Skip to main content Accessibility help
×
  • Cited by 11
Publisher:
Cambridge University Press
Online publication date:
January 2017
Print publication year:
2016
Online ISBN:
9781316795699

Book description

Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. Special attention is given to recent work on modules of constant Jordan type, and the methods involve producing and examining vector bundles on projective space and their Chern classes. Extensive background material is provided, which will help the reader to understand vector bundles and their Chern classes from an algebraic point of view, and to apply this to modular representation theory of elementary abelian groups. The final section, addressing problems and directions for future research, will also help to stimulate further developments in the subject. With no similar books on the market, this will be an invaluable resource for graduate students and researchers working in representation theory.

Reviews

'In summary, this book provides a thorough introduction to the theory of the correspondence between modular representations of elementary abelian groups and vector bundles over projective space. In it the reader will find results from the literature, as well as new contributions to the field. It provides all of the background necessary to understand the material, and provides a lot of interesting examples as well as open problems.'

Alan Koch Source: Mathematical Reviews

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×

Contents

References
[1] K, Akin, D. A, Buchsbaum, and J, Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), 207–278.
[2] G, Almkvist, The number of nonfree components in the decomposition ofsymmetric powers in characteristic p, Pacific J. Math. 77 (1978), 293–301.
[3] G, Almkvist, Reciprocity theorems for representations in characteristic p, Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin, 32ème année (Paris, 1979), Lecture Notes in Mathematics, vol. 795, Springer-Verlag, Berlin/New York, 1980, pp. 1–9.
[4] G, Almkvist, Representations of Z/pZ in characteristic p and reciprocitytheorems, J. Algebra 68 (1981), 1–27.
[5] G, Almkvist, Some formulas in invariant theory, J. Algebra 77 (1982), 338–359.
[6] G, Almkvist and R, Fossum, Decomposition of exterior and symmetric powers ofindecomposable Z/pZ-modules in characteristic p and relations to invariants, Séminaire d'Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Mathematics, vol. 641, Springer-Verlag, Berlin/New York, 1978, pp. 1–111.
[7] J. L, Alperin and L, Evens, Representations, resolutions, and Quillen's dimensiontheorem, J. Pure & Applied Algebra 22 (1981), 1–9.
[8] J. L, Alperin and L, Evens, Varieties and elementary abelian subgroups, J. Pure & Applied Algebra 26 (1982), 221–227.
[9] V, Ancona and G, Ottaviani, Stability of special instanton bundles on P2n+1, Trans. Amer. Math. Soc. 341 (1994), 677–693.
[10] V, Ancona and G, Ottaviani, On moduli of instanton bundles on P2n+1, Pacific J. Math. 171 (1995), 343–351.
[11] V, Ancona and G, Ottaviani, The Horrocks bundles of rank three on P5, J. Reine & Angew. Math. 460 (1995), 69–92.
[12] A, Aramova, L. L, Avramov, and J, Herzog, Resolutions of monomial ideals andcohomology over exterior algebras, Trans. Amer. Math. Soc. 352 1999, no.2, 579–594.
[13] E, Arrondo, Schwarzenberger bundles of arbitrary rank on the projective space, J. London Math. Soc. 82 (2010), 697–716.
[14] M. F, Atiyah, On the Krull–Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307–317.
[15] M. F, Atiyah, V. G, Drinfeld, N. J, Hitchin and Yu. I., Manin, Construction ofinstantons, Phys. Lett. 65A (1978), 185–187. 312
[16] M. F, Atiyah and I. G, Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969.
[17] M. F, Atiyah and E, Rees, Vector bundles on projective 3-space, Invent. Math. 35 (1976), 131–153.
[18] M. F, Atiyah and D. O, Tall, Group representations, λ-rings and theJ -homomorphism, Topology 8 (1969), 253–297.
[19] M. D, Atkinson and R, Westwick, Spaces of linear transformations of equalrank, Linear & Multilinear Algebra 13 (1983), 231–239.
[20] A. B, Aure, Surfaces on quintic threefolds associated to the Horrocks–Mumfordbundle, Arithmetic of complex manifolds (Erlangen, 1988), Lecture Notes in Mathematics, vol. 1399, Springer-Verlag, Berlin/New York, 1989, pp. 1–9.
[21] M, Auslander and R.-O., Buchweitz, The homological theory of maximal Cohen–Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37.
[22] L. L, Avramov, R.-O., Buchweitz, S. B., Iyengar, and C, Miller, Homology ofperfect complexes, Adv. in Math. 223 (2010), 1731–1781.
[23] L. L, Avramov, V, Gasharov, and I, Peeva, Complete intersection dimension, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 67–114.
[24] G. S, Avrunin and L. L, Scott, Quillen stratification for modules, Invent. Math. 66 (1982), 277–286.
