Abstract
A relative version of Rickard's theorem is proved, namely, if ω is a quasi-Frobenius proper class of short sequences in an Abelian category \(\mathcal{A}\), then the ω-stable category of the category \(\mathcal{A}\) is a quotient category of the relative bounded derived category \(D_\omega ^b \)(\(\mathcal{A}\)). Bibliography: 20 titles.
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REFERENCES
J. Rickard, “Derived categories and stable equivalence,” J. Pure Appl. Alg., 61,No. 1, 303–317 (1989).
B. Keller and D. Vossieck, “Sous les catégories dérivées,” Comp. Rend. Acad. Sci. Paris, 305,No. 6, 225–228 (1987).
D. J. Benson and W. W. Wheeler, “The Green correspondence for infinitely generated modules,” J. London Math. Soc., 63, 69–82 (2001).
A. I. Generalov, “Relative homological algebra in pre-Abelian categories, I: derived categories,” Algebra Analiz, 4,No. 1, 98–119 (1992).
A. Neeman, “The derived category of an exact category,” J. Algebra, 135,No. 2, 388–394 (1990).
A. I. Generalov, “Derived categories of an additive category,” Algebra Analiz, 4,No. 5, 98–119 (1992).
D. Quillen, “Higher algebraic K-theory,” Lect. Notes Math., 341, 85–147 (1973).
E. G. Sklyarenko, “Relative homological algebra in categories of modules,” Usp. Mat. Nauk, 33,No. 3, 85–120 (1978).
A. I. Generalov, “Relative homological algebra. Cohomology of categories, posets, and coalgebras,” in: Handbook of Algebra, Vol. 1, Elsevier Sci., Amsterdam (1996), pp. 611–638.
A. P. Mishina and L. A. Skornyakov, Abelian Groups and Modules [in Russian], Nauka, Moscow (1969).
S. MacLane, Homology, Spriger-Verlag, Berlin/New York (1963).
D. Happel, “On the derived category of a finite-dimensional algebra,” Comm. Math. Helv., 62,No. 3, 339–389 (1987).
H. Bass, Algebraic K-Theory, Benjamin, New York (1968).
A. I. Generalov, “Inductively closed proper classes over bounded hnp-rings,” Algebra Logika, 25,No. 4, 384–404 (1986).
J.-L. Verdier, “Catégories dérivées,” Lect. Notes Math., 569, 262–311 (1977).
S. I. Gel'fand and Yu. I. Manin, Methods of Homological Algebra [in Russian], V. 1, Nauka, Moscow (1988).
J.-L. Verdier, “Des catégories dérivées des catégories abéliennes,” Astérisque, No. 239 (1996).
A. I. Generalov, “Localization of pre-triangulated categories,” Algebra Analiz, 11,No. 3, 20–52 (1999).
M. Linckelmann, “La catégorie stable d'une algèbre auto-injective est triangulée,” Compt. Rend. Acad. Sci. Paris, 305,No. 10, 403–406 (1987).
C. M. Ringel, “Recent advances in the representation theory of finite-dimensional algebras,” in: Representation Theory of Finite Groups and Finite-Dimensional Algebras, Vol. 95, Burkhäuser Verlag (1991), pp. 141–192.
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Generalov, A.I. QF-Proper Classes and Relative Stable Categories. Journal of Mathematical Sciences 120, 1563–1574 (2004). https://doi.org/10.1023/B:JOTH.0000017885.83073.3e
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DOI: https://doi.org/10.1023/B:JOTH.0000017885.83073.3e