Compressed algebras.

*(English)*Zbl 0558.13007
Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 121-151 (1984).

[For the entire collection see Zbl 0539.00006.]

The purpose of the work is twofold. Firstly we want to give a presentation of A. Iarrobino’s construction of graded compressed algebras, that is algebras of maximal length among those of a given socle type [A. Iarrobino, ”Compressed algebras: Artin algebras having given socle degrees and maximal length”, Trans. Am. Math. Soc. 285, 337- 378 (1984)]. Secondly we want to generalize the notion of compressed algebras, introduced by Iarrobino for artinian rings, to the class of graded Cohen-Macaulay algebras and to generalize Iarrobino’s results to this class of rings. Our approach is completely within the framework of algebras and avoids the duality, used by Iarrobino, between graded algebras and the corresponding algebra of differential operators. We show that our class of compressed algebras contains e.g. rings of surfaces with rational or elliptic singularities, Cohen-Macaulay and Gorenstein rings of maximal embedding dimension and certain classes of determinantal and Pfaffian coordinate rings. We also generalize results by D. Buchsbaum - D. Eisenbud - A. Iarrobino (”Appendix to the above mentioned article by Iarrobino”) on resolutions of compressed algebras to the Cohen-Macaulay case which implies results by Schenzel, Sally and Wahl.

The purpose of the work is twofold. Firstly we want to give a presentation of A. Iarrobino’s construction of graded compressed algebras, that is algebras of maximal length among those of a given socle type [A. Iarrobino, ”Compressed algebras: Artin algebras having given socle degrees and maximal length”, Trans. Am. Math. Soc. 285, 337- 378 (1984)]. Secondly we want to generalize the notion of compressed algebras, introduced by Iarrobino for artinian rings, to the class of graded Cohen-Macaulay algebras and to generalize Iarrobino’s results to this class of rings. Our approach is completely within the framework of algebras and avoids the duality, used by Iarrobino, between graded algebras and the corresponding algebra of differential operators. We show that our class of compressed algebras contains e.g. rings of surfaces with rational or elliptic singularities, Cohen-Macaulay and Gorenstein rings of maximal embedding dimension and certain classes of determinantal and Pfaffian coordinate rings. We also generalize results by D. Buchsbaum - D. Eisenbud - A. Iarrobino (”Appendix to the above mentioned article by Iarrobino”) on resolutions of compressed algebras to the Cohen-Macaulay case which implies results by Schenzel, Sally and Wahl.

##### MSC:

13C13 | Other special types of modules and ideals in commutative rings |

13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |