On Euler systems of rank $r$ and their Kolyvagin systems Kazim Buyukboduk 11G0511G1011G4011R2314G10Euler systemsKolyvagin systemsIwasawa Theory$p%-adic $L$-functionsBloch-Kato conjecture In this paper we set up a general Kolyvagin system machinery for Euler systems of rank $r$ (in the sense of Perrin-Riou) associated to a large class of Galois representations, building on our previous work on Kolyvagin systems of Rubin-Stark units and generalizing the results of Kato, Rubin and Perrin-Riou. Our machinery produces a bound on the size of the classical Selmer group attached to a Galois representation $T$ (that satisfies certain technical hypotheses) in terms of a certain $r \times r$ determinant; a bound which remarkably goes hand in hand with Bloch-Kato conjectures. At the end, we present an application based on a conjecture of Perrin-Riou on $p$-adic $L$-functions, which lends further evidence to Bloch-Kato conjectures. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4237 10.1512/iumj.2010.59.4237 en Indiana Univ. Math. J. 59 (2010) 1277 - 1332 state-of-the-art mathematics http://iumj.org/access/