Saturation and Castelnuovo-Mumford regularity. (English) Zbl 1105.13010

Formulae for the (Castelnuovo-Mumford) regularity reg\((I)\) and for other cohomological invariants of a homogeneous ideal \(I\) of a polynomial ring over a field \(k\) are proven; the methods are effective, i. e. can be realized in a computer algebra system - this was done by the authors (SINGULAR) and is also explained in the paper. Bayer and Stillman have shown that, in generic coordinates, one has \(\text{reg}(I)=\text{reg}(\text{in}(I))\) where \(\text{in}(I)\) is the initial ideal of \(I\) with respect to reverse lexicographic order.
In this paper a monomial ideal \(N(I)\) is constructed (which is in general not equal to \(\text{in}(I)\)) such that \(\text{reg}(I)=\text{reg}(N(I));\) depth\((R/I)=\text{depth}(R/N(I))=:\text{depth}\); \(\text{end}(H^{\text{depth}}_m(R/I))=\text{end}(H^{\text{depth}}_m(R/(N(I)))\); and \(a(R/I)\leq a(R/N(I))\) hold (where end is the highest non-vanishing degree of a graded module, \(m\) is the maximal homogeneous ideal of the polynomial ring and \(a(R/I):=\text{end}(H^{\dim(R/I)}_m(R/I))\)). A class of certain monomial ideals is defined to which all \(N(I)\) belong (for arbitrary homogeneous ideals \(I\)); for ideals \(J\) belonging to this class formulae for \(\text{reg}(J)\), \(\text{depth}(R/J)\), \(\text{end}(H^{\text{depth}(R/J)}_m(R/J))\) and \(a(R/J)\) are proven. A combination of the results explained in the preceding paragraphs leads to formulae for \(\text{reg}(I)\), \(\text{depth}(R/I)\), \(\text{end}(H^{\text{depth}(R/I)}_m(R/I))\) and upper bounds for \(a(R/I)\) for any homogeneous ideal \(I\).


13D02 Syzygies, resolutions, complexes and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D45 Local cohomology and commutative rings
Full Text: DOI


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