[25] S, Baland, Modules of constant Jordan type with two non-projective blocks, J. Algebra 346 (2011), 343–350.
[26] S, Baland, On the generic kernel filtration for modules of constant Jordan type, Arch. Math. (Basel) 99 (2012), 305–314.
[27] E, Ballico, Uniform vector bundles of rank (n + 1) on Pn, Tsukuba J. Math. 7 1983, 215–226.
[28] E, Ballico, Vector spaces of matrices of low rank and vector bundles onprojective spaces, Beiträge Algebra Geom. 36 (1995), 119–122.
[29] V, Baranovsky and J, Pecharich, On equivalences of derived and singularcategories, Cent. Eur. J. Math. 8 2010, no.1, 1–14.
[30] M. J. J., Barry, Decomposing tensor products and exterior and symmetricsquares, J. Group Theory 14 (2011), 59–82.
[31] M. J. J., Barry, Generators for decompositions of tensor products of modules, Arch. Math. (Basel) 97 (2011), 503–512.
[32] W, Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 1977, 63–91.
[33] W, Barth, Kummer surfaces associated with the Horrocks–Mumford bundle, Journées de Géométrie Algébrique d'Angers, juillet 1979 (A. Beauville, ed.), Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, pp. 29–48.
[34] W, Barth, Irreducibility of the space of mathematical instanton bundles withrank 2 and c2 = 4, Math. Ann. 258 (1981), 81–106.
[35] W, Barth, K, Hulek, and R, Moore, Shioda's modular surface S(5) and theHorrocks–Mumford bundle, Vector bundles on algebraic varieties (Bombay 1984) (M. F. Atiyah et al., ed.), Oxford University Press, 1987, pp. 35–106.
[36] W, Barth and R, Moore, Geometry in the space of Horrocks–Mumford surfaces, Topology 28 (1989), 231–245.
[37] L. B, Beasley, Spaces of matrices of equal rank, Linear Algebra Appl. 38 (1981), 227–237.
[38] A. A, Beilinson, Coherent sheaves on Pn and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 214–216.
[39] D. J, Benson, Representations and cohomology I: Basic representation theory offinite groups and associative algebras, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, 1991, reprinted in paperback, 1998.
[40] D. J, Benson, Representations and cohomology II: Cohomology of groups andmodules, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, 1991, reprinted in paperback, 1998.
[41] D. J, Benson, Modules for elementary abelian p-groups, Proceedings of the ICM (Hyderabad 2010), vol. II, 2010, pp. 113–124.
[42] D. J, Benson, Modules of constant Jordan type with one non-projective block, Algebras and Representation Theory 13 (2010), 315–318.
[43] D. J, Benson, Modules of constant Jordan type with small non-projective part, Algebras and Representation Theory 16 (2013), 29–33.
[44] D. J, Benson, Modules of constant Jordan type and a conjecture of Rickard, J. Algebra 398 (2014), 343–349.
[45] D. J, Benson, Modules for elementary abelian groups and hypersurface singularities, Commutative Algebra and Noncommutative Algebraic Geometry II, MSRI Publications, vol. 68, Cambridge University Press, 2015, pp. 19–42.
[46] D. J, Benson and J. F, Carlson, Diagrammatic methods for modular representationsand cohomology, Commun. in Algebra 15 (1987), 53–121.
[47] D. J, Benson and J. F, Carlson, Projective resolutions and Poincaré dualitycomplexes, Trans. Amer. Math. Soc. 132 (1994), 447–488.
[48] D. J, Benson, J. F, Carlson, and J, Rickard, Complexity and varieties for infinitelygenerated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615.
[49] D. J, Benson, J. F, Carlson, and J, Rickard, Thick subcategories of the stablemodule category, Fundamenta Mathematicae 153 (1997), 59–80.
[50] D. J, Benson and E, Green, Nonprincipal blocks with one simple module, Quarterly Journal of Math (Oxford) 55 (2004), 1–11.
[51] D. J, Benson, S. B, Iyengar, and H, Krause, Stratifying modular representationsof finite groups, Ann. of Math. 174 (2011), 1643–1684.
[52] D. J, Benson and R, Kessar, Blocks inequivalent to their Frobenius twists, J. Algebra 315 (2007), 588–599.
[53] D. J, Benson and J, Pevtsova, A realization theorem for modules of constantJordan type and vector bundles, Trans. Amer. Math. Soc. 364 (2012), 6459–6478.
[54] D. J, Benson and F, Reid, Modules with small Loewy length, J. Algebra 414 2014, 288–299.
[55] I. N, Bernstein, I. M, Gelfand, and S. I, Gelfand, Algebraic vector bundles on Pn and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 1978, no.3, 66–67.
[56] A. I, Bondal and M. M, Kapranov, Homogeneous bundles, Helices and Vector Bundles, London Math. Soc. Lecture Note Series, vol. 148, Cambridge University Press, 1990, pp. 45–55.
