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Horrocks’ question for monomially graded modules. (English) Zbl 0692.13009

The paper under review contains a new approach to Horrock’s question, which asks whether \(\dim_ kTor^ R_ i(k,M)\geq \binom{n}{i}\) for every regular local ring \((R,m,k)\) of dimension n and every finite length module \(M,\) \(i=1,...,n\). The main result gives a general inequality relating values of an additive function defined on a class of \(R[X_ 1,...,X_ d]\)-modules satisfying certain conditions. The author’s techniques allow R to be a commutative ring with the price of imposing additional constraints on the modules themselves. This approach is fruitful indeed: there is an application where the additive function is defined via Serre’s intersection multiplicity; one gets a general result on iterated Koszul homology; one arrives at a significant addition to the results known about Horrock’s question. Moreover, the essential elements of the techniques used here seem to be of more importance than their utility in this paper.
Reviewer: M.Cipu

MSC:

13D05 Homological dimension and commutative rings
14A05 Relevant commutative algebra
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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