## 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 $$I$$.

### MSC:

 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

### Software:

SINGULAR; mregular; Macaulay2; CoCoA
Full Text:

### References:

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