# A module associated to an endomorphism of a vector bundle

Let $$E$$ be a vector bundle over a compact connected Hausdorff space $$X$$. To an endomorphism $$\alpha \in End(E)$$, we associate a $$C(X)-$$ module $$\Gamma(E,\alpha)$$ consisting of all $$\beta\in End(E)$$ such that there exists a $$\gamma \in End(E)$$ with $$\beta=\alpha \circ \gamma$$.

$$\require{AMScd} \def\diaguparrow#1{\smash{\raise.6em\rlap{\scriptstyle #1} \lower.3em{\mathord{\diagup}}\raise.52em{\!\mathord{\nearrow}}}} \begin{CD} && E\\ & \diaguparrow{\exists \gamma} @VV \alpha V \\ E @>> \beta> E \end{CD}$$

Is $$\Gamma(E,\alpha)$$ necessarily a projective $$C(X)-$$ module? Is it necessarilly a finitly generated module? Is it true to say that $$\Gamma(E, \alpha)$$ is a finitely generated projective module if and only if $$\alpha$$ is a strict morphism, namely $$\dim( \ker \alpha_x)$$ is independent of $$x\in X$$?

Remark 1: If $$\dim( \ker \alpha_x)$$ is independent of $$x\in X$$, then $$\Gamma(E,\alpha)$$ is a finitely generated projective module. Because it is isomorphic to $$hom(E,F)=\Gamma (Hom(E,F))$$, the module of sections of the Home bundle $$Home (E,F)$$ where $$F=\text{Range}\; \alpha$$.

Remark 2: in this question we actually work with a pair $$(E, \alpha)$$ in the category of vector bundles and considered the module of "Liftable" morphisms $$\beta$$. One cane repeat the same consideration but on other categories. For examples, in the category of Banach space, one obtain a Banach algebra $$\Gamma(E, \alpha)$$ associated to a pair $$(E, \alpha)$$ where $$\alpha$$ is a bounded operator on Banach space $$E$$. So Is there a categorical terminology for such construction?

It seems that your $$C(X)$$-module $$\Gamma(E,\alpha)$$ is just the image of $$\alpha_*\colon \hom(E,E)\rightarrow \hom(E,E)$$. The module $$\Gamma(E,\alpha)$$ is a direct summand of $$\Gamma(E',\alpha')$$ for $$\alpha'\colon E'\rightarrow E'$$ an extension of $$\alpha$$ to a trivialization $$E\subset E'$$ of $$E$$. Therefore, we can assume without loss of generality that $$E$$ is a trivial bundle. Then $$\Gamma(E,\alpha)$$ is a direct sum of $$n=\dim E$$ copies of the $$n\times n$$ matrix in $$C(X)$$ induced by $$\alpha$$, which can be anyone. Therefore the cokernel of $$\alpha_*$$ is the direct sum $$n$$ copies of any given finitely presented $$C(X)$$-module. This proves that your first question is equivalent to whether any finitely presented $$C(X)$$-module has projective dimension $$\leq 1$$. This would mean that the weak global dimension of $$C(X)$$ is $$\leq 1$$. I don't know much about these things and haven't been able to find much in internet about this topic, but you may open here a question on the weak global dimension of function rings. The answer to the second question is positive, as we have just seen.