Abstract
We prove a result of cohomology and base change for families of coherent systems over a curve. We use that in order to prove the existence of (non-split, non-degenerate) universal families of extensions for families of coherent systems (in the spirit of the paper "Universal families of extensions" by H. Lange). Such results will be applied in subsequent papers in order to describe the wallcrossing for some moduli spaces of coherent systems.
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