Geodesic
On dimension theory for complexes
Sbornik. Mathematics, Tome 36 (1980) no. 4, pp. 469-481 
A. F. Ivanov. On dimension theory for complexes. Sbornik. Mathematics, Tome 36 (1980) no. 4, pp. 469-481. http://geodesic.mathdoc.fr/item/SM_1980_36_4_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Passage from the category of modules to the derived category gives insight into some classical results of homological dimension theory, and also yields a proof of the nondegeneracy of the Yoneda multiplication $\operatorname{Ext}_A^p(k,\,\cdot\,)\times\operatorname{Ext}_A^{n-p}(\,\cdot\,,k)\to\operatorname{Ext}_A^n(k,k)=k$, where the argument is a noetherian module (or a finite complex with noetherian homology) and $A$ is a regular local ring. Bibliography: 9 titles.

[1] J.-P. Serre, “Algébre locale”, Multiplicités, Lecture Notes in Math., 11, Springer-Verlag, Berlin–New York, 1965 | MR | Zbl

[2] M. Auslander, D. A. Buchsbaum, “Homological dimension in local rings. I”, Trans. Amer. Math. Soc., 85:2 (1957), 390–405 | DOI | MR | Zbl

[3] M. Auslander, D. A. Buchsbaum, “Codimension and Multiplicity”, Ann. Math., 68:3 (1958), 625–657 | DOI | MR | Zbl

[4] R. Hartshorne, “Residues and Duality”, Lecture Notes in Math., 20, Springer-Verlag, Berlin–New York, 1966 | MR | Zbl

[5] Chr. Peskine, L. Szpiro, Dimension projective finie et cohomologie locale, Thèse ..., Univ. Paris, 1971; Publ. Math., Inst. Hautes Étud. Sci., 42 (1972), 47–119 | Zbl

[6] S. Maklein, Gomologiya, izd-vo “Mir”, Moskva, 1966

[7] A. Kartan, S. Eilenberg, Gomologicheskaya algebra, IL, Moskva, 1960

[8] P. Gabriel, “Objets injectifs dans les categories abeliennes”, Sem. Dubreil–Pisot, exp. 17, 1958–1959

[9] M. Auslander, “Modules over unramified local rings”, Illinois J. Math., 5 (1961), 631–647 | MR | Zbl