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

Bermejo, Isabel; Gimenez, Philippe

J. Algebra 303, No. 2, 592-617 (2006).

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 \(I\).

Reviewer: Michael Hellus (Leipzig)

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

Keywords:

Castelnuovo-Mumford regularity; effective methods; depth; \(a\)-invariant

Software:

Macaulay2; SINGULAR; CoCoA; mregular

References:

[1]	D. Bayer, The division algorithm and the Hilbert scheme, PhD thesis, Harvard University, 1982
[2]	Bayer, D.; Mumford, D., What can be computed in algebraic geometry?, (), 1-48 · Zbl 0846.13017
[3]	Bayer, D.; Stillman, M., A criterion for detecting m-regularity, Invent. math., 87, 1-11, (1987) · Zbl 0625.13003
[4]	Bermejo, I.; Gimenez, P., On castelnuovo – mumford regularity of projective curves, Proc. amer. math. soc., 128, 1293-1298, (2000) · Zbl 0944.13007
[5]	Bermejo, I.; Gimenez, P., Computing the castelnuovo – mumford regularity of some subschemes of \(\mathbb{P}_K^n\) using quotients of monomial ideals, J. pure appl. algebra, 164, 23-33, (2001) · Zbl 0989.13008
[6]	I. Bermejo, P. Gimenez, G.-M. Greuel, mregular.lib, a library for computing the Castelnuovo-Mumford regularity, {\scSingular} 3.0.0, 2005
[7]	{\sccocoa}, A system for doing computation in commutative algebra
[8]	Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms. an introduction to computational algebraic geometry and commutative algebra, (1997), Springer New York
[9]	Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Grad. texts in math., vol. 150, (1995), Springer · Zbl 0819.13001
[10]	Eisenbud, D.; Goto, S., Linear free resolutions and minimal multiplicities, J. algebra, 88, 89-133, (1984) · Zbl 0531.13015
[11]	Eisenbud, D.; Grayson, D.R.; Stillman, M.; Sturmfels, B., Computations in algebraic geometry with Macaulay 2, Algorithms comput. math., vol. 8, (2002), Springer · Zbl 0973.00017
[12]	Galligo, A., À propos du théorème de préparation de weirstrass, (), 543-579
[13]	Grayson, D.R.; Stillman, M., Macaulay 2, A software system for research in algebraic geometry
[14]	Green, M., Generic initial ideals, (), 119-186 · Zbl 0933.13002
[15]	Greuel, G.-M.; Pfister, G., A {\scsingular}. introduction to commutative algebra, (2002), Springer Berlin
[16]	Greuel, G.-M.; Pfister, G.; Schoenemann, H., {\scsingular} 3.0.0, A computer algebra system for polynomial computations, (2005), Center for Computer Algebra, University of Kaiserslautern
[17]	Hartshorne, R., Algebraic geometry, Grad. texts in math., vol. 52, (1977), Springer · Zbl 0367.14001
[18]	M. Lejeune-Jalabert, Effectivité de calculs polynomiaux, Cours de DEA, Institut Fourier, Grenoble, 1984-1985
[19]	Miller, E., Resolutions and duality for monomial ideals, (2000), PhD Thesis, Berkeley
[20]	Miller, E.; Sturmfels, B., Combinatorial commutative algebra, Grad. texts in math., vol. 227, (2004), Springer
[21]	Seidenberg, A., Constructions in algebra, Trans. amer. math. soc., 197, 273-313, (1974) · Zbl 0356.13007
[22]	Sturmfels, B., Four counterexamples in combinatorial algebraic geometry, J. algebra, 230, 282-294, (2000) · Zbl 1018.14022
[23]	Villarreal, R.H., Monomial algebras, Pure and appl. math., vol. 238, (2001), Dekker · Zbl 1002.13010