Local algebra and rational homotopy

Avramov, Luchezar L.

Homotopie algébrique et algèbre locale, Astérisque, no. 113-114 (1984), pp. 15-43.

@incollection{AST_1984__113-114__15_0,
     author = {Avramov, Luchezar L.},
     title = {Local algebra and rational homotopy},
     booktitle = {Homotopie alg\'ebrique et alg\`ebre locale},
     series = {Ast\'erisque},
     pages = {15--43},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {113-114},
     year = {1984},
     mrnumber = {749041},
     zbl = {0552.13003},
     language = {en},
     url = {http://www.numdam.org/item/AST_1984__113-114__15_0/}
}

TY  - CHAP
AU  - Avramov, Luchezar L.
TI  - Local algebra and rational homotopy
BT  - Homotopie algébrique et algèbre locale
AU  - Collectif
T3  - Astérisque
PY  - 1984
SP  - 15
EP  - 43
IS  - 113-114
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_1984__113-114__15_0/
LA  - en
ID  - AST_1984__113-114__15_0
ER  -

%0 Book Section
%A Avramov, Luchezar L.
%T Local algebra and rational homotopy
%B Homotopie algébrique et algèbre locale
%A Collectif
%S Astérisque
%D 1984
%P 15-43
%N 113-114
%I Société mathématique de France
%U http://www.numdam.org/item/AST_1984__113-114__15_0/
%G en
%F AST_1984__113-114__15_0

Avramov, Luchezar L. Local algebra and rational homotopy, dans Homotopie algébrique et algèbre locale, Astérisque, no. 113-114 (1984), pp. 15-43. http://www.numdam.org/item/AST_1984__113-114__15_0/

Bibliographie
Cité par

[1] M. André, Hopf algebras with divided powers, J. Algebra 18 (1971), 19-50. | DOI | MR | Zbl

[2] M. André, Homologie des Algèbres Commutatives, Springer-Verlag, Berlin, 1979. | MR | Zbl

[3] M. André, Algèbre homologique des anneaux locaux à corps résiduel de caractéristique deux, Lecture Notes in Mathematics 740, 237-242. | MR | Zbl

[4] M. André, Produits de Massey et $(2 p + l)$ -ièmes déviations, Lecture Notes in Mathematics 795, 341-359, Springer-Verlag, Berlin, 1980. | MR | Zbl

[5] M. André, Le caractère additif des déviations des anneaux locaux, Comment. Math. Helv. (to appear). | EuDML | MR | Zbl

[6] P. Andrews and M. Arkowitz, Sullivan's minimal models and higher order Whitehead products, Canad. J. Math. 30 (1978), 961-982. | DOI | MR | Zbl

[7] D. J. Anick, Constructions d'espaces de lacets et d'anneaux locaux à séries de Poincaré-Betti non-rationelles, C. R. Acad. Sci. Paris 290(1980), A. 729-733. | MR | Zbl

[8] D. J. Anick, A counterexample to a conjecture of Serre, Annals Math, 115, (1982), 1-33. | MR | Zbl

[9] E. F. Assmus, Jr., On the homology of local rings, Illinois J. Math. 3(1959), 187-199. | MR | Zbl

[10] M. Auslander and D. Buchsbaum, Codimension and multiplicity, Annals Math. 68 (1958), 625-657. | DOI | MR | Zbl

[11] L. L. Avramov, On the Hopf algebra of a local ring, Isv. Akad. Nauk. SSSR, Ser. mat., 38 (1974), 253-277 | MR | Zbl

L. L. Avramov, On the Hopf algebra of a local ring English translation : Math. USSR, Izv. 8(1974). | DOI | MR | Zbl

[12] L. L. Avramov, Homology of local flat extensions and complete intersection defects, Math. Ann. 228(1977), 27-37. | DOI | EuDML | MR | Zbl

[13] L. L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), 400-453. | DOI | MR | Zbl

[14] L. L. Avramov, Sur la croissance des nombres de Betti d'un anneau local, C. R. Acad. Sci. Paris 289 (1979), A. 369-372. | MR | Zbl

[15] L. L. Avramov, Free Lie subalgebras of the cohomology of local rings, Trans. Amer. Math. Soc. 270(1982), 589-608. | DOI | MR | Zbl

[16] L. L. Avramov, Differential graded models for local rings, RIMS Symposium on Commutative Algebra and Algebraic Geometry (Sept. 10-12, 1981, org. Tadao Oda), | Zbl

L. L. Avramov, Differential graded models for local rings, RIMS Symposium on Commutative Algebra and Algebraic Geometry RIMS Kokyuroku 446, 80-88, Kyoto University, 1981. | Zbl

[17] L. L. Avramov, The homology of a tensor product diagram, Reports, Dept. of Maths. Univ. of Stockholm, N° 2, 1982.

