References
Published online by Cambridge University Press: 12 November 2021
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Homological Theory of Representations , pp. 457 - 468Publisher: Cambridge University PressPrint publication year: 2021
References
Adámek, J. and Rosický, J., Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, 189, Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
Akin, K. and Buchsbaum, D. A., Characteristic-free representation theory of the general linear group. II. Homological considerations, Adv. Math. 72 (1988), no. 2, 171–210.Google Scholar
Akin, K., Buchsbaum, D. A. and Weyman, J., Schur functors and Schur complexes, Adv. Math. 44 (1982), no. 3, 207–278.CrossRefGoogle Scholar
Alonso Tarrío, L., Jeremías López, A. and Souto Salorio, M. J., Localization in categories of complexes and unbounded resolutions, Can. J. Math. 52 (2000), no. 2, 225–247.Google Scholar
Angeleri Hügel, L., Happel, D. and Krause, H. (eds.), Handbook of tilting theory, London Mathematical Society Lecture Note Series, 332, Cambridge University Press, Cambridge, 2007.Google Scholar
Assem, I. and Skowroński, A., Iterated tilted algebras of type Ãn, Math. Z. 195 (1987), no. 2, 269–290.CrossRefGoogle Scholar
Auslander, M., Coherent functors, in Proc. Conf. categorical algebra (La Jolla, Calif., 1965), 189–231, Springer, New York, 1966.Google Scholar
Auslander, M., Representation theory of Artin algebras II, Commun. Algebra 1 (1974), 269–310.Google Scholar
Auslander, M., Large modules over Artin algebras, in Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), 1–17, Academic Press, New York, 1976.Google Scholar
Auslander, M., Functors and morphisms determined by objects, in Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), 1–244, Lecture Notes in Pure and Applied Mathematics, 37, Dekker, New York, 1978.Google Scholar
Auslander, M. and Buchsbaum, D. A., Homological dimension in noetherian rings II, Trans. Am. Math. Soc. 88 (1958), 194–206.Google Scholar
Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen– Macaulay approximations, Mém. Soc. Math. Fr. 38 (1989), 5–37.Google Scholar
Auslander, M., Platzeck, M. I. and Reiten, I., Coxeter functors without diagrams, Trans. Am. Math. Soc. 250 (1979), 1–46.Google Scholar
Auslander, M. and Reiten, I., Representation theory of Artin algebras. III. Almost split sequences, Commun. Algebra 3 (1975), 239–294.Google Scholar
Auslander, M. and Reiten, I., Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111–152.Google Scholar
Auslander, M. and Reiten, I., Cohen–Macaulay and Gorenstein Artin algebras, in Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 221–245, Progress in Mathematics, 95, Birkhäuser, Basel, 1991.Google Scholar
Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of Artin algebras, corrected reprint of the 1995 original, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1997.Google Scholar
Azumaya, G., Corrections and supplemetaries to my paper concerning Krull– Remak–Schmidt’s theorem, Nagoya Math. J. 1 (1950), 117–124.Google Scholar
Baer, R., Erweiterung von Gruppen und ihre Isomorphismen, Math. Z. 38 (1934), no. 1, 375–416.Google Scholar
Baer, R., Abelian groups that are direct summands of every containing abelian group, Bull. Am. Math. Soc. 46 (1940), 800–806.CrossRefGoogle Scholar
Bass, H. and Murthy, M. P., Grothendieck groups and Picard groups of abelian group rings, Ann. Math. (Ser. 2) 86 (1967), 16–73.Google Scholar
Baumann, P. and Kassel, C., The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. 533 (2001), 207–233.Google Scholar
Beilinson, A. A., Coherent sheaves on Pn and problems in linear algebra, Funktsional. Anal. Prilozhen. 12 (1978), no. 3, 68–69.Google Scholar
Beilinson, A. A., Bernšteĭn, J. and Deligne, P., Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Societé Mathématique de France, Paris, 1982.Google Scholar
Beilinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory, J. Am. Math. Soc. 9 (1996), no. 2, 473–527.CrossRefGoogle Scholar
