Notes on the module of Euler systems

In this paper, we study the module of Euler systems. We determine the ideal of an Iwasawa algebra associated with Euler systems of rank $0$. We also show that the module of higher rank Euler systems for $\mathbb{G}_{m}$ over a totally real field is free of rank $1$ under the assumptions that Greenberg conjecture holds true and that the $\mu$-invariant of a certain Iwasawa module vanishes.