[57] V. M, Bondarenko and Y. A, Drozd, Representation type of finite groups, (transl. from Russian) J. Soviet Math. 20 (1982), 2515–2528.
[58] V. M, Bondarenko and I. V, Lytvynchuk, The representation type of elementaryabelian p-groups with respect to the modules of constant Jordan type, Algebra and Discrete Mathematics 14 2012, no.1, 29–36.
[59] A, Boralevi, The Horrocks–Mumford bundle restricted to planes, Collect. Math. 58 (2007), 101–117.
[60] A, Boralevi, D, Faenzi, and E, Mezzetti, Linear spaces of matrices of constantrank and instanton bundles, Adv. in Math. 248 (2013), 895–920.
[61] C, Borcea, On desingularizated Horrocks–Mumford quintics, J. Reine & Angew. Math. 421 (1991), 23–41.
[62] A, Borel and J.-P., Serre, Le théorème de Riemann–Roch, Bull. Soc. Math. France 86 (1958), 97–136.
[63] W, Bosma, J, Cannon, and C, Playoust, The Magma algebra system, I. The userlanguage, J. Symbolic Comput. 24 (1997), 235–265.
[64] R, Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203–248.
[65] R. M, Bryant and M, Johnson, Adams operations on the Green ring of a cyclicgroup of prime-power order, J. Algebra 323 (2010), 2818–2833.
[66] R.-O, Buchweitz, Maximal Cohen–Macaulay modules and Tate cohomologyover Gorenstein rings, Unpublished preprint, 1986.
[67] R.-O, Buchweitz, G.-M., Greuel, and F.-O., Schreyer, Cohen–Macaulay moduleson hypersurface singularities II, Invent. Math. 88 (1987), 165–182.
[68] T, Bühler, Exact categories, Expo. Math. 28 (2010), 1–69.
[69] J, Burke and G, Stevenson, The derived category of a graded Gorenstein ring, Commutative Algebra and Noncommutative Algebraic Geometry II, MSRI Publications, vol. 68, Cambridge University Press, 2015, pp. 93–123.
[70] J, Burke and M. E, Walker, Matrix factorizations in higher codimension, Trans. Amer. Math. Soc. 367 2015, no.5, 3323–3370.
[71] J. F, Carlson, The modular representation ring of a cyclic 2-group, J. London Math. Soc. 11 (1975), 91–92.
[72] J. F, Carlson, Complexity and Krull dimension, Representations of Algebras, Puebla,Mexico, 1980, Lecture Notes in Mathematics, vol. 903, Springer-Verlag, Berlin/New York, 1981, pp. 62–67.
[73] J. F, Carlson, The complexity and varieties of modules, Integral representations and their applications, Oberwolfach, 1980, Lecture Notes in Mathematics, vol. 882, Springer-Verlag, Berlin/New York, 1981, pp. 415–422.
[74] J. F, Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 1983, 104–143.
[75] J. F, Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), 291–299.
[76] J. F, Carlson, The cohomology ring of a module, J. Pure & Applied Algebra 36 1985, 105–121.
[77] J. F, Carlson, Varieties for modules, The Arcata Conference on Representations of Finite Groups (Arcata, California, 1986), Proc. Symp. Pure Math., vol. 47, American Math. Society, 1987, pp. 37–44.
[78] J. F, Carlson, Varieties and modules of small dimension, Arch.Math. (Basel) 60 1993, 425–430.
[79] J. F, Carlson, Cohomology and induction from elementary abelian subgroups, Quarterly Journal of Math (Oxford) 51 (2000), 169–181.
[80] J. F, Carlson and E. M, Friedlander, Exact category of modules of constant Jordantype, Algebra, arithmetic and geometry: Manin Festschrift, Progr. in Math., vol. 269, Birkhäuser Verlag, Basel, 2009, pp. 259–281.
[81] J. F, Carlson, E. M, Friedlander, and J, Pevtsova, Modules of constant Jordantype, J. Reine & Angew. Math. 614 (2008), 191–234.
[82] J. F, Carlson, E. M, Friedlander, and A. A, Suslin, Modules for Z/p × Z/p, Comment. Math. Helvetici 86 (2011), 609–657.
[83] H, Cartan and S, Eilenberg, Homological algebra, Princeton Mathematical Series, no. 19, Princeton Univ. Press, 1956.
[84] P, Cascini, Weighted Tango bundles on Pn and their moduli spaces, Forum Math. 13 (2001), 251–260.
[85] A, Causa, R, Re, and T, Teodorescu, Some remarks on linear spaces of nilpotentmatrices, Le Matematiche 53(Suppl.) (1998), 23–32.
[86] C.-Y. J., Chan, A correspondence between Hilbert polynomials and Chernpolynomials over projective spaces, Illinois J. Math. 48 (2004), no.2, 451–462.
[87] X.-W., Chen, Unifying two results of Orlov on singularity categories, Abh. Math. Sem. Univ. Hamburg 80 (2010), no.2, 207–212.
[88] X.-W., Chen, The singularity category of an algebra with radical square zero, Doc. Math. 16 (2011), 921–936.
[89] L, Chouinard, Projectivity and relative projectivity over group rings, J. Pure & Applied Algebra 7 (1976), 278–302.
[90] W.-L., Chow, On equivalence classes of cycles in an algebraic variety, Ann. of Math. 64 (1956), 450–479.
[91] I, Coanda, On the Bernstein–Gel'fand–Gel'fand correspondence and a result of Eisenbud, Fløystad, and Schreyer, J. Math. Kyoto Univ. 43 (2003), 429–439.
[92] I, Coanda, Infinitely stably extendable vector bundles on projective space, Arch. Math. (Basel) 94 (2010), 539–545.
[93] I, Coanda and G, Trautmann, The splitting criterion of Kempf and the Babyloniantower theorem, Commun. in Algebra 34 (2006), 2485–2488.
[94] L, Costa and G, Ottaviani, Group actions on instanton bundles over P3, Math. Nachr. 246–247 (2002), 31–46.
[95] L, Costa and G, Ottaviani, Nondegenerate multidimensional matrices andinstanton bundles, Trans. Amer. Math. Soc. 355 (2003), 49–55.
[96] W.W, Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. 56 (1988), 451–483.
[97] E. C, Dade, Endo-permutation modules over p-groups, I, Ann. of Math. 107 1978, 459–494.
[98] E. C, Dade, Endo-permutation modules over p-groups, II, Ann. of Math. 108 1978, 317–346.
[99] W, Decker, Das Horrocks–Mumford-Bündel und das Modul-Schema für stabile 2-Vektorbündel über P4 mit c1 = -1, c2 = 4, Math. Zeit. 188 (1984), 101–110.
[100] W, Decker, Stable rank 2 vector bundles with Chern-classes c1 = -1, c2 = 4, Math. Ann. 275 (1986), 481–500.
[101] W, Decker, N, Manolache, and F.-O., Schreyer, Geometry of the Horrocks bundleon P5, Complex projective geometry (G. Elligsrud et al., ed.), London Math. Soc. Lecture Note Series, vol. 179, Cambridge University Press, 1992, pp. 128–148.
[102] W, Decker and F.-O, Schreyer, On the uniqueness of the Horrocks–Mumfordbundle, Math. Ann. 273 (1986), 415–443.
[103] W, Decker and F.-O, Schreyer, Pullbacks of the Horrocks–Mumford bundle, J. Reine & Angew. Math. 382 (1987), 215–220.
[104] I, Dolgachev and M, Kapranov, Arrangements of hyperplanes and vectorbundles on Pn, Duke Math. J. 71 (1993), 633–664.
[105] S. R, Doty, The submodule structure of certain Weyl modules for groups of typeAn, J. Algebra 95 (1985), 373–383.
[106] P, Dowbor and H, Meltzer, Classical vector bundles and representations ofquivers, Bol. Soc. Mat. Mexicana 11 2005, no.2, 205–220.
[107] Y. A, Drozd, Tame and wild matrix problems, Representation Theory II, Lecture Notes in Mathematics, vol. 832, Springer-Verlag, Berlin/New York, 1980, pp. 242–258.
[108] L, Ein, Stable vector bundles on projective spaces in Char p > 0, Math. Ann. 254 (1980), 53–72.
[109] L, Ein, Some stable vector bundles on P4 and P5, J. Reine & Angew. Math. 337 (1982), 142–153.
[110] D, Eisenbud, Homological algebra on a complete intersection, with an applicationto group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.
[111] D, Eisenbud, G, Fløystad, and F.-O. Schreyer, Sheaf cohomology and free resolutionsover exterior algebras, Trans. Amer. Math. Soc. 355 (2003), 4397–4426.
[112] D, Eisenbud and J, Harris, Vector spaces of matrices of low rank, Adv. in Math. 70 (1988), 135–155.
[113] P, Ellia, Sur les fibrés uniformes de rang (n+1) sur Pn, Mém. Soc. Math. France (N.S.), vol. 7, Gauthier–Villars, 1982.
[114] G, Ellingsrud and A, Strømme, Stable rank-2 vector bundles on P3 with c1 = 0 and c2 = 3, Math. Ann. 255 (1981), 123–135.
[115] L, Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 1961, 224–239.
[116] D, Faenzi, A geometric construction of Tango bundle on P5, Kodai Math. J. 27 (2004), 1–6.
[117] M. L, Fania and E, Mezzetti, Vector spaces of skew-symmetric matrices ofconstant rank, Linear Algebra Appl. 434 (2011), 2383–2403.