[18] L. L. Avramov, Descente des déviations par homomorphismes locaux et generation des idéaux de dimension projective finie, CR. Acad. Sci. Paris 295 (1982), I. 665-668. | MR | Zbl

[19] L. L. Avramov, Homotopy Lie algebras for commutative rings and $D G$ algebras, to appear.

[20] I. K. Babenko, On real homotopy properties of complete intersections, Izv. Akad. Nauk SSSR, Ser. mat. 43 (1979), 1004-1024; | MR | Zbl

I. K. Babenko, On real homotopy properties of complete intersections English translation : Math. USSR, Izv. 13(1979). | MR | Zbl

[21] N. Bourbaki, Algèbre Commutative, Chapitre 3, Hermann, Paris, 1961.

[22] A. K. Bousfield and V. K. A. M. Gugenheim, On the PL De Rham Theory and rational homotopy type, Memoirs Amer. Math. Soc. 179(1976). | MR | Zbl

[23] W. Bruns, « Jede » endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals, J. Algebra 39(1976), 429-439. | DOI | MR | Zbl

[24] H. Cartan, Algèbres d'Eilenberg-MacLane, Séminaire H. Cartan, Ecole Normale Supérieure, 1954-1955, Exposés 2 à 11

H. Cartan, Algèbres d'Eilenberg-MacLane, reprinted in Oeuvres, v. III, 1309-1394, Springer-Verlag, Berlin, 1979. | MR

[25] A. Clark and L. Smith, The rational homotopy of a wedge, Pacific J. Math. 24 (1968), 241-246. | DOI | MR | Zbl

[26] P. Deligne, Ph. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29(1975), 245-274. | DOI | EuDML | MR | Zbl

[27] D. Eisenbud, Homological algebra on a complete intersection with an application to group representations, Trans. Amer. Math. Soc. 260(1980), 35-64. | DOI | MR | Zbl

[28] E. G. Evans and Ph. Griffith, The syzygy problem, Annals Math. 114(1981), 323-333. | DOI | MR | Zbl

[29] Y. Felix and S. Halperin, Rational L.-S. Category and its applications, Trans. Amer. Math. Soc. 273(1982), 1-37. | DOI | MR | Zbl

[30] Y. Felix, S. Halperin and J. C. Thomas, The homotopy Lie algebra for finite complexes, Publ. Math. IHES 56 (1982), 179-202. | DOI | EuDML | Numdam | MR | Zbl

[31] Y. Felix and J. C. Thomas, The radius of convergence of Poincaré series of loop spaces, Invent. Math. 68(1982), 257-274. | DOI | EuDML | MR | Zbl

[32] F. Ghione and T. H. Gulliksen, Some reduction formulas for the Poincaré series of modules, Atti. Accad. naz. Lincei. VIII Ser., Rend. Cl. Sci. Fis.mat.natur. 58(1975), 82-91. | MR | Zbl

[33] E. H. Gover and M. Ramras, Increasing sequences of Betti numbers, Pacific J. Math. 87(1980), 65-68. | DOI | MR | Zbl

[34] T. H. Gulliksen, A homological characterization of local complete intersections, Compositio Math. 23 (1971), 251-255. | EuDML | Numdam | MR | Zbl

[35] T. H. Gulliksen, A change of rings theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34(1974), 167-183. | DOI | EuDML | MR | Zbl

[36] T. H. Gulliksen, On the deviations of a local ring, Math. Scand. 47(1980), 5-20. | DOI | EuDML | MR | Zbl

[37] T. H. Gulliksen and G. Levin, Homology of Local Rings, Queen's papers in Pure and Applied Mathematics - No. 20, Queen's University, Kingston, Ontario, 1969. | MR | Zbl

[38] S. Halperin, Rational fibrations, minimal models, and fibrings of homogeneous spaces, Trans. Amer. Math. Soc. 244(1978), 199-224. | DOI | MR | Zbl

[39] S. Halperin, Lectures on Minimal Models, Pub. IRMA-Lille, V. 3, Fasc. 4 (1981), (3rd Edition). | MR | Zbl

[40] C. Jacobsson, On flat homomorphisms and the Yoneda Ext-algebra of the fibre, these Proceedings.

