Arithmetical Rank Research Articles

The complete intersection locus of certain ideals

and dimension d. The most primitive quantities attached to an ideal Z are its height, ht(Z), and its minimum number of generators, u(Z). The fundamental relation between ht(Z) and u(Z) is pro- vided by Krull’s theorem: (1) u(Z) > ht(Z). The case of equality is noteworthy enough and Z will be called a complete intersec- tion if it so happens. (Strictly, this terminology is used when the ideal is generated by a regular sequence, but the two notions coincide in what will be our standard con- text - that of Cohen-Macaulay rings.) To bridge the gap between o(Z) and ht(Z) several other measures have been in- troduced. We single out: (i) ara(Z) = arithmetical rank of Z : = the least integer I such that the radical of Z is the radical of an r-generated ideal. (ii) I(Z) = analytic spread of Z := Krull dimension of the algebra @I’@

Über den arithmetischen Rang quadratfreier Potenzproduktideale

In this paper we give an algorithm to compute an upper bound for the arithmetical rank of squarefree monomial ideals, i.e. the minimal number of hypersurfaces which cut out set-theoretically the variety of such an ideal. An apriori bound N – a=b+2 is obtained, where N means the number of variables, a the lowest degree in the ideal and b the lowest degree of syzygies in the first syzygy module (Thm. 2). These results sharpen more general results of [2] for the considered class of ideals by methods different from [1], [7].