Zum Nachweis arithmetischer Cohen-Macaulay Varietäten

There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example 4], 10], 11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner 3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner 7], p. 35 and, using our methods, we rather easily obtain the result of 7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay 5] (see also 6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.

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Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet