# zbMATH — the first resource for mathematics

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)

##### MSC:
 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