[118] R, Farnsteiner, Jordan types for indecomposable modules of finite groupschemes, J. Eur. Math. Soc. 16 (2014), 925–989.
[119] W, Feit, The representation theory of finite groups, North Holland, Amsterdam, 1982.
[120] G, Fløystad, Monads on projective spaces, Commun. in Algebra 28 2000, no.12, 5503–5516.
[121] G, Fløystad, Describing coherent sheaves on projective space, arXiv:math/0012263, 2001.
[122] R. M, Fossum, Invariants and Schur functors in characteristic p > 0, Young tableaux and Schur functors in algebra and geometry (Torun, 1980), Astérisque, vol. 87, Soc. Math. France, 1981, pp. 85–95.
[123] R. M, Fossum, Decompositions revisited, Séminaire d'Algèbre Paul Dubreil and Marie-Paule Malliavin, 34ème Année (Paris, 1981), Lecture Notes in Mathematics, vol. 924, Springer-Verlag, Berlin/New York, 1982, pp. 260–295.
[124] E. M, Friedlander and J, Pevtsova, Representation theoretic support spaces forfinite group schemes, Amer. J. Math. 127 (2005), 379–420, correction: AJM 128 2006, 1067–1068.
[125] E. M, Friedlander and J, Pevtsova, !-supports for modules for finite groupsschemes, Duke Math. J. 139 (2007), 317–368.
[126] E. M, Friedlander and J, Pevtsova, Generalized support varieties for finite groupschemes, Documenta Math. Extra Volume Suslin (2010), 197–222.
[127] E. M, Friedlander and J, Pevtsova, Constructions for infinitesimal groupschemes, Trans. Amer. Math. Soc. 363 (2011), 6007–6061.
[128] E. M, Friedlander, J, Pevtsova, and A, Suslin, Generic and maximal Jordantypes, Invent. Math. 168 (2007), 485–522.
[129] W, Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, Band 2, Springer-Verlag, Berlin/New York, 1984.
[130] M, Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices I, Amer. J. Math. 80 (1958), 614–622.
[131] M, Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices III, Ann. of Math. 70 (1959), 167–205.
[132] M, Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960), 21–31.
[133] M, Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. 73 (1961), 324–348.
[134] M, Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices IV, Ann. of Math. 75 (1962), 382–418.
[135] S. P, Glasby, C. E, Praeger, and B, Xia, Decomposing modular tensor products:'Jordan partitions', their parts and p-parts, Israel J. Math. 209 (2015), 215–233.
[136] E. S, Golod, The cohomology ring of a finite p-group, Dokl. Akad. Nauk. SSSR 125 (1959), 703–706, (Russian).
[137] A. L, Gorodentsev and A. N, Rudakov, Exceptional vector bundles on projectivespaces, Duke Math. J. 54 (1987), 115–130.
[138] J. A, Green, The modular representation algebra of a finite group, Illinois J. Math. 6 (1962), 607–619.
[139] J. P. C, Greenlees, Commutative algebra in group cohomology, J. Pure & Applied Algebra 98 (1995), 151–162.
[140] A, Grothendieck, Sur la classification des fibrés holomorphes sur la sphère deRiemann, Amer. J. Math. 79 (1957), 121–138.
[141] A, Grothendieck, Éléments de géométrie algébrique I, Publ.Math. IHES (1960), no.4, 1–228.
[142] A, Grothendieck, Éléments de géométrie algébrique II, Publ. Math. IHES (1961), no.8, 1–222.
[143] T. A, Hannula, T. G, Ralley, and I, Reiner, Modular representation algebras, Bull. Amer. Math. Soc. 73 (1967), 100–101.
[144] D, Happel, Triangulated categories in the representation theory of finite dimensionalalgebras, London Math. Soc. Lecture Note Series, vol. 119, Cambridge University Press, 1988.
[145] R, Hartshorne, Complete intersections and connectedness, Amer. J. Math. 84 1962, 497–508.
[146] R, Hartshorne, Residues and duality, Lecture Notes in Mathematics, vol. 20, Springer-Verlag, Berlin/New York, 1966.
[147] R, Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, Berlin/New York, 1977.
[148] R, Hartshorne, Stable vector bundles and instantons, Comm. Math. Phys. 59 1978, 1–15.
[149] R, Hartshorne, Stable vector bundles of rank 2 on P3, Math. Ann. 238 (1978), 229–280.
[150] R, Hartshorne, Algebraic vector bundles on projective spaces: a problem list, Topology 18 (1979), 117–128.
[151] J, Herzog and D, Popescu, Thom–Sebastiani problems for maximal Cohen–Macaulay modules, Math. Ann. 309 (1997), 677–700.