Beligiannis, A. and Reiten, I., Homological and homotopical aspects of torsion theories, Memoirs of the American Mathematical Society, 188, American Mathematical Society, Providence, RI, 2007, no. 883.Google Scholar
Benson, D. J., Representations and cohomology II, Cambridge Studies in Advanced Mathematics, 31, Cambridge University Press, Cambridge, 1991.Google Scholar
Benson, D. J., Representations of elementary abelian p-groups and vector bundles, Cambridge Tracts in Mathematics, 208, Cambridge University Press, Cambridge, 2017.Google Scholar
Benson, D., Iyengar, S. B., Krause, H. and Pevtsova, J., Local duality for representations of finite group schemes, Compos. Math. 155 (2019), no. 2, 424–453.Google Scholar
Benson, D. and Krause, H., Pure injectives and the spectrum of the cohomology ring of a finite group, J. Reine Angew. Math. 542 (2002), 23–51.Google Scholar
Bergman, G. M., Coproducts and some universal ring constructions, Trans. Am. Math. Soc. 200 (1974), 33–88.CrossRefGoogle Scholar
Bernšteĭn, I. N., Gel’fand, I. M. and Gel’fand, S. I., A certain category of g-modules, Funktsional. Anal. Prilozhen. 10 (1976), no. 2, 1–8.Google Scholar
Bernšteĭn, I. N., Gel’fand, I. M. and Gel’fand, S. I., Algebraic vector bundles on Pn and problems of linear algebra, Funktsional. Anal. Prilozhen. 12 (1978), no. 3, 66–67.Google Scholar
Bernšteĭn, I. N., Gel’fand, I. M. and Ponomarev, V. A., Coxeter functors, and Gabriel’s theorem, Usp. Mat. Nauk 28 (1973), no. 2(170), 19–33.Google Scholar
Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compos. Math. 86 (1993), no. 2, 209–234.Google Scholar
Bondal, A. I. and Kapranov, M. M., Representable functors, Serre functors, and mutations, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337; translation in Math. USSR-Izv. 35 (1990), no. 3, 519–541.Google Scholar
Bourbaki, N., Éléments de mathématique: Algèbre, Chapters 4–7, Lecture Notes in Mathematics, 864, Masson, Paris, 1981.Google Scholar
Brauer, R. and Nesbitt, C., On the regular representations of algebras, Proc. Natl. Acad. Sci. U.S.A. 23 (1937), 236–240.Google Scholar
Brenner, S. and Butler, M. C. R., Generalizations of the Bernšteĭn Gel’fand Ponomarev reflection functors, in Representation theory, II (Proc. Second Int. Conf., Carleton Univ., Ottawa, Ont., 1979), 103–169, Lecture Notes in Mathematics, 832, Springer, Berlin, 1980.Google Scholar
Buan, A. B. and Krause, H., Tilting and cotilting for quivers and type Ãn, J. Pure Appl. Algebra 190 (2004), 1–21.Google Scholar
Buchweitz, R.-O., Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, Universität Hannover, 1986.Google Scholar
Cartan, H., Algèbres d’Eilenberg–MacLane, Exposés 2–11, Sém. Cartan, H., Éc. Normale Sup. (1954–1955), Sécrétariat Mathématique, Paris, 1956.Google Scholar
Cartan, H. and Eilenberg, S., Homological algebra, Princeton University Press, Princeton, NJ, 1956.Google Scholar
Carter, R. W. and Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.Google Scholar
Cauchy, A.-L., Mémoire sur les fonctions alternées et sur les sommes alternées, Exercices d’analyse et de physique mathématique, ii (1841), 151–159; Œuvres complètes, 2ème série xii, 173–182, Gauthier-Villars, Paris, 1916.Google Scholar
Chase, S. U., On direct sums and products of modules, Pacific J. Math. 12 (1962), 847–854.CrossRefGoogle Scholar
Church, T., Ellenberg, J. S., Farb, B. and Nagpal, R., FI-modules over Noetherian rings, Geom. Topol. 18 (2014), no. 5, 2951–2984.Google Scholar
Cline, E., Parshall, B. and Scott, L., Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99.Google Scholar
Corry, L., Modern algebra and the rise of mathematical structures, second edition, Birkhäuser, Basel, 2004.CrossRefGoogle Scholar
Crawley-Boevey, W., Tame algebras and generic modules, Proc. London Math. Soc. (Ser. 3) 63 (1991), no. 2, 241–265.Google Scholar
Crawley-Boevey, W., Modules of finite length over their endomorphism rings, in Representations of algebras and related topics (Tsukuba, 1990), 127–184, London Mathematical Society Lecture Note Series, 168, Cambridge University Press, Cambridge, 1992.Google Scholar
