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Syntomic cohomology here defined yields a link between rigid cohomology andetale cohomology, viewing the last one as the fixed points under Frobenius ofthe former one. Let V be a complete discrete valuation ring, with perfectresidue field k = V/m of characteristic p > 0 and fraction field K ofcharacteristic 0. Having defined syntomic cohomology with compact supports ofan abelian sheaf G on a k-scheme X, we show that it coincides with etalecohomology with compact supports when G is a lisse sheaf. If moreover theconvergent F-isocrystal associated to G comes from an overconvergent isocrystalE, then the rigid cohomology of E expresses as a limit of syntomiccohomologies: then the etale cohomology with compact supports of G is the fixedpoints of Frobenius acting on the rigid cohomology of E.

Cohomologie syntomique: liens avec les cohomologies \'etale et rigide