  Let $p$ be a prime, $T$ a $p$-adic representation over a number field $K$ and
$\mathcal{K}$ an arbitrary Galois extension of $K$. Then for each non-negative
integer $r$ we define a natural notion of a `non-commutative Euler system of
rank $r$' for $T$ relative to the extension $\mathcal{K}/K$. We prove that if
$p$ is odd and $T$ and $\mathcal{K}/K$ satisfy certain mild hypotheses, then
there exist non-commutative Euler systems that control the Galois structure of
cohomology groups of $T$ over intermediate fields of $\mathcal{K}/K$ and have
rank that depends explicitly on $T$. As a first concrete application of this
approach, we (unconditionally) extend the classical Euler system of cyclotomic
units to the setting of arbitrary totally real Galois extensions of
$\mathbb{Q}$ and describe explicit links between this extended cyclotomic Euler
system, the values at zero of derivatives of Artin $L$-series and the Galois
structures of ideal class groups. As an important preliminary to the
formulation and proof of these results, we introduce natural non-commutative
generalizations of several standard constructions in commutative algebra
including higher Fitting invariants, higher exterior powers and the
Grothendieck-Knudsen-Mumford determinant functor on perfect complexes that are
of independent interest.