Crawley-Boevey, W., Locally finitely presented additive categories, Commun. Algebra 22 (1994), no. 5, 1641–1674.Google Scholar
Crawley-Boevey, W., Infinite-dimensional modules in the representation theory of finite-dimensional algebras, in Algebras and modules, I (Trondheim, 1996), 29–54, CMS Conference Proceedings, 23, AmericanMathematical Society, Providence, RI, 1998.Google Scholar
Crawley-Boevey, W., Classification of modules for infinite-dimensional string algebras, Trans. Am. Math. Soc. 370 (2018), no. 5, 3289–3313.Google Scholar
de Concini, C., Eisenbud, D. and Procesi, C., Young diagrams and determinantal varieties, Invent. Math. 56 (1980), no. 2, 129–165.Google Scholar
Deligne, P., Cohomologie à supports propres, in SGA 4, Théorie des Topos et Cohomologie Etale des Schémas, vol. 3, 250–480, Lecture Notes in Mathematics, 305, Springer, Heidelberg, 1973.Google Scholar
Djament, A., La propriété noethérienne pour les foncteurs entre espaces vectoriels [d’après Putman, A., Sam, S. et Snowden, A.], Astérisque, 380, Séminaire Bourbaki, Vol. 2014/2015 (2016), Exp. No. 1090, 35–60.Google Scholar
Dlab, V. and Ringel, C. M., Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291.Google Scholar
Dlab, V. and Ringel, C. M., The module theoretical approach to quasi-hereditary algebras, in Representations of algebras and related topics (Kyoto, 1990), 200–224, London Mathematical Society Lecture Note Series, 168, Cambridge University Press, Cambridge, 1992.Google Scholar
Donkin, S., On Schur algebras and related algebras I, J. Algebra 104 (1986), no. 2, 310–328.Google Scholar
Donkin, S., On tilting modules for algebraic groups, Math. Z. 212 (1993), no. 1, 39–60.Google Scholar
Doubilet, P., Rota, G.-C. and Stein, J., On the foundations of combinatorial theory IX. Combinatorial methods in invariant theory, Stud. Appl. Math. 53 (1974), 185–216.Google Scholar
Eckmann, B. and Schopf, A., Über injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78.Google Scholar
Eilenberg, S., Homological dimension and syzygies, Ann. Math. (Ser. 2) 64 (1956), 328–336.Google Scholar
Eilenberg, S. and MacLane, S., Group extensions and homology, Ann. Math. (Ser. 2) 43 (1942), 757–831.Google Scholar
Eilenberg, S. and Nakayama, T., On the dimension of modules and algebras II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. 9 (1955), 1–16.Google Scholar
Franke, J., On the Brown representability theorem for triangulated categories, Topology 40 (2001), no. 4, 667–680.CrossRefGoogle Scholar
Freyd, P., Representations in abelian categories, in Proc. Conf. categorical algebra (La Jolla, Calif., 1965), 95–120, Springer, New York, 1966.Google Scholar
Friedlander, E. M. and Suslin, A., Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270.Google Scholar
Fuchs, L., Infinite abelian groups, vol. I, Pure and Applied Mathematics, 36, Academic Press, New York, 1970.Google Scholar
Fulton, W., Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997.Google Scholar
Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. Fr. 90 (1962), 323–448.CrossRefGoogle Scholar
Gabriel, P., Auslander–Reiten sequences and representation-finite algebras, in Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), 1–71, Lecture Notes in Mathematics, 831, Springer, Berlin, 1980.Google Scholar
Gabriel, P., Un jeu? Les nombres de Catalan, UniZürich, Mitteilungsblatt des Rektorates 6 (1981), 4–5.Google Scholar
Gabriel, P. and Oberst, U., Spektralkategorien und reguläre Ringe im von-Neumannschen Sinn, Math. Z. 92 (1966), 389–395.Google Scholar
Gabriel, P. and Roiter, A. V., Representations of finite-dimensional algebras [with a chapter by Keller, B.], Encyclopaedia of Mathematical Sciences, 73, Algebra, VIII, Springer, Berlin, 1992.Google Scholar
Gabriel, P. and Ulmer, F., Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, 221, Springer, Berlin, 1971.Google Scholar
Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, Springer, New York, 1967.Google Scholar
Garavaglia, S., Decomposition of totally transcendental modules, J. Symbolic Logic 45 (1980), no. 1, 155–164.Google Scholar
