| 释义 |
- Dual and transpose
- References
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin. Every Artin algebra is an Artin ring. Dual and transposeThere are several different dualities taking finitely generated modules over Λ to modules over the opposite algebra Λop. - If M is a left Λ module then the right Λ-module M is defined to be HomΛ(M,Λ).
- The dual D(M) of a left Λ-module M is the right Λ-module D(M) = HomR(M,J), where J is the dualizing module of R, equal to the sum of the injective envelopes of the non-isomorphic simple R-modules or equivalently the injective envelope of R/rad R. The dual of a left module over Λ does not depend on the choice of R (up to isomorphism).
- The transpose Tr(M) of a left Λ-module M is a right Λ-module defined to be the cokernel of the map Q → P, where P → Q → M → 0 is a minimal projective presentation of M.
References- {{Citation | last1=Auslander | first1=Maurice | last2=Reiten | first2=Idun | last3=Smalø | first3=Sverre O. | title=Representation theory of Artin algebras | origyear=1995 | url=https://books.google.com/books?isbn=0521599237 | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | volume=36 | year=1997 | isbn=978-0-521-59923-8 | mr=1314422 | zbl=0834.16001 }}
1 : Ring theory |