Abstract
We review the history of several homological conjectures, both from a chronological and a methodological point of view.
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References
J. L. Alperin: ‘Diagrams for modules’, J. Pure Appl. Algebra 16 (1980) 111–119.
D. J. Anick and E. L. Green: ‘On the homology of path algebras’, Comm. Algebra 15 (1987) 309–341.
M. Auslander and D. Buchsbaum: ‘Homological dimension in noetherian rings’, Proc. Nat. Acad. Sci. USA 42 (1956) 36–38.
M. Auslander and D. Buchsbaum: ‘Homological dimension in regular local rings’, Trans. Amer. Math. Soc. 85 (1957) 390–405.
M. Auslander and D. Buchsbaum: ‘Unique factorization in regular local rings’, Proc. Nat. Acad. Sci. USA 45 (1959) 733–734.
M. Auslander and R.-O.Buchweitz: ‘The homological theory of maximal Cohen-Macaulay approximations’, Memoire Soc. Math. France 38 (1989) 5–37.
M. Auslander and I. Reiten: ‘Applications of contravariantly finite subcategories’, Advances in Math. 86 (1991) 111–152.
M. Auslander and S. Smala: ‘Preprojective modules over artin algebras’, J. Algebra 66 (1980) 61–122.
H. Bass: ‘Finitistic dimension and a homological generalization of semiprimary rings’, Trans. Amer. Math. Soc. 95 (1960) 466–488.
H. Bass: ‘Injective dimension in noetherian rings’, Trans. Amer. Math. Soc. 102 (1962) 18–29.
W. D. Burgess and B. Zimmermann Huisgen: ‘Approximating modules by modules of finite projective dimension’, preprint.
M. Butler: Unpublished notes, 1992.
C. Cibils: ‘The syzygy quiver and the finitistic dimension’, Comm. Algebra 21 (1993) 4167–4171.
P. Dräxler and D. Happel: ‘A proof of the generalized Nakayama conjecture for algebras with J21+1 = 0 and A/J’ representation finite’, J. Pure Appl. Algebra 78 (1992) 161–164.
D. R. Farkas, C. D. Feustel, and E. L. Green: ‘Synergy in the theories of Gröbner bases and path algebras’, Canad. J. Math. 45 (1993) 727–739.
H. Fujita: ‘Tiled orders of finite global dimension’, Trans. Amer. Math. Soc. 322 (1990) 329–341. Erratum: Trans. Amer. Math. Soc. 327 (1991) 919–920.
K. R. Fuller: ‘Algebras from diagrams’, J. Pure Appl. Algebra 48 (1987) 23–37.
K. R. Fuller and M. Saorin: ‘On the finitistic dimension conjecture for artinian rings’, Manuscripta Math. 74 (1991) 117–132.
K. R. Fuller and Y. Wang: ‘Redundancy in resolutions and finitistic dimensions of Noetherian rings’, Comm. Algebra 21 (1993) 2983–2994.
K. R. Goodearl and B. Zimmermann Huisgen: ‘The syzygy type of finite dimensional algebras and classical orders’, in preparation.
E. L. Green, E. E. Kirkman, and J. J. Kuzmanovich: ‘Finitistic dimension of finite dimensional monomial algebras’, J. Algebra 136 (1991) 37–51.
E. L. Green and B. Zimmermann Huisgen: ‘Finitistic dimension of artinian rings with vanishing radical cube’, Math. Z. 206 (1991) 505–526.
L. Gruson and L. Raynaud: ‘Critères de platitude et de projectivité. Techniques de “platification” d’un module’, Invent. Math. 13 (1971) 1–89.
D. Happel and B. Zimmermann Huisgen: ‘Viewing finite dimensional representations through infinite dimensional ones’, in preparation.
D. Hilbert: ‘über die Theorie der algebraischen Formen’, Math. Ann. 36 (1890) 473–534.
B. Zimmermann Huisgen: ‘Predicting syzygies over monomial relation algebras’, Manuscripta Math. 70 (1991) 157–182.
B. Zimmermann Huisgen: ‘Homological domino effects and the first finitistic dimension conjecture’, Invent. Math. 108 (1992) 369–383.
B. Zimmermann Huisgen: ‘Bounds on finitistic and global dimension for artinian rings with vanishing radical cube’, J. Algebra 161 (1993) 47–68.
B. Zimmermann Huisgen: ‘Field dependent homological behavior of finite dimensional algebras’, Manuscripta Math. 82 (1994) 15–29.
K. Igusa, S. Smala and G. Todorov: ‘Finite projectivity and contravariant finiteness’, Proc. Amer. Math. Soc. 109 (1990) 937–941.
K. Igusa and G. Todorov: ‘On the finitistic global dimension conjecture for artin algebras’, preprint.
K. Igusa and D. Zacharia: ‘Syzygy pairs in a monomial algebra’, Proc. Amer. Math. Soc. 108 (1990) 601–604.
J. P. Jans: ‘Some generalizations of finite projective dimension’, Illinois J. Math. 5 (1961) 334–344.
W. Jansen and C. Odenthal: work in progress.
V. A. Jategaonkar: ‘Global dimension of triangular orders over a discrete valuation ring’, Proc. Amer. Math. Soc. 38 (1973) 8–14.
V. A. Jategaonkar: ‘Global dimension of tiled orders over a discrete valuation ring’, Trans. Amer. Math. Soc. 196 (1974) 313–330.
C. U. Jensen and H. Lenzing: ‘Model Theoretic Algebra’, New York-London (1989) Gordon and Breach.
E. E. Kirkman and J. J. Kuzmanovich: ‘Global dimensions of a class of tiled orders’, J. Algebra 127 (1989) 57–72.
H. Mochizuki: ‘Finitistic global dimension for rings’, Pacific J. Math. 15 (1965) 249–258.
M. Nagata: ‘A general theory of algebraic geometry over Dedekind domains II’, Amer. J. Math. 80 (1958) 382–420.
A. Schofield: ‘Bounding the global dimension in terms of the dimension’, Bull. London Math. Soc. 17 (1985) 393–394.
J.-P. Serre: ‘Sur la dimension homologique des anneaux et des modules Noethériens’, in Proc. Internat. Symposium on Algebraic Number Theory, Tokyo and Nikko 1955, pp. 175–189, Tokyo (1956) Science Council of Japan.
L. W. Small: ‘A change of rings theorem’, Proc. Amer. Math. Soc. 19 (1968) 662–666.
H. Tachikawa: ‘Quasi-Frobenius rings and generalizations’, Lecture Notes in Math. 351, Berlin-Heidelberg-New York (1973) Springer-Verlag.
R. B. Tarsy: ‘Global dimension of orders’, Trans. Amer. Math. Soc. 151 (1970) 335–340.
Y. Wang: ‘A note on the finitistic dimension conjecture’, Comm. Algebra 22 (1994) 2525–2528.
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Huisgen, B.Z. (1995). The Finitistic Dimension Conjectures — A Tale of 3.5 Decades. In: Facchini, A., Menini, C. (eds) Abelian Groups and Modules. Mathematics and Its Applications, vol 343. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0443-2_41
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DOI: https://doi.org/10.1007/978-94-011-0443-2_41
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