By Ams-Ims-Siam Summer Research Conference on Commutative Algebra, Craig L. Huneke, William J. Heinzer, Judith D. Sally

This quantity comprises refereed papers on topics explored on the AMS-IMS-SIAM summer time examine convention, Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, held at Mount Holyoke university in 1992. The convention featured a chain of one-hour invited lectures on fresh advances in commutative algebra and interactions with such parts as algebraic geometry, illustration thought, and combinatorics. the main subject matters of the convention have been tight closure Hilbert features, birational algebra, unfastened resolutions and the homological conjectures, Rees algebras, and native cohomology. With contributions by way of a number of top specialists within the box, this quantity offers a great survey of present study in commutative algebra.

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And F are geomet2 rically different, since F preserves orientation while F does not. 2 THEOREM. There are exactly seven geometrically distinct types of frieze groups, with presentations as given above. They fall into the four isomorphism types of C^, D^, C^ x C ? , D^ * C . In the following figure we show friezes with symmetry groups of the seven types. In these figures, x is a horizontal translation. lines indicate axes of reflections or glide reflections. Broken In the figures 1 2 for F , F , and F_, small circles mark centers of rotational symmetry.

To £ 1 and £ ? , hence must be a translation. In the figure, c K P ^ P ^ ) = 2d(P1,P2>. If £. and £ meet in 0, then the orien- tation preserving transformation a = P-,P9 P, fixes 0, hence must be a rotation about 0. In the figure, the angle P 0(P a) is twice the angle PiOP," E '/o 'V ' i, 26 COROLLARY. Every orientation preserving transformation is a product of two reflections, and every orientation reversing transformation is a product of three reflections. Proof. Thus, E is generated by reflections.

Thus it is true for all parallelograms. The argument just given can be viewed as a generalization of the procedure common in analytic geometry where, to prove a geometric theorem, one first chooses a coordinate system judiciously. Problems Problem 1. When is the product of two glide reflections a rotation? When is it a translation? tation? When is the product of four reflections a ro- When a translation? Problem 2. If a is an element of a group G, define the conjugation map