Partially supported by an NSF post-doctoral fellowship that the equivariant groups A G * (X) depend only on the stack [X/G] and not on its presentation as a quotient. If X is smooth, then A 1 G (X) is isomorphic to Mumford's Picard group of the stack, and the ring A * G (X) can naturally be identified as an integral Chow ring of [X/G] (Section 5.3).These results imply that equivariant Chow groups are a useful tool for computing Chow groups of quotient spaces and stacks. For example, Pandharipande ([Pa1], [Pa2]) has used equivariant methods to compute Chow rings of moduli spaces of maps of projective spaces as well as the Hilbert scheme of rational normal curves. In this paper, we compute the integral Chow rings of the stacks M 1,1 and M 1,1 of elliptic curves, and obtain a simple proof of Mumford's result ([Mu]) that P ic f un (M 1,1 ) = Z/12Z. In an appendix to this paper, Angelo Vistoli computes the Chow ring of M 2 , the moduli stack of smooth curves of genus 2.Equivariant Chow groups are also useful in proving results about intersection theory on quotients. It is easy to show that if X is smooth then there is an intersection product on A G * (X). The theorem on quotients therefore implies that there exists an intersection product on the rational Chow groups of a quotient of a smooth algebraic space by a proper action. The existence of such an intersection product was shown by Gillet and Vistoli, but only under the assumption that the stabilizers are reduced. This is automatic in characteristic 0, but typically fails in characteristic p. The equivariant approach does not require this assumption and therefore extends the work of Gillet and Vistoli to arbitrary characteristic. Furthermore, by avoiding the use of stacks, the proof becomes much simpler.Finally, equivariant Chow groups define invariants of quotient stacks which exist in arbitrary degree, and associate to a smooth quotient stack an integral intersection ring which when tensored with Q agrees with rings defined by Gillet and Vistoli. By analogy with quotient stacks, this suggests that there should be an integer intersection ring associated to an arbitrary smooth stack, which could be nonzero in degrees higher than the dimension of the stack.We remark that besides the properties mentioned above, the equivariant Chow groups we define are compatible with other equivariant theories such as cohomology and K-theory. For instance, if X is smooth then there is a cycle map from equivariant Chow theory to equivariant cohomology (Section 2.8). In addition, there is a map from equivariant K-theory to equivariant Chow groups, which is an isomorphism after completion; and there is a localization theorem for torus actions, which can be used to give an intersection theoretic proof of residue formulas of Bott and Kalkman. These topics will be treated elsewhere.

Supplementary Material Available: Detailed results of the X-ray crystal structure of (r/-C5Me5)2Os2(CO)2(Ai-H)2, tables of experimental details, positional and thermal parameters, general temperature factor expressions (U,B), bond distances, and bond angles, and the structure of (7/-C5Me5)2Os2(CO)2^-H)2 (8 pages). Ordering information is given on any current masthead page.

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We prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties.

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The inherent sensitivity and simplicity make the new experiment generally applicable to a number of biological and chemical systems where observation of 15N chemical shifts has proven time consuming or impossible because of the low natural abundance and NMR sensitivity of 15N. Pulse sequences with more than one pulse applied to the observed nuclei5 make suppression of unwanted signals harder and do, in practice, not provide the improvement that might be expected.16Acknowledgment. The sample of gramacidin S was kindly provided by Professor D. H. Live (Rockefeller University). We are indebted to professor Gary Maciel for many valuable comments in preparing the manuscript and further gratefully acknowledge use of the Colorado State University Regional NMR Center, supported by National Science Foundation Grant CHE-8208821. A.B. acknowledges support from the Department of Energy (Laramie Energy Technology Center), and R.H.G. was supported by NSF Grant PCM 79-16861, awarded to C. D.Poulter.

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