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Intersection multiplicities and Grothendieck spaces

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1. Chain complexes 2. Derived functors 3. Tor and Ext 4. Homological dimensions 5. Spectral sequences 6. Group homology and cohomology 7. Lie algebra homology and cohomology 8. Simplicial methods in homological algebra 9. Hothschild and cyclic homology 10. The derived category Appendix: category theory language.
The main aim of this paper is to discuss the relation between Serre's intersection multiplicity and the Euler form. The Euler form is defined to be an alternating sum of the length of Ext-modules and is used by Mori and Smith to develop intersection theory over noncommutative rings. We show that they differ by a sign and that this relation is closely related to Serre's vanishing theorem.
The Rees algebra is the homogeneous coordinate ring of a blowing- up. The present paper gives a necessary and sucient condition for a Noe- therian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal bers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sucient condition. By using it, however, we obtain the armative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a nite-dimensional Gorenstein ring.