Bulletin of the American Mathematical Society

MathSciNet review: 1567832
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: Dieter Happel
Title: Triangulated categories in the representation theory of finite dimensional algebras
Additional book information: Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, 1988. (London Mathematical Society Lecture Notes Series, vol. 119), ix + 208 pp., $24.95. ISBN 0-521-33922-7.

Ibrahim Assem and Dieter Happel, Generalized tilted algebras of type $A_{n}$, Comm. Algebra 9 (1981), no. 20, 2101–2125. MR 640613, DOI 10.1080/00927878108822697

Ibrahim Assem, Dieter Happel, and Oscar Roldán, Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33 (1984), no. 3, 235–242. MR 761629, DOI 10.1016/0022-4049(84)90058-6

Ibrahim Assem and Andrzej Skowroński, On some classes of simply connected algebras, Proc. London Math. Soc. (3) 56 (1988), no. 3, 417–450. MR 931509, DOI 10.1112/plms/s3-56.3.417

Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. IV. Invariants given by almost split sequences, Comm. Algebra 5 (1977), no. 5, 443–518. MR 439881, DOI 10.1080/00927877708822180

I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387

A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966

7.

I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Algebraic bundles over P and problems of linear algebra, Functional Anal. Appl. 12 (1978), 212-214.

I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065

Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel′fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 103–169. MR 607151

E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), no. 2, 397–409. MR 866784, DOI 10.1016/0021-8693(86)90224-3

P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125

Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265–297. MR 915180, DOI 10.1007/BFb0078849

Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. MR 910167, DOI 10.1007/BF02564452

Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126. MR 1035222, DOI 10.1007/BFb0084073

Dieter Happel, Jeremy Rickard, and Aidan Schofield, Piecewise hereditary algebras, Bull. London Math. Soc. 20 (1988), no. 1, 23–28. MR 916069, DOI 10.1112/blms/20.1.23

Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2

Dieter Happel and Claus Michael Ringel, The derived category of a tubular algebra, Representation theory, I (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1177, Springer, Berlin, 1986, pp. 156–180. MR 842465, DOI 10.1007/BFb0075264

Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093

David Hughes and Josef Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 347–364. MR 693045, DOI 10.1112/plms/s3-46.2.347

Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9

M. M. Kapranov, The derived category of coherent sheaves on Grassmann varieties, Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 78–79 (Russian). MR 705052

Yoichi Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. MR 852914, DOI 10.1007/BF01163359

Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436

Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. MR 1027750, DOI 10.1016/0022-4049(89)90081-9

Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870

26.

J.-L. Verdier, Catégories dérivées, état 0, Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin and New York, 1977, pp. 262-311.

Takayoshi Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, J. Algebra 134 (1990), no. 2, 298–325. MR 1074331, DOI 10.1016/0021-8693(90)90055-S

1.

I. Assem and D. Happel, Generalized tilted algebras of type A, Comm. Algebra 9 (1981), 2101-2125. MR 0640613

2.

I. Assem, D. Happel and O. Roldán, Representation-finite trivial extension algebras, J. Pure. Appl. Algebra 33 (1984), 235-242. MR 761629

3.

I. Assem and A. Skowroński, Algebras with cycle-finite derived categories, Math. Ann. 280 (1988), 441-463. MR 936322

4.

M. Auslander and I. Reiten, Representation theory of artin algebras. III and IV, Comm. Algebra 3 (1975), 239-294 and 5 (1977), 443-518. MR 439881

5.

A. A. Beilinson, Coherent sheaves on P and problems of linear algebra, Functional Anal. Appl. 12 (1978), 214-216. MR 509388

6.

A. A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisque 100 (1982). MR 751966

7.

I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Algebraic bundles over P and problems of linear algebra, Functional Anal. Appl. 12 (1978), 212-214.

8.

I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Russian Math. Surveys 28 (1973), 17-32. MR 393065

9.

S. Brenner and M. C. R. Butler, Generalization of the Bernstein-Gelfand-Ponomarev reflection functors, Proc. ICRA II (Ottawa, 1979) Lecture Notes in Math., vol. 832, Springer-Verlag, Berlin and New York, 1980, pp. 103-169. MR 607151

10.

E. Cline, B. Parshall and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), 397-409. MR 866784

11.

P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergeb. Math. Grenzgeb. Band 35, Springer-Verlag, Berlin and New York, 1967. MR 210125

12.

W. Geigle and H. Lenzing, A class of weighted projective curves arising in the representation theory of finite dimensional algebras, Proc. Lambrecht 1985, Lecture Notes in Math., vol. 1273, Springer-Verlag, Berlin and New York, 1987, pp. 265-297. MR 915180

13.

D. Happel, On the derived category of a finite dimensional algebra, Comment. Math. Helv. 62 (1987), 339-389. MR 910167

14.

D. Happel, Hochschild cohomology of finite dimensional algebras, preprint, University of Bielefeld, 1988. MR 1035222

15.

D. Happel, J. Rickard and A. Schofield, Piecewise hereditary algebras, Bull. London Math. Soc. 20 (1988), 23-28. MR 916069

16.

D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443. MR 675063

17.

D. Happel and C. M. Ringel, The derived category of a tubular algebra, Proc. ICRA IV (Ottawa 1984) Lecture Notes in Math., vol. 1177, Springer-Verlag, Berlin and New York, 1986, pp. 156-180. MR 842465

18.

R. Hartshorne, Residues and duality, Lecture Notes in Math., vol. 20, Springer-Verlag, Berlin and New York, 1966. MR 222093

19.

D. Hugues and J. Washbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. 46 (1983), 347-364. MR 693045

20.

B. Iversen, Cohomology of sheaves, Springer-Verlag, 1986. MR 842190

21.

M. M. Kapranov, Derived category of coherent sheaves on Grassman manifolds, Functional Anal. Appl. 17 (1983), 145-146. MR 705052

22.

Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113-146. MR 852914

23.

J. Rickard, A Morita theory for derived categories, preprint, University College London, 1987. MR 1002456

24.

J. Rickard, Derived categories and stable equivalence, preprint, University College London, 1987. MR 1027750

25.

C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer-Verlag, Berlin and New York, 1984. MR 774589

26.

J.-L. Verdier, Catégories dérivées, état 0, Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin and New York, 1977, pp. 262-311.

27.

T. Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, preprint, Jōbu University, 1988. MR 1074331