Complete intersections and mod p cochains

Benson, David J¹ ; Greenlees, John P C² ; Shamir, Shoham³

¹Department of Mathematics, University of Aberdeen, Meston Building, Aberdeen AB24 3UE, UK
²School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
³Department of Mathematics, University of Bergen, 5008 Bergen, Norway

Algebraic and Geometric Topology, Tome 13 (2013) no. 1, pp. 61-114

Résumé

We give homotopy invariant definitions corresponding to three well-known properties of complete intersections, for the ring, the module theory and the endomorphisms of the residue field, and we investigate them for the mod p cochains on a space, showing that suitable versions of the second and third are equivalent and that the first is stronger. We are particularly interested in classifying spaces of groups, and we give a number of examples. The case of rational homotopy theory is treated in [J. Pure Appl. Algebra 217 (2013) 636–663], and there are some interesting contrasts.

Détail

DOI : 10.2140/agt.2013.13.61

Classification : 13C40, 55P43, 13D99, 20J06, 55N99, 14M10, 55P42, 55U35, 20J05
Keywords: complete intersection, commutative ring spectrum, derived category, group cohomology, mod $p$ cochains

Bibliographie

[1] J Aguadé, C Broto, D Notbohm, Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke, I, Topology 33 (1994) 455

[2] J L Alperin, R Brauer, D Gorenstein, Finite groups with quasi-dihedral and wreathed Sylow 2–subgroups, Trans. Amer. Math. Soc. 151 (1970) 1

[3] K K S Andersen, J Grodal, The classification of 2–compact groups, J. Amer. Math. Soc. 22 (2009) 387

[4] K K S Andersen, J Grodal, J M Møller, A Viruel, The classification of p–compact groups for p odd, Ann. of Math. 167 (2008) 95

[5] L L Avramov, Infinite free resolutions, from: "Six lectures on commutative algebra" (editors J Elias, J M Giral, R M Miró-Roig, S Zarzuela), Progr. Math. 166, Birkhäuser (1998) 1

[6] L L Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Ann. of Math. 150 (1999) 455

[7] L L Avramov, R O Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000) 285

[8] D Benson, An algebraic model for chains on ΩBGp∧, Trans. Amer. Math. Soc. 361 (2009) 2225

[9] D J Benson, J F Carlson, Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. 342 (1994) 447

[10] D J Benson, J P C Greenlees, Complete intersections and derived categories (2009)

[11] N Castellana, The homotopic adjoint representation for the exotic p–compact groups, preprint (1999)

[12] N Castellana, On the p–compact groups corresponding to the p–adic reflection groups G(q,r,n), Trans. Amer. Math. Soc. 358 (2006) 2799

[13] W G Dwyer, Strong convergence of the Eilenberg–Moore spectral sequence, Topology 13 (1974) 255

[14] W G Dwyer, J P C Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002) 199

[15] W G Dwyer, J P C Greenlees, S Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357

[16] W Dwyer, J P C Greenlees, S Iyengar, Finiteness in derived categories of local rings, Comment. Math. Helv. 81 (2006) 383

[17] W G Dwyer, C W Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. 139 (1994) 395

[18] D Eisenbud, Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970) 247

[19] D Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980) 35

[20] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, 47, American Mathematical Society (1997)

[21] K Erdmann, Algebras and semidihedral defect groups, II, Proc. London Math. Soc. 60 (1990) 123

[22] Y Félix, S Halperin, J C Thomas, Elliptic Hopf algebras, J. London Math. Soc. 43 (1991) 545

[23] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, 205, Springer (2001)

[24] Y Felix, J C Thomas, M Vigué-Poirrier, The Hochschild cohomology of a closed manifold, Publ. Math. Inst. Hautes Études Sci. (2004) 235

[25] Z Fiedorowicz, S Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, 674, Springer (1978)

[26] J P C Greenlees, Spectra for commutative algebraists, from: "Interactions between homotopy theory and algebra" (editors L L Avramov, J D Christensen, W G Dwyer, M A Mandell, B E Shipley), Contemp. Math. 436, Amer. Math. Soc. (2007) 149

[27] J P C Greenlees, K Hess, S Shamir, Complete intersections in rational homotopy theory, J. Pure Appl. Algebra 217 (2013) 636

[28] T H Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974) 167

[29] T H Gulliksen, On the deviations of a local ring, Math. Scand. 47 (1980) 5

[30] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149

[31] N R Kitchloo, Topology of Kac–Moody groups, Ph.D. thesis, Massachusetts Institute of Technology (1998)

[32] S N Kleinerman, The cohomology of Chevalley groups of exceptional Lie type, Mem. Amer. Math. Soc. 39 (1982)

[33] R Levi, On finite groups and homotopy theory, Mem. Amer. Math. Soc. 118 (1995)

[34] R Levi, A counter-example to a conjecture of Cohen, from: "Algebraic topology : New trends in localization and periodicity" (editors C Broto, C Casacuberta, G Mislin), Progr. Math. 136, Birkhäuser (1996) 261

[35] R Levi, On homological rate of growth and the homotopy type of ΩBGp∧, Math. Z. 226 (1997) 429

[36] R Levi, On p–completed classifying spaces of discrete groups and finite complexes, J. London Math. Soc. 59 (1999) 1064

[37] T Y Lin, Duality and Eilenberg–Mac Lane spectra, Proc. Amer. Math. Soc. 56 (1976) 291

[38] M A Mandell, E∞ algebras and p–adic homotopy theory, Topology 40 (2001) 43

[39] J M Møller, N–determined 2–compact groups, I, Fund. Math. 195 (2007) 11

[40] J M Møller, N–determined 2–compact groups, II, Fund. Math. 196 (2007) 1

[41] B Oliver, Equivalences of classifying spaces completed at the prime two, Mem. Amer. Math. Soc. 180 (2006)

[42] D Quillen, The mod 2 cohomology rings of extra-special 2–groups and the spinor groups, Math. Ann. 194 (1971) 197

[43] D Quillen, On the cohomology and K–theory of the general linear groups over a finite field, Ann. of Math. 96 (1972) 552

[44] J P Serre, Sur la dimension homologique des anneaux et des modules noethériens, from: "Proceedings of the international symposium on algebraic number theory", Science Council of Japan (1956) 175

[45] J Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969) 453

[46] S Shamir, Hochschild cohomology of cochain-algebras and local coefficients, In preparation

[47] B Shipley, HZ–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351

[48] K Ziemiański, A faithful unitary representation of the 2–compact group DI(4), J. Pure Appl. Algebra 213 (2009) 1239