[152] J, Herzog, B, Ulrich, and J, Backelin, Linear maximal Cohen–Macaulay modulesover strict complete intersections, J. Pure & Applied Algebra 71 (1991), 187–202.
[153] W, Hesselink, Singularities in the nilpotent scheme of a classical group, Trans. Amer. Math. Soc. 222 (1976), 1–32.
[154] L, Hille, Homogeneous vector bundles and Koszul algebras, Math. Nachr. 191 1998, 189–195.
[155] F, Himstedt and P, Symonds, Exterior and symmetric powers of modules forcyclic 2-groups, J. Algebra 410 (2014), 393–420.
[156] F, Hirzebruch, Topological methods in algebraic geometry, Grundlehren der mathematischen Wissenschaften, vol. 131, Springer-Verlag, Berlin/New York, 1978.
[157] N, Hoffmann, Independent parameters for special instanton bundles on P2 n+1, Journal of Geometry and Physics 61 (2011), 2321–2330.
[158] M, Holloway and R, Kessar, Quantum complete rings and blocks with onesimple module, Quarterly Journal of Math (Oxford) 56 (2005), 209–221.
[159] G, Horrocks, A construction of locally free sheaves, Topology 7 (1968), 117– 120.
[160] G, Horrocks, Examples of rank three vector bundles on five dimensionalprojective space, J. London Math. Soc. 18 (1978), 15–27.
[161] G, Horrocks and D, Mumford, A rank 2 vector bundle on P4 with 15, 000 symmetries, Topology 12 (1973), 63–81.
[162] I, Hughes and G, Kemper, Symmetric powers of modular representations,Hilbert series and degree bounds, Commun. in Algebra 28 (2000), 2059–2088.
[163] K, Hulek, Geometry of the Horrocks–Mumford bundle, Algebraic geometry (Bowdoin 1985) (S. J. Bloch, ed.), Proc. Symp. Pure Math., vol. 46, Part II, American Math. Society, 1987, pp. 69–85.
[164] K, Hulek, The Horrocks–Mumford bundle, Vector bundles in algebraic geometry (Durham 1993), London Math. Soc. Lecture Note Series, vol. 208, Cambridge University Press, 1995, pp. 139–177.
[165] K, Hulek and A. Van de Ven, The Horrocks–Mumford bundle and the Ferrandconstruction, Manuscripta Math. 50 (1985), 313–335.
[166] B, Ilic and J. M, Landsberg, On symmetric degeneracy loci, spaces of symmetricmatrices of constant rank and dual varieties, Math. Ann. 314 (1999), 159–174.
[167] S., Iyengar (ed.), Twenty-four hours of local cohomology, Graduate Studies in Mathematics, American Math. Society, 2008.
[168] S, Iyengar and R, Takahashi, Annihilation of cohomology and strong generationof module categories, Int. Math. Res. Not. 2016 2016, no.2, 499–535.
[169] S, Jackowski and J. E, McClure, Homotopy decomposition of classifying spacesvia elementary abelian subgroups, Topology 31 (1992), 113–132.
[170] K, Jaczewski, M, Szurek, and J, Wisniewski, Geometry of the Tango bundle, Proceedings of the conference on algebraic geometry (Berlin, 1985) (Leipzig), Teubner-Texte Math., vol. 92, Teubner, 1986, pp. 177–185.
[171] G. D, James and A, Kerber, The representation theory of the symmetric group, Cambridge University Press, 1981.
[172] H, Kaji, Example of σ-transition matrices defining the Horrocks–Mumfordbundle, Tokyo J. Math. 12 (1989), 21–32.
[173] S. O, Kaptanoglu, Structure and detection theorems for k[C2 × C2]-modules, Rend. Sem. Mat. Univ. Padova 123 (2010), 169–189.
[174] S. O, Kaptanoglu, p-power points and modules of constant p-power Jordantype, Commun. in Algebra 39 (2011), 3781–3800.
[175] B, Keller, Chain complexes and stable categories,Manuscripta Math. 67 (1990), 379–417.
[176] B, Keller and D, Vossieck, Sous les catégories dérivées, Comptes Rendus Acad. Sci. Paris, Série I 305 (1987), 225–228.
[177] G, Kempf, A criterion for the splitting of a vector bundle, Forum Math. 2 (1990), 477–480.
[178] H, Knörrer, Cohen–Macaulay modules on hypersurface singularities I, Invent. Math. 80 (1987), 153–164.
[179] F. M, Kouwenhoven, The λ-structure of the Green ring of cyclic p-groups, The Arcata Conference on the Representation Theory of Finite Groups (P. Fong, ed.), Proc. Symp. Pure Math., vol. 47, 1987, pp. 451–466.
[180] D, Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431– 449.
[181] O, Kroll, Complexity and elementary abelian p-groups, J. Algebra 88 (1984), 155–172.
