nLab dualizable module

Idea

Definition

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

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

Furthermore, the abstract evaluation map

$$Hom_R(M,R)\otimes_R M\to R$$

must coincide with the map induced by the bilinear map

$$Hom_R(M,R)\times_R M\to R$$

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

Characterization

Theorem

An $R$-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 $R^n$ for some $n\ge0$, so to show that finitely generated projective modules are dualizable?, it suffices to observe that $R$ is dualizable as an $R$-module.

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

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

and the evaluation map

$$ev: Hom_R(M,R)\otimes M\to R$$

satisfy the triangle identities:

$$(id_M \otimes ev)\circ (coev\otimes id_M) = id_M,$$

$$(ev \otimes id_{Hom(M,R)})\circ (id_{Hom(M,R)}\otimes coev) = id_{Hom(M,R)}.$$

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

$$\sum_{i\in I} m_i\otimes f_i.$$

The triangle identities now read

$$\sum_{i\in I} m_i f_i(p) = p,\qquad p\in M$$

$$\sum_{i\in I} r(m_i) f_i = r, \qquad r\in Hom_R(M,R).$$

The first identity implies that $m_i$ ($i\in I$) generate $M$ as an $R$-module, i.e., $M$ is finitely generated.

Consider the map $a: R^I\to M$ that sends $(r_i)_{i\in I}$ to $\sum_{i\in I} m_i r_i$. Consider also the map $b: M\to R^I$ that sends $p\in M$ to $(f_i(p))_{i\in I}\in R^I$. The first triangle identity now reads $b a = id_M$. Thus, $M$ is a retract of $R^I$, i.e., $M$ 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 $R$ 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 $R$.

Theorem

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

Theorem

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

Related concepts

• dualizable object

• perfect chain complex

• perfect module

• Serre–Swan theorem

• smooth Serre–Swan theorem

• duality between geometry and algebra