dual basis


Linear algebra

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Basic facts




The concept of a dual basis is a way of characterizing projective modules, alternative to their characterization as direct summands of free modules. The terminology derives from a similarity with a situation involving dual vector spaces, see below.


Let RR be a ring or, more generally, an associative algebra over a unital commutative ring kk.

Assuming the axiom of choice, one of the standard characterizations of the projective modules NN (say left) over RR is that there is an epimorphism FNF\to N from a free module over RR which is split (“NN is a direct summand of a free module”, this prop.).

Equivalently, an RR-module MM is projective iff it has a dual basis {(x i,x i *)M×Hom k(M,R)} iI\{(x_i,x_i^*)\in M\times Hom_k(M,R)\}_{i\in I} in the following sense. There is a free RR-module F= iIRF = \oplus_{i\in I} R, an epimorphism of RR-modules FMF\to M iR ir i ir ix i\oplus_i R \ni \sum_i r_i\mapsto \sum_i r_i x_i which is split, i.e. has a right inverse, where this inverse, by the universal property of the direct sum, must be of the form x ix i *(x)x i iRx\mapsto \sum_i x_i^*(x)x_i\in \oplus_i R where x i *Hom k(M,R)x_i^*\in Hom_k(M,R) and (x)(\forall x) x i *(x)0x_i^*(x)\neq 0 for only finitely many iIi\in I. In particular, x i *(x j)=δ i jx_i^*(x_j) = \delta_i^j (Kronecker delta).

Relation to dual vector spaces

This terminology is related to but a bit different than in the case of kk-vector spaces (cf. at dual vector space). If VV has a vector space basis (x j) jI(x_j)_{j\in I} is a linear basis of VV then x i *x_i^* defined by x i *(x j)=δ i jx_i^*(x_j) = \delta_i^j is not necessarily a basis of V *=Hom k(V,k)V^* = Hom_k(V,k); it is if VV is finite dimensional. In the case of projective module MM, x ix_i do not form a basis in a free sense, but only a set of generators and with the property that there exist another set x i *x_i^* in M *M^* such that they together form a “dual basis”. (Still, one sometimes says that x i *x_i^* form a basis dual to x ix_i.)

category: algebra

Revised on May 21, 2017 19:35:17 by Anonymous (