[182] N. M, Kumar, Construction of rank two victor bundles on P4 in positivecharacteristic, Invent. Math. 130 (1997), 277–286.
[183] N. M, Kumar, C, Peterson, and A. P., Rao, Construction of low rank vectorbundles on P4 and P5, J. Alg. Geometry 11 (2002), 203–217.
[184] N. M, Kumar, C, Peterson, and A. P., Rao, Degenerating families of rank twobundles, Proc. Amer. Math. Soc. 131 (2003), 3681–3688.
[185] N. M, Kumar, C, Peterson, and A. P., Rao, Monads on projective spaces, Manuscripta Math. 112 (2003), 183–189.
[186] E, Lee, A modular quintic Calabi–Yau threefold of level 55, Canadian J. Math. 63 (2011), 616–633.
[187] K. J, Lim, The varieties for some Specht modules, J. Algebra 321 (2009), 2287– 2301.
[188] M, Linckelmann, Quillen stratification for block varieties, J. Pure & Applied Algebra 172 (2002), 257–270.
[189] I. G, Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1995, 2ed.
[190] S., Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, Berlin/New York, 1971.
[191] L, Manivel and E, Mezzetti, On linear spaces of skew-symmetric matrices ofconstant rank, Manuscripta Math. 117 (2005), 319–331.
[192] J. D, McFall, How to compute the elementary divisors of the tensor product oftwo matrices, Linear & Multilinear Algebra 7 1979, no.3, 193–201.
[193] J. D, McFall, On elementary divisors of the tensor product of two matrices, Linear Algebra Appl. 33 (1980), 67–86.
[194] R. M, Miro-Roig and J. A, Orus-Lacort, On the smoothness of the moduli spaceof mathematical instanton bundles, Compositio Math. 105 (1997), 109–119.
[195] T, Muir and W. H, Metzler, A treatise on the theory of determinants, 2ed, Dover, 1960.
[196] S.-Y., Nam and Y.-T., Oh, λ-ring structure of the Green ring of a cyclic p-group, J. Algebra 338 (2011), 92–113.
[197] S.-Y., Nam and Y.-T., Oh, Symmetric powers of the p + 1-dimensional indecomposablemodule of a cyclic p-group and the λ-structure of its Green ring, J. Algebra 368 (2012), 75–91.
[198] A, Neeman, Algebraic and analytic geometry, Cambridge University Press, 2007.
[199] C. W, Norman, On the Jordan form of the tensor product over fields of primecharacteristic, Linear & Multilinear Algebra 38 (1995), 351–371.
[200] C. W, Norman, On Jordan bases for the tensor product and Kronecker sumand their elementary divisors over fields of prime characteristic, Linear & Multilinear Algebra 56 (2008), 415–451.
[201] J, O'Halloran, A simple proof of the Gerstenhaber–Hesselink theorem fornilpotent matrices, Commun. in Algebra 15 (1987), 2017–2023.
[202] C, Okonek, M, Schneider, and H, Spindler, Vector bundles on complex projectivespaces, Birkhäuser Verlag, Basel, 1980; 3ed. 2011.
[203] C, Okonek and H, Spindler, Mathematical instanton bundles on P2n+1, J. Reine & Angew. Math. 364 (1986), 35–50.
[204] D. O, Orlov, Triangulated categories of singularities and equivalences betweenLandau–Ginzburg models, Sbornik Math 197 (2006), 1827–1840.
[205] G, Ottaviani and E, Rubei, Quivers and the cohomology of homogeneous vectorbundles, Duke Math. J. 132 (2006), 459–508.
[206] D. G, Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549–572.
[207] D. G, Quillen, The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573–602.
[208] D. G, Quillen, Higher algebraic K-theory I, Algebraic K-theory I: Higher K-theories (H. Bass, ed.), Lecture Notes in Mathematics, vol. 341, Springer- Verlag, Berlin/New York, 1973, pp. 85–147.
[209] D. G, Quillen, Projective modules over polynomial rings, Invent. Math. 36 1976, 167–171.
[210] D. G, Quillen and B. B, Venkov, Cohomology of finite groups and elementaryabelian subgroups, Topology 11 (1972), 317–318.
[211] T, Ralley, Decomposition of products of modular representations, J. London Math. Soc. 44 (1969), 480–484.
[212] A. P, Rao, A note on cohomology modules of rank two bundles, J. Algebra 86 1984, 23–34.
[213] A. P, Rao, Mathematical instantons in characteristic two, Compositio Math. 119 1999, 169–184.
[214] F, Reid, Varieties for modules of small dimension, Ph.D. thesis, University of Aberdeen, 2014.
[215] J.-C, Renaud, The characters and structure of a class of modular representationalgebras of cyclic p-groups, J. Austral. Math. Soc. 26 (1978), 410–418.
[216] J.-C, Renaud, The decomposition of products in the modular representation ringof a cyclic group of prime power order, J. Algebra 58 (1979), 1–11.