Geigle, W., The Krull–Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences, Manuscripta Math. 54 (1985), no. 1–2, 83–106.Google Scholar
Geigle, W. and Lenzing, H., A class of weighted projective curves arising in representation theory of finite-dimensional algebras, in Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265–297, Lecture Notes in Mathematics, 1273, Springer, Berlin, 1987.Google Scholar
Geigle, W. and Lenzing, H., Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273–343.Google Scholar
Geiß, Ch. and Reiten, I., Gentle algebras are Gorenstein, in Representations of algebras and related topics, 129–133, Fields Institute Communications, 45, American Mathematical Society, Providence, RI, 2005.Google Scholar
Green, E. L. and Zacharia, D., The cohomology ring of a monomial algebra, Manuscripta Math. 85 (1994), no. 1, 11–23.Google Scholar
Green, J. A., Polynomial representations of GLn, Lecture Notes in Mathematics, 830, Springer, Berlin, 1980.Google Scholar
Green, J. A., Combinatorics and the Schur algebra, J. Pure Appl. Algebra 88 (1993), no. 1–3, 89–106.Google Scholar
Grothendieck, A., Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221.Google Scholar
Grothendieck, A., The cohomology theory of abstract algebraic varieties, in Proc. Int. Congress of mathematicians (Edinburgh, 1958), 103–118, Cambridge University Press, New York, 1960.Google Scholar
Grothendieck, A., Groupes de classes des categories abéliennes et triangulées, complexes parfaits, in Cohomologie l-adique et fonctions L, 351–371, Lecture Notes in Mathematics, 589, Springer, Berlin, 1977.Google Scholar
Grothendieck, A. and Dieudonné, J. A., Eléments de géométrie algébrique. I, Grundlehren der Mathematischen Wissenschaften, 166, Springer, Berlin, 1971.Google Scholar
Grothendieck, A. and Verdier, J. L., Préfaisceaux, in SGA 4, Théorie des Topos et Cohomologie Etale des Schémas, vol1, Théorie des Topos, 1–217, Lecture Notes in Mathematics, 269, Springer, Heidelberg, 1972–1973.Google Scholar
Gruson, L. and Jensen, C. U., Deux applications de la notion de L-dimension, C. R. Acad. Sci. Paris Sér. A–B 282 (1976), no. 1, A23–A24.Google Scholar
Gruson, L. and Jensen, C. U., Dimensions cohomologiques reliées aux foncteurs lim(i) , in Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1980), 234–294, Lecture Notes in Mathematics, 867, Springer, Berlin, 1981.Google Scholar
Happel, D., On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389.Google Scholar
Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988.Google Scholar
Happel, D., Auslander–Reiten triangles in derived categories of finite-dimensional algebras, Proc. Am. Math. Soc. 112 (1991), no. 3, 641–648.Google Scholar
Happel, D., A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), no. 2, 381–398.CrossRefGoogle Scholar
Happel, D., Reiten, I. and Smalø, S. O., Tilting in abelian categories and quasitilted algebras, Memoirs of the American Mathematical Society, 120, American Mathematical Society, Providence, RI, 1996, no. 575.Google Scholar
Hartshorne, R., Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64 [with an appendix by Deligne, P.], Lecture Notes in Mathematics, 20, Springer, Berlin, 1966.Google Scholar
Heller, A., Homological algebra in abelian categories, Ann. Math. (Ser. 2) 68 (1958), 484–525.Google Scholar
Heller, A., The loop-space functor in homological algebra, Trans. Am. Math. Soc. 96 (1960), 382–394.Google Scholar
Henn, H.-W., Lannes, J. and Schwartz, L., The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Am. J. Math. 115 (1993), no. 5, 1053–1106.Google Scholar
Herzog, I., Elementary duality of modules, Trans. Am. Math. Soc. 340 (1993), no. 1, 37–69.Google Scholar
Herzog, I., The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. (Ser. 3) 74 (1997), no. 3, 503–558.Google Scholar
Hilbert, D., Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890), no. 4, 473–534.Google Scholar
Hochster, M., Prime ideal structure in commutative rings, Trans. Am. Math. Soc. 142 (1969), 43–60.Google Scholar
