nLab
dualizable module

Idea

Definition

A module MM over a commutative ring RR is dualizable if it is a dualizable object in the symmetric monoidal category of RR-modules equipped with the tensor product over RR.

Since this symmetric monoidal category is a closed monoidal category, the dual object to MM is necessarily Hom R(M,R)Hom_R(M,R).

Furthermore, the abstract evaluation map

Hom R(M,R) RMRHom_R(M,R)\otimes_R M\to R

must coincide with the map induced by the bilinear map

Hom R(M,R)× RMRHom_R(M,R)\times_R M\to R

that sends (f,m)(f,m) to f(m)f(m).

Characterization

Theorem

An RR-module is dualizable if and only if it is a finitely generated projective module.

Proof

First, dualizable objects are closed under retracts and finite direct sums. Any finitely generated projective module is a retract of R nR^n for some n0n\ge0, so to show that finitely generated projective modules are dualizable?, it suffices to observe that RR is dualizable as an RR-module.

Conversely, we show that dualizable objects are finitely generated projective modules. Unfolding the definition of a dualizable object, an RR-module MM is dualizable if the coevaluation map

coev:RMHom R(M,R)coev: R \to M\otimes Hom_R(M,R)

and the evaluation map

ev:Hom R(M,R)MRev: Hom_R(M,R)\otimes M\to R

satisfy the triangle identities:

(id Mev)(coevid M)=id M,(id_M \otimes ev)\circ (coev\otimes id_M) = id_M,
(evid Hom(M,R))(id Hom(M,R)coev)=id Hom(M,R).(ev \otimes id_{Hom(M,R)})\circ (id_{Hom(M,R)}\otimes coev) = id_{Hom(M,R)}.

The coevaluation map sends 1R1\in R to a finite sum

iIm if i.\sum_{i\in I} m_i\otimes f_i.

The triangle identities now read

iIm if i(p)=p,pM\sum_{i\in I} m_i f_i(p) = p,\qquad p\in M
iIr(m i)f i=r,rHom R(M,R).\sum_{i\in I} r(m_i) f_i = r, \qquad r\in Hom_R(M,R).

The first identity implies that m im_i (iIi\in I) generate MM as an RR-module, i.e., MM is finitely generated.

Consider the map a:R IMa: R^I\to M that sends (r i) iI(r_i)_{i\in I} to iIm ir i\sum_{i\in I} m_i r_i. Consider also the map b:MR Ib: M\to R^I that sends pMp\in M to (f i(p)) iIR I(f_i(p))_{i\in I}\in R^I. The first triangle identity now reads ba=id Mb a = id_M. Thus, MM is a retract of R IR^I, i.e., MM is a projective module.

Geometric interpretation

See also Serre–Swan theorem and smooth Serre–Swan theorem.

Theorem

(Serre, 1955.) The category of dualizable modules over a commutative ring RR is equivalent to the category of algebraic vector bundles (defined as locally free sheaves? over the structure sheaf of rings?) over the Zariski spectrum of RR.

Theorem

(Swan, 1962.) Given a compact Hausdorff space XX, the category of dualizable modules over the real algebra of continuous maps XRX\to\mathbf{R} is equivalent to the category of finite-dimensional continuous vector bundles over XX.

Theorem

(See, e.g., Nestruev 2003, 11.33.) Given a smooth manifold XX, the category of dualizable modules over the real algebra of smooth maps XRX\to\mathbf{R} is equivalent to the category of finite-dimensional smooth vector bundles over XX.

Last revised on April 14, 2021 at 17:04:14. See the history of this page for a list of all contributions to it.