[217] J.-C, Renaud, On modular representation algebras and a class of matrixalgebras, J. Austral. Math. Soc. 33 (1982), 351–355.
[218] J.-C, Renaud, Recurrence relations in a modular representation algebra, Bull. Austral. Math. Soc. 26 (1982), 215–219.
[219] J, Rickard, Derived categories and stable equivalence, J. Pure & Applied Algebra 61 (1989), 303–317.
[220] C. M, Ringel, The representation type of local algebras, Representations of Algebras, Lecture Notes in Mathematics, vol. 488, Springer-Verlag, Berlin/New York, 1974.
[221] A. N, Rudakov, The Markov numbers and exceptional bundles on P2, Izvestiya 32 (1989), 99–112.
[222] E, Sato, On the decomposability of infinitely extendable vector bundles onprojective spaces and Grassmann varieties, J. Math. Kyoto Univ. 17 (1977), 127–150.
[223] H, Schoutens, Projective dimension and the singular locus, Commun. in Algebra 31 2003, no.1, 217–239.
[224] J.-P., Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 1955, no.2, 197–278.
[225] J.-P., Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955–1956), 1–42.
[226] J.-P., Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot (1958), Fasc. 2, Exp. 23.
[227] I. R, Shafarevich, Basic algebraic geometry, Springer-Verlag, Berlin/New York, 1977.
[228] B, Srinivasan, The modular representation ring of a cyclic p-group, Proc. London Math. Soc. 14 (1964), 677–688.
[229] Robert, Steinberg, Générateurs, relations et revêtements de groupes algébriques, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain, 1962, pp. 113–127.
[230] A. A, Suslin, Projective modules over polynomial rings are free (Russian), Dokl. Akad. Nauk SSSR 229 (1976), 1063–1066.
[231] J, Sylvester, On the dimension of spaces of linear transformations satisfyingrank conditions, Linear Algebra Appl. 78 (1986), 1–10.
[232] P, Symonds, The complexity of a module and elementary abelian subgroups: ageometric approach, Proc. Amer. Math. Soc. 113 (1991), 27–29.
[233] P, Symonds, Cyclic group actions on polynomial rings, Bull. London Math. Soc. 39 (2007), 181–188.
[234] H, Tango, An example of indecomposable vector bundle of rank n - 1 on Pn, J. Math. Kyoto Univ. 16 (1976), 137–141.
[235] H, Tango, On morphisms from projective space Pn to the Grassmann variety Gr(n, d), J. Math. Kyoto Univ. 16 (1976), 201–207.
[236] H, Tango, On vector bundles on Pn which have σ-transition matrices, Tokyo J. Math. 16 (1993), 1–29.
[237] J, Tate, Homology of Noetherian rings and local rings, Illinois J.Math. 1 (1957), 14–27.
[238] A. N, Tyurin, Finite dimensional vector bundles over infinite varieties, Math. USSR Izv. 10 (1976), 1187–1204.
[239] A. N, Tyurin, Collected works I: Vector bundles, Universitätsverlag Göttingen, 2008, F. Bogomolov et al., eds.
[240] V. K, Vedernikov, A family of rank-2 mathematical instanton bundles on P3, Publ. RIMS, Kyoto Univ. 33 (1997), 573–598.
[241] A. Van de, Ven, On uniform vector bundles, Math. Ann. 195 (1972), 245–248.
[242] B. B, Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSSR 127 (1959), 943–944.
[243] P. J, Webb, Bounding the ranks of ZG-modules by their restrictions to elementaryabelian subgroups, J. Pure & Applied Algebra 23 (1982), 311–318.
[244] D. L, Wehlau, Invariants for the modular cyclic group of prime order viaclassical invariant theory, J. Eur. Math. Soc. 15 (2013), 775–803.
[245] R, Westwick, Spaces of linear transformations of equal rank, Linear Algebra Appl. 5 (1972), 49–64.
[246] R, Westwick, Spaces of matrices of fixed rank, Linear & Multilinear Algebra 20 1987, 171–174.
[247] R, Westwick, Examples of constant rank spaces, Linear & Multilinear Algebra 28 (1990), 155–174.
[248] R, Westwick, Spaces of matrices of fixed rank II, Linear Algebra Appl. 235 1996, 163–169.
[249] K, Wolffhardt, The Hochschild homology of complete intersections, Trans. Amer. Math. Soc. 171 (1972), 51–66.
[250] J, Worch, Categories of modules for elementary abelian p-groups and generalizedBeilinson algebras, J. London Math. Soc. 88 (2013), 649–668.
[251] Y, Yoshino, Cohen–Macaulay modules over Cohen–Macaulay rings, London Math. Soc. Lecture Note Series, vol. 146, Cambridge University Press, 1990.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.