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Let's say I have a field k and a flat family of k[t]-modules M over the formal disk Spec k[h]k[[h]].

Now, assume that M/hM is torsion as a k[t]-module (but NOT finitely generated). Can I conclude that M/(h-i)M M[h^{-1}] is torsion as a k[tk((h))[t] modulefor any i in k?
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are deformations of torsion modules always torsion?

Let's say I have a field k and a flat family of k[t]-modules M over the disk Spec k[h].

Now, assume that M/hM is torsion as a k[t]-module (but NOT finitely generated). Can I conclude that M/(h-i)M is torsion as a k[t] module for any i in k?