Hughes, D. and Waschbüsch, J., Trivial extensions of tilted algebras, Proc. London Math. Soc. (Ser. 3) 46 (1983), no. 2, 347–364.CrossRefGoogle Scholar
Humphreys, J. E., Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, 94, American Mathematical Society, Providence, RI, 2008.Google Scholar
Iwanaga, Y., On rings with finite self-injective dimension, Commun. Algebra 7 (1979), no. 4, 393–414.Google Scholar
Jans, J. P., On the indecomposable representations of algebras, Ann. Math. (Ser. 2) 66 (1957), 418–429.Google Scholar
Jensen, C. U. and Lenzing, H., Model-theoretic algebra with particular emphasis on fields, rings, modules, Gordon and Breach Science Publishers, New York, 1989.Google Scholar
Kaplansky, I., Infinite abelian groups, University of Michigan Press, Ann Arbor, MI, 1954, revised 1969.Google Scholar
Keller, B., Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379–417.Google Scholar
Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (Sér. 4) 27 (1994), no. 1, 63–102.Google Scholar
Keller, B. and Krause, H., Tilting preserves finite global dimension, C. R. Math. Acad. Sci. Paris 358 (2020), no. 5, 563–571.Google Scholar
Keller, B. and Vossieck, D., Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228.Google Scholar
Kiełpiński, R., On Γ-pure injective modules, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 127–131.Google Scholar
Krause, H., The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), no. 3, 259–271.Google Scholar
Krause, H., A Brown representability theorem via coherent functors, Topology 41 (2002), no. 4, 853–861.Google Scholar
Krause, H., Coherent functors and covariantly finite subcategories, Algebras Represent. Theory 6 (2003), no. 5, 475–499.Google Scholar
Krause, H., The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162.Google Scholar
Krause, H., Koszul, Ringel and Serre duality for strict polynomial functors, Compos. Math. 149 (2013), no. 6, 996–1018.CrossRefGoogle Scholar
Krause, H., Krull–Schmidt categories and projective covers, Expo. Math. 33 (2015), no. 4, 535–549.Google Scholar
Krause, H., Completing perfect complexes [with appendices by Barthel, T. and Keller, B.], Math. Z. 296 (2020), 1387–1427.Google Scholar
Krause, H. and Ringel, C. M., Infinite length modules, Trends in Mathematics, Birkhäuser, Basel, 2000.Google Scholar
Krause, H. and Saorín, M., On minimal approximations of modules, in Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997), 227–236, Contemporary Mathematics, 229, American Mathematical Society, Providence, RI, 1998.Google Scholar
Krause, H. and Vossieck, D., Length categories of infinite height, in Geometric and topological aspects of the representation theory of finite groups, 213–234, Springer Proceedings in Mathematics and Statistics, 242, Springer, Cham, 2018.CrossRefGoogle Scholar
Lam, T. Y., A first course in noncommutative rings, Graduate Texts in Mathematics, 131, Springer, New York, 1991.Google Scholar
Lenzing, H., Über die Funktoren Ext1 (·,E) und Tor1 (·,E), Dissertation, FU Berlin, 1964.Google Scholar
Lenzing, H., Auslander’s work on Artin algebras, in Algebras and modules, I (Trondheim, 1996), 83–105, CMS Conference Proceedings, 23, American Mathematical Society, Providence, RI, 1998.Google Scholar
Macdonald, I. G., Symmetric functions and Hall polynomials, second edition, Oxford Mathematical Monographs, Oxford University Press, New York, 1995.Google Scholar
Mac Lane, S., Homology, Die Grundlehren der mathematischen Wissenschaften, 114, Academic Press, New York; Springer, Berlin, 1963.Google Scholar
Mac Lane, S., Categories for the working mathematician, second edition, Graduate Texts in Mathematics, 5, Springer, New York, 1998.Google Scholar
Matlis, E., Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528.Google Scholar
Nagata, M., Local rings, Interscience Tracts in Pure and Applied Mathematics, 13, Interscience Publishers, New York, 1962.Google Scholar
Neeman, A., The derived category of an exact category, J. Algebra 135 (1990), no. 2, 388–394.Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Am. Math. Soc. 9 (1996), no. 1, 205–236.Google Scholar
