Article

Algebras and Representation Theory

, Volume 12, Issue 1, pp 19-52

Rigid Dualizing Complexes Over Commutative Rings

  • Amnon YekutieliAffiliated withDepartment of Mathematics, Ben Gurion University Email author 
  • , James J. ZhangAffiliated withDepartment of Mathematics, University of Washington

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Abstract

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes.

Keywords

Commutative rings Derived categories Dualizing complexes Rigid complexes

Mathematics Subject Classifications (2000)

Primary: 14F05 Secondary: 14B25 14F10 13D07 18G10 16E45