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Book Information:
Author:
Sarah Witherspoon
Title:
Hochschild cohomology for algebras
Additional book information:
Graduate Studies in Mathematics, Vol. 204,
American Mathematical Society,
Providence, RI,
2019,
xi+250 pp.,
ISBN 978-1-4704-4931-5
Michael J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), no. 1, 69–89. MR 1432347, DOI 10.1006/jabr.1996.6813
Petter Andreas Bergh and Steffen Oppermann, Cohomology of twisted tensor products, J. Algebra 320 (2008), no. 8, 3327–3338. MR 2450729, DOI 10.1016/j.jalgebra.2008.08.005
Alexander Braverman and Dennis Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), no. 2, 315–328. MR 1383469, DOI 10.1006/jabr.1996.0122
B. Briggs and V. Gélinas, The $A_{\infty }$ centre of the Yoneda algebra and the characteristic action of Hochschild cohomology on the derived category, (2017), arXiv:1702.00721.
Ragnar-Olaf Buchweitz, Edward L. Green, Dag Madsen, and Øyvind Solberg, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005), no. 5-6, 805–816. MR 2189240, DOI 10.4310/MRL.2005.v12.n6.a2
Joachim Cuntz and Daniel Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 251–289. MR 1303029, DOI 10.1090/S0894-0347-1995-1303029-0
Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras, $K$-Theory 33 (2004), no. 1, 67–87. MR 2199789, DOI 10.1007/s10977-004-0838-7
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
Victor Ginzburg and Shrawan Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179–198. MR 1201697, DOI 10.1215/S0012-7094-93-06909-8
Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126. MR 1035222, DOI 10.1007/BFb0084073
G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58–67. MR 11076, DOI 10.2307/1969145
Bernhard Keller, $A$-infinity algebras in representation theory, Representations of algebra. Vol. I, II, Beijing Norm. Univ. Press, Beijing, 2002, pp. 74–86. MR 2067371
Stefan Schwede, An exact sequence interpretation of the Lie bracket in Hochschild cohomology, J. Reine Angew. Math. 498 (1998), 153–172. MR 1629858, DOI 10.1515/crll.1998.048
Stephen F. Siegel and Sarah J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. (3) 79 (1999), no. 1, 131–157. MR 1687539, DOI 10.1112/S0024611599011958
Nicole Snashall and Øyvind Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732. MR 2044054, DOI 10.1112/S002461150301459X
Øyvind Solberg, Support varieties for modules and complexes, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 239–270. MR 2258047, DOI 10.1090/conm/406/07659
Michel van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348. MR 1443171, DOI 10.1090/S0002-9939-98-04210-5
Yury Volkov, Gerstenhaber bracket on the Hochschild cohomology via an arbitrary resolution, Proc. Edinb. Math. Soc. (2) 62 (2019), no. 3, 817–836. MR 3974969, DOI 10.1017/s0013091518000901
References
- Michael J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), no. 1, 69–89. MR 1432347, DOI 10.1006/jabr.1996.6813
- Petter Andreas Bergh and Steffen Oppermann, Cohomology of twisted tensor products, J. Algebra 320 (2008), no. 8, 3327–3338. MR 2450729, DOI 10.1016/j.jalgebra.2008.08.005
- Alexander Braverman and Dennis Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), no. 2, 315–328. MR 1383469, DOI 10.1006/jabr.1996.0122
- B. Briggs and V. Gélinas, The $A_{\infty }$ centre of the Yoneda algebra and the characteristic action of Hochschild cohomology on the derived category, (2017), arXiv:1702.00721.
- Ragnar-Olaf Buchweitz, Edward L. Green, Dag Madsen, and Øyvind Solberg, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005), no. 5-6, 805–816. MR 2189240, DOI 10.4310/MRL.2005.v12.n6.a2
- Joachim Cuntz and Daniel Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 251–289. MR 1303029, DOI 10.2307/2152819
- Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras, $K$-Theory 33 (2004), no. 1, 67–87. MR 2199789, DOI 10.1007/s10977-004-0838-7
- Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
- Victor Ginzburg and Shrawan Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179–198. MR 1201697, DOI 10.1215/S0012-7094-93-06909-8
- Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126. MR 1035222, DOI 10.1007/BFb0084073
- G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58–67. MR 11076, DOI 10.2307/1969145
- Bernhard Keller, $A$-infinity algebras in representation theory, Representations of algebra. Vol. I, II, Beijing Norm. Univ. Press, Beijing, 2002, pp. 74–86. MR 2067371
- Stefan Schwede, An exact sequence interpretation of the Lie bracket in Hochschild cohomology, J. Reine Angew. Math. 498 (1998), 153–172. MR 1629858, DOI 10.1515/crll.1998.048
- Stephen F. Siegel and Sarah J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. (3) 79 (1999), no. 1, 131–157. MR 1687539, DOI 10.1112/S0024611599011958
- Nicole Snashall and Øyvind Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732. MR 2044054, DOI 10.1112/S002461150301459X
- Øyvind Solberg, Support varieties for modules and complexes, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 239–270. MR 2258047, DOI 10.1090/conm/406/07659
- Michel van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348. MR 1443171, DOI 10.1090/S0002-9939-98-04210-5
- Yury Volkov, Gerstenhaber bracket on the Hochschild cohomology via an arbitrary resolution, Proc. Edinb. Math. Soc. (2) 62 (2019), no. 3, 817–836. MR 3974969, DOI 10.1017/s0013091518000901
Review Information:
Reviewer:
Karin Erdmann
Affiliation:
Mathematical Institute, University of Oxford, Oxford, United Kingdom
Email:
erdmann@maths.ox.ac.uk
Journal:
Bull. Amer. Math. Soc.
58 (2021), 143-150
DOI:
https://doi.org/10.1090/bull/1712
Published electronically:
November 4, 2020
Review copyright:
© Copyright 2020
American Mathematical Society
