NOVELTY - Hollow fiber polymer membrane comprises polyimide polymer membrane material made from the monomers (A1+B1+C1), where the ratio of B1 to C1 is 1:4-4:1 and the hollow fiber polymer membrane material further comprises at least some covalent ester crosslinks.    USE - The hollow fiber polymer membrane is useful for a gas/separation, a gas/liquid separation, a liquid/liquid separation or a liquid/solid separation, where the carbon dioxide is separated from methane, and carbon dioxide is removed from natural gas comprising carbon dioxide, methane and other gases (all claimed).    ADVANTAGE - The hollow fiber polymer membrane has improved selectivity and permeability properties.    DETAILED DESCRIPTION - Hollow fiber polymer membrane comprises polyimide polymer membrane material made from the monomers of formula (A1+B1+C1), where the ratio of B1 to C1 is 1:4-4:1 and the hollow fiber polymer membrane material further comprises at least some covalent ester crosslinks.    A1 = dianhydride group of formula (I);    X1, X2 = halogenated alkyl, phenyl or halo group;    R1-R6 = H, alkyl or halo;    B1 = diamino cyclic compound without carboxylic acid functionality; and    C1 = diamino cyclic compound with carboxylic acid functionality.    INDEPENDENT CLAIMS are included for making a crosslinked hollow fiber membrane, comprising (a) preparing a polyimide polymer comprising a predetermined quantity of crosslinkable sites, (b) treating the polyimide polymer with a diol comprising ethylene glycol, propylene glycol, 1,3-propanediol, 1,4-butanediol, 1,2-butanediol or 1,3-butanediol, at esterification conditions to form a monoesterified polymer, (c) forming a monoesterified hollow fiber from the monoesterified polymer, and (d) subjecting the monoesterified hollow fiber to transesterification conditions to form a covalently crosslinked hollow fiber membrane; or the step of (a) as above per se, forming a hollow fiber from the polyimide polymer, the step of (c) as above per se, and subjecting the monoesterified hollow fiber to transesterification conditions to form a crosslinked hollow fiber membrane.