[41] D. Lehmann, Théorie homotopique des formes différentielles, Astérisque 45(1977). | MR | Zbl

[42] J. M. Lemaire, Anneaux locaux et espaces de lacets à séries de Poincaré non-rationnelles, Séminaire Bourbaki, 1980-81, No. 570, Lecture Notes in Mathematics 901, Springer-Verlag, Berlin, 1981. | EuDML | Numdam | MR | Zbl

[43] G. Levin, Homology of local rings, Ph.D. Thesis, Univ. of Chicago, 1965. | MR

[44] G. Levin, Local rings and Golod homomorphisms, J. Algebra 37(1975), 266-289. | DOI | MR | Zbl

[45] G. Levin, Finitely-generated Ext-algebras, Math. Scand. 49 (1981), 161-180. | DOI | EuDML | MR | Zbl

[46] G. Levin and L. L. Avramov, Factoring out the socle of a local Gorenstein ring, J. Algebra 55(1978), 74-83. | DOI | MR | Zbl

[47] C. Löfwall, On the subalgebra generated by the one-dimensional elements of the Yoneda Ext-algebra, Reports, Dept. of Maths., Univ. of Stockholm, No. 5, 1976. | Zbl

[48] C. Löfwall et J. E. Roos, Cohomologie des algèbres de Lie graduées et séries de Poincaré-Betti non rationelles, C. R. Acad. Sci. Paris 290(1980), A. 733-736. | MR | Zbl

[49] H. Matsumura, Commutative Algebra (2nd edition), Benjamin/Cummings, Reading, Mass, 1980. | MR

[50] T. Miller, On the formality of $(k - 1)$ -connected compact manifolds of dimension less than or equal to 4k - 2, Illinois J. Math. 23 (1979), 253-258. | MR | Zbl

[51] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Annals Math. 81(1965), 211-264. | DOI | MR | Zbl

[52] J. C. Moore, Algèbre homologique et homologie des espaces classifiants, Séminaire H. Cartan, Ecole Normale Supérieure, 1959-1960, Exposé 7, | EuDML | Numdam | Zbl

J. C. Moore, Algèbre homologique et homologie des espaces classifiants, Secrétariat Math., Paris, 1961. | Numdam | Zbl

[53] J. Neissendorfer, The rational homotopy groups of complete intersections, Illinois J. Math. 23 (1979), 175-182. | MR | Zbl

[54] V. P. Palamodov, Deformations of Complex spaces, Uspehi Mat. Nauk 31, No. 3 (1976), 129-194, (in Russian) | MR | Zbl

V. P. Palamodov, Deformations of Complex spaces English translation: Rusian Math. Surveys 31(1976). | MR | Zbl

[55] D. Quillen, On the (co-)homology of commutative rings, Proc. Symp. Pure Math. 17, 65-87, | DOI | MR | Zbl

D. Quillen, On the (co-)homology of commutative rings, Amer. Math. Soc., Providence, 1970. | MR | Zbl

[56] M. Ramras, Betti numbers and reflexive modules, in Ring Theory (R. Gordon, Ed.), 297-308, Academic Press, NY, 1972. | DOI | MR | Zbl

[57] M. Ramras, Sequences of Betti numbers, J. Algebra 66(1980), 193-204. | DOI | MR | Zbl

[58] J. E. Roos, Relations between the Poincaré-Betti series of loop spaces and of local rings, Lecture Notes in Mathematics 740, 285-322, Springer-Verlag, Berlin, 1979. | MR | Zbl

[59] J. E. Roos, Homology of loop spaces and of local rings, in Proceedings 18th Scandinavian Congress of Mathematics, 1980 | Zbl

J. E. Roos, Homology of loop spaces and of local rings, Progress in Mathematics 11, 441-468, Birkhäuser, Basel, 1981. | MR | Zbl

[60] J. E. Roos, The use of graded Lie algebras in the theory of local rings, Proc. Durham Conf. in Commutative Algebra, 1981, | Zbl

J. E. Roos, The use of graded Lie algebras in the theory of local rings London Math. Soc. Lecture Notes, 72,204-230, Cambridge University Press, 1982. | MR | Zbl

[61] C. Schoeller, Homologie des anneaux locaux noethériens, C. R. Acad. Sci. Paris 265(1967), A. 768-771. | MR | Zbl

[62] J. P. Serre, Algèbre locale.Multiplicités, Lecture Notes in Mathematics 11 (3e edition), Springer-Verlag, Berlin, 1975. | Zbl

[63] G. Sjödin, A set of generators for Ext $_{R} (k, k)$ , Math. Scand. 38(1967), 1-12. | EuDML | MR | Zbl

[64] G. Sjödin, Hopf algebras and derivations, J. Algebra 64(1980), 218-229. | DOI | MR | Zbl

[65] J. Stasheff, Rational Poincaré duality spaces, Preprint, 1981. | MR | Zbl

[66] D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES 47 (1978), 269-331. | DOI | EuDML | Numdam | MR | Zbl