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Hilbert functions, analytic spread, and Koszul homology. (English) Zbl 0803.13012
Heinzer, William J. (ed.) et al., Commutative algebra: syzygies, multiplicities, and birational algebra: AMS-IMS-SIAM summer research conference on commutative algebra, held July 4-10, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 159, 401-422 (1994).
Let \((R,m)\) be a Cohen-Macaulay local ring, \(I\) an ideal in \(R\), and \(J\) a reduction of \(I\), i.e. \(I^{n+1} = JI^ n\) for large \(n \in \mathbb{N}\). \(R[It]\) denotes the Rees ring of \(I\). In the first part of the paper (sections 2 and 3), the author emphasizes, mainly for \(m\)-primary ideals \(I\), the ubiquity of the socalled Sally module \(IR[It]/IR[Jt]\) of \(I\) with respect to \(J\) in questions like the Cohen-Macaulayness of the form ring \(gr_ I (R) = R[It]/IR[It]\) of \(I\), or the behaviour of the coefficients of the Hilbert polynomial of \(I\). In these two sections the author provides new (transparent) proofs for certain already known results. The main results of the paper are contained in section 4 and 5. The most important result (theorem 5.1) provides a class of ideals \(I\) with a Cohen-Macaulay Rees ring; the ideals \(I\) in this class are described by restrictions on the depth of its Koszul homologies (i.e. sliding depth), on the reduction exponent \((I^ 2=JI)\), and on the local number of generators of \(I\) in certain codimensions. The main ideas for the proof are that the sliding depth property for \(I\) lifts to \(J\) (theorem 4.11), that in this case there is an upper bound for the \(a\)- invariant \(a(gr_ R(J))\) of the form ring of \(J\) (by proposition 5.2, \(a(gr_ y(R)) \leq-\)height \((I))\), and, finally, that this upper bound can be used in two natural exact sequences, relating the form of \(I\) with that of \(J\), for showing that the \(a\)-invariant of \(gr_ I(R)\) is negative (which implies in the given situation the Cohen-Macaulayness of \(R[It])\).
For the entire collection see [Zbl 0790.00007].
Reviewer: J.Ribbe (Köln)

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C40 Linkage, complete intersections and determinantal ideals