Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Gorenstein derived functors


Author: Henrik Holm
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1913-1923
MSC (2000): Primary 13D02, 13D05, 13D07, 13H10, 16E05, 16E10, 16E30
DOI: https://doi.org/10.1090/S0002-9939-04-07317-4
Published electronically: February 13, 2004
MathSciNet review: 2053961
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Over any associative ring $R$ it is standard to derive $\mathrm{Hom}_R(-,-)$using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains $\mathrm{Ext}_R^n(-,-)$ in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product $-\otimes_R-$ using Gorenstein flat modules.


References [Enhancements On Off] (What's this?)

  • 1. M. Auslander, Anneaux de Gorenstein et torsion en algèbre commutative, Séminaire d'algèbre commutative 1966/67, notes by M. Mangeney, C. Peskine and L. Szpiro, École Normale Supérieure de Jeunes Filles, Paris, Secrétariat mathématique, 1967. MR 37:1435
  • 2. M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. 94, American Mathematical Society, Providence, RI, 1969. MR 42:4580
  • 3. L. L. Avramov and A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. 85, Part 2 (2002), $393-440$. MR 2003g:16009
  • 4. L. W. Christensen, Gorenstein dimensions, Lecture Notes in Math. 1747, Springer-Verlag, Berlin, 2000. MR 2002e:13032
  • 5. E. E. Enochs and O. M. G. Jenda, Balanced functors applied to modules, J. Algebra 92 (1985), $303-310$. MR 86h:16025
  • 6. E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), $611-633$. MR 97c:16011
  • 7. E. E. Enochs and O. M. G. Jenda, Gorenstein balance of hom and tensor, Tsukuba J. Math. 19, No. 1 (1995), $1-13$. MR 97a:16019
  • 8. E. E. Enochs and O. M. G. Jenda, Gorenstein injective and flat dimensions, Math. Japonica 44, No. 2 (1996), $261-268$. MR 97k:13019
  • 9. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Expositions in Math. 30, Walter de Gruyter, Berlin, 2000. MR 2001h:16013
  • 10. L. Gruson and M. Raynaud, Critères de platitude et de projectivité. Techniques de ``platification'' d'un module, Invent. Math. 13 (1971), $1-89$. MR 46:7219
  • 11. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York-Heidelberg, 1977. MR 57:3116
  • 12. H. Holm, Gorenstein homological dimensions, J. Pure and Appl. Algebra (to appear).
  • 13. C. U. Jensen, On the vanishing of $\varprojlim^{(i)}$, J. Algebra 15 (1970), $151-166$. MR 41:5460
  • 14. C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994. MR 95f:18001
  • 15. J. Xu, Flat covers of modules, Lecture Notes in Math. 1634, Springer-Verlag, Berlin, 1996. MR 98b:16003

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13D02, 13D05, 13D07, 13H10, 16E05, 16E10, 16E30

Retrieve articles in all journals with MSC (2000): 13D02, 13D05, 13D07, 13H10, 16E05, 16E10, 16E30


Additional Information

Henrik Holm
Affiliation: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, 2100 K\obenhavnØ, DK–Danmark
Email: holm@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9939-04-07317-4
Keywords: Gorenstein dimensions, homological dimensions, derived functors, Tor-modules, Ext-modules
Received by editor(s): May 14, 2002
Received by editor(s) in revised form: April 16, 2003
Published electronically: February 13, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society