A module associated to an endomorphism of a vector bundle

Question

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?

Answer 1

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.