Neeman, A., Brown representability for the dual, Invent. Math. 133 (1998), no. 1, 97–105.Google Scholar
Neeman, A., Triangulated categories, Annals of Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001.Google Scholar
Ore, O., Linear equations in non-commutative fields, Ann. Math. (Ser. 2) 32 (1931), no. 3, 463–477.Google Scholar
Orlov, D. O., Triangulated categories of singularities and D-branes in Landau– Ginzburg models, Proc. Steklov Inst. Math. 2004, no. 3(246), 227–248; translated from Mat, Tr.. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi Prilozh., 240–262.Google Scholar
Orlov, D. O., Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), no. 1, 206–217.Google Scholar
Osofsky, B. L., Homological dimension and cardinality, Trans. Am. Math. Soc. 151 (1970), 641–649.Google Scholar
Pareigis, B., Categories and functors, translated from the German, Pure and Applied Mathematics, 39, Academic Press, New York, 1970.Google Scholar
Parshall, B. J., Finite-dimensional algebras and algebraic groups, in Classical groups and related topics (Beijing, 1987), 97–114, Contemporary Mathematics, 82, American Mathematical Society, Providence, RI,1989.Google Scholar
Parshall, B. J. and Scott, L. L., Derived categories, quasi-hereditary algebras, and algebraic groups, in Proc. Ottawa—Moosonee Workshop in algebra (Ottawa, 1987), 1–104, Carleton Mathematical Lecture Notes, 3, Carleton University, Ottawa, Ont., 1988.Google Scholar
Popescu, N. and Gabriel, P., Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, C. R. Acad. Sci. Paris 258 (1964), 4188–4190.Google Scholar
Prest, M., Remarks on elementary duality, Ann. Pure Appl. Logic 62 (1993), no. 2, 185–205.Google Scholar
Prest, M., Ziegler spectra of tame hereditary algebras, J. Algebra 207 (1998), no. 1, 146–164.Google Scholar
Prest, M., Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications, 121, Cambridge University Press, Cambridge, 2009.Google Scholar
Prüfer, H., Untersuchungen über die Zerlegbarkeit der abzählbaren primären Abelschen Gruppen, Math. Z. 17 (1923), no. 1, 35–61.Google Scholar
Puppe, D., On the structure of stable homotopy theory, in Colloquium on algebraic topology, 65–71, Aarhus Universitet Matematisk Institut, Aarhus, 1962.Google Scholar
Quillen, D., Higher algebraic K-theory. I, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 85–147, Lecture Notes in Mathematics, 341, Springer, Berlin, 1973.Google Scholar
Ranicki, A., Non-commutative localization in algebra and topology, Proc. Workshop, Edinburgh, April 29–30, 2002, London Mathematical Society Lecture Note Series, 330, Cambridge University Press, Cambridge, 2006.Google Scholar
Ravenel, D. C., Localization with respect to certain periodic homology theories, Am. J. Math. 106 (1984), no. 2, 351–414.Google Scholar
Reiten, I. and Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Am. Math. Soc. 15 (2002), no. 2, 295–366.Google Scholar
Richter, G., Noetherian semigroup rings with several objects, in Group and semigroup rings (Johannesburg, 1985), 231–246, North-Holland Mathematical Studies, 126, North-Holland, Amsterdam, 1986.Google Scholar
Rickard, J., Morita theory for derived categories, J. London Math. Soc. (Ser. 2) 39 (1989), no. 3, 436–456.Google Scholar
Ringel, C. M., Infinite-dimensional representations of finite-dimensional hereditary algebras, in Symposia Mathematica, vol. XXIII (Conf. abelian groups and their relationship to the theory of modules, INDAM, Rome, 1977), 321–412, Academic Press, London, 1979.Google Scholar
Ringel, C. M., The canonical algebras, in Topics in algebra, Part 1 (Warsaw, 1988), 407–432, Banach Center Publications, 26, Part 1, PWN, Warsaw, 1990.Google Scholar
Ringel, C. M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209– 223.Google Scholar
Ringel, C. M., The Ziegler spectrum of a tame hereditary algebra, Colloq. Math. 76 (1998), no. 1, 105–115.Google Scholar
Roby, N., Lois polynomes et lois formelles en théorie des modules, Ann. Sci. Éc. Norm. Supér. (Sér. 3) 80 (1963), 213–348.Google Scholar
