, behave either like a perfect ideal of codimension two or a Gorenstein ideal of codimension three: If R/I is not Gorenstein then (R,I) is smoothable in codimension three but not four; and if it is Gorenstein but not a complete intersection, then (R,I) is smoothable in codimension six but not seven. Further applications are to homogeneous ideals in \(R=k[X_ 1,..,X_ n]\), expressing linkage properties in terms of the twists that appear in a minimal homogeneous resolution \(0\to F_ g\to...\to F_ 1\to F_ 0\) of R/I. If \(F_ i=\otimes^{\beta_ i}_{j=1}R(-n_{ij}),\quad then\) (after localizing at the irrelevant maximal ideal) I is not licci if \(\max_ j\{n_{gj}\}\leq (g-1)d\), where \(d=\min_ j\{n_{1j}\}\). With some additional assumptions the authors also prove that the multiplicity e(R/I) is minimal over all ideals in the even linkage class of I. If, on the other hand, I is licci, \(n_{1j}=d\) for all j, and \(n_{gj}\) are also independent of j, then R/I has small class group, i.e. \(Cl(R/I)={\mathbb{Z}}[K_{R/I}]\), if the completion \((R/I)^{\wedge}\) is rigid.

The utility of the authors’ result has two main aspects. First, since it is an explicit construction, they are able to actually compute the non- Gorenstein and non-complete intersection loci of the second universal link (this is the critical step) and to prove that the defining ideals of these loci have prime radicals. This enables them to establish sharp smoothability results. Secondly, there is the philosophy, amply demonstrated here, that if (S,J) is essentially a deformation of (R,I) then S/J is “at least as good” as R/I. The authors prove that if R/I is Cohen-Macaulay, Gorenstein, complete intersection, \((R_ k)\), respectively, then so is S/J. The multiplicity, Cohen-Macaulay type, embedding codimension of S/J, and deviation of J are all less than or equal to the corresponding values for R/I and I.

Applications are to ideals in a regular local ring, which for definiteness we shall take to be power series over a field. The authors demonstrate that with respect to smoothability all licci ideals, independent of codimension, behave either like a perfect ideal of codimension two or a Gorenstein ideal of codimension three: If R/I is not Gorenstein then (R,I) is smoothable in codimension three but not four; and if it is Gorenstein but not a complete intersection, then (R,I) is smoothable in codimension six but not seven. Further applications are to homogeneous ideals in \(R=k[X_ 1,..,X_ n]\), expressing linkage properties in terms of the twists that appear in a minimal homogeneous resolution \(0\to F_ g\to...\to F_ 1\to F_ 0\) of R/I. If \(F_ i=\otimes^{\beta_ i}_{j=1}R(-n_{ij}),\quad then\) (after localizing at the irrelevant maximal ideal) I is not licci if \(\max_ j\{n_{gj}\}\leq (g-1)d\), where \(d=\min_ j\{n_{1j}\}\). With some additional assumptions the authors also prove that the multiplicity e(R/I) is minimal over all ideals in the even linkage class of I. If, on the other hand, I is licci, \(n_{1j}=d\) for all j, and \(n_{gj}\) are also independent of j, then R/I has small class group, i.e. \(Cl(R/I)={\mathbb{Z}}[K_{R/I}]\), if the completion \((R/I)^{\wedge}\) is rigid.

Reviewer: M.Miller

### MSC:

13A15 | Ideals and multiplicative ideal theory in commutative rings |

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

13D99 | Homological methods in commutative ring theory |

13D10 | Deformations and infinitesimal methods in commutative ring theory |

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |