proceedings of the american mathematical society Volume 117, Number 3, March 1993 EXAMPLES OF BUCHSBAUMQUASI-GORENSTEINRINGS MANFRED HERRMANN AND NGO VIET TRUNG (Communicated by Louis J. Rafliff, Jr.) Abstract. The paper shows the existence of Buchsbaum quasi-Gorenstein rings of any admissible depth. Introduction Until now few examples of non-Cohen-Macaulay prime almost complete in- tersections were known. One possibility to find such examples is to go through linkages. According to an idea of Peskine and Szpiro [6], every prime ideal linked to a quasi-Gorenstein ideal is an almost complete intersection, cf. [9]. Recall that an ideal 7 of a ring A is called quasi-Gorenstein if the factor ring A/I is quasi-Gorenstein, i.e., the canonical module of A/I is isomorphic to A/1. However, to find non-Cohen-Macaulay quasi-Gorenstein ideals is usually also hard. For instance, Schenzel [8] used a result of Mumford on abelian va- rieties to give a class of non-Cohen-Macaulay quasi-Gorenstein rings that are Buchsbaum with depth 2. The aim of this paper is to construct Buchsbaum quasi-Gorenstein rings of any admissible depth that are generated by monomials. Note that from the description of the local cohomology of the canonical module of a Buchs- baum ring [8] once can easily deduce that the depth t of a Buchsbaum quasi- Gorenstein ring A is either dim A, i.e., A is a Cohen-Macaulay ring, or 2 < t < [(dim A + l)/2]. Rings generated by monomials are, in other terms, affine semigroup rings whose structures can be described well by means of the underlying affine semigroups [11,7]. For instance, there exist sufficient (and nec- essary) conditions for such rings to be Cohen-Macaulay, Gorenstein, or Buchs- baum. Using the theory of affine semigroup rings, we can translate the prob- lem of constructing quasi-Gorenstein rings generated by monomials to one of finding certain kinds of systems of diophantine homogeneous linear equations. From this we then derive examples of (non-Cohen-Macaulay) Buchsbaum quasi- Gorenstein rings and, therefore, of Buchsbaum almost complete intersection rings of any admissible depth. Compared with Schenzel's results, our method Received by the editors January 4, 1991 and, in revised form, June 27, 1991. 1991 Mathematics Subject Classification. Primary 13H10. The second author was supported by the Alexander von Humboldt Foundation. © 1993 American Mathematical Society 0002-9939/93 $1.00 + 5.25 per page 619 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use