We introduce and study -persistence on the homology, with coefficients in a field, of a filtration of topological spaces. This is a family, one for each , of homological invariants which provide information not readily available by the (persistent) Betti numbers of the given filtration. This may help to detect noise, not just in the simplicial structure of the filtration but in further geometrical properties in which the higher codiagonals of the -structure are translated. Based in the classification of zigzag modules, a characterization of the -persistence in terms of its associated barcode is given.
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