Roos, J.-E., Sur la décomposition bornée des objets injectifs dans les catégories de Grothendieck, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A449–A452.Google Scholar
Roos, J. E., Locally Noetherian categories and generalized strictly linearly compact rings. Applications, in Category theory, homology theory and their applications, II (Battelle Institute Conf., Seattle, Wash., 1968, Vol. Two), 197–277, Springer, Berlin, 1969.Google Scholar
Salce, L., Cotorsion theories for abelian groups, in Symposia Mathematica, vol. XXIII (Conf. abelian groups and their relationship to the theory of modules, INDAM, Rome, 1977), 11–32, Academic Press, London, 1979.Google Scholar
Sam, S. V. and Snowden, A., Gröbner methods for representations of combinatorial categories, J. Am. Math. Soc. 30 (2017), no. 1, 159–203.Google Scholar
Schlichting, M., Negative K-theory of derived categories, Math. Z. 253 (2006), no. 1, 97–134.Google Scholar
Schofield, A. H., Representation of rings over skew fields, London Mathematical Society Lecture Note Series, 92, Cambridge University Press, Cambridge, 1985.Google Scholar
Schubert, H., Categories [translated from the German by Eva Gray], Springer, New York, 1972.Google Scholar
Schur, I., Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen. Dissertation, Berlin, 1901. In I. Schur, Gesammelte Abhandlungen I, 1–70, Springer, Berlin, 1973.Google Scholar
Schur, I., Über die rationalen Darstellungen der allgemeinen linearen Gruppe, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. (1927), 58–75. In I. Schur, Gesammelte Abhandlungen III, 68–85, Springer, Berlin, 1973.Google Scholar
Schwede, S., Algebraic versus topological triangulated categories, in Triangulated categories, 389–407, London Mathematical Society Lecture Note Series, 375, Cambridge University Press, Cambridge, 2010.Google Scholar
Scott, L. L., Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories, in The Arcata Conf. on representations of finite groups (Arcata, Calif., 1986), 271–281, Proceedings of Symposia in Pure Mathematics, 47, Part 2, American Mathematical Society, Providence, RI, 1987.Google Scholar
Serre, J.-P., Faisceaux algébriques cohérents, Ann. Math. (Ser. 2) 61 (1955), 197–278.Google Scholar
Serre, J.-P., Cohomologie et géométrie algébrique, in Proc. Int. Congress of mathematicians, 1954, Amsterdam, vol. III, 515–520, Erven P. Noordhoff, Groningen, 1956.Google Scholar
Serre, J.-P., Sur les modules projectifs, in Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 14ième année: 1960/61. Algèbre et théorie des nombres. Fasc. 1, 1–16, Faculté des Sciences de Paris, Secrétariat mathématique, Paris, 1963.Google Scholar
Simson, D., Pure semisimple categories and rings of finite representation type, J. Algebra 48 (1977), no. 2, 290–296.Google Scholar
Simson, D., On pure semi-simple Grothendieck categories. I, Fund. Math. 100 (1978), no. 3, 211–222.Google Scholar
Simson, D., On right pure semisimple hereditary rings and an Artin problem, J. Pure Appl. Algebra 104 (1995), no. 3, 313–332.Google Scholar
Spaltenstein, N., Resolutions of unbounded complexes, Compos. Math. 65 (1988), no. 2, 121–154.Google Scholar
Stanley, R. P., Enumerative combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999.Google Scholar
Totaro, B., Projective resolutions of representations of GL(n), J. Reine Angew. Math. 482 (1997) 1–13.Google Scholar
Verdier, J.-L., Des catégories dérivées des catégories abéliennes, Astérisque, 239, Société Mathématique de France, 1996.Google Scholar
Warfield, R. B., Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719.Google Scholar
Yoneda, N., On the homology theory of modules, J. Fac. Sci. Univ. Tokyo Sect. I 7 (1954), 193–227.Google Scholar
Ziegler, M., Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213.Google Scholar
Zimmermann, W., Rein injektive direkte Summen vonModuln, Commun. Algebra 5 (1977), no. 10, 1083–1117.Google Scholar
Zimmermann-Huisgen, B., Rings whose right modules are direct sums of indecomposable modules, Proc. Am. Math. Soc. 77 (1979), no. 2, 191–197.Google Scholar