  We extend the well known notion of \textit{coideal} on $\mathbb{N}$ to
families of block sequences on $FIN_k$ and prove that if a coideal of block
sequences is \textit{semiselective} and satisfies a local version of Gowers'
theorem \cite{Gow} then the local Ramsey property relative to it can be
characterized in terms of the abstract Baire property, and the family of all
sets having the local Ramsey property relative to one such coideal is closed
under the Suslin operation. We also prove that these coideals satisfy a sort of
\emph{canonical partition property} in the sense of Taylor \cite{taylor},
L\'opez-Abad \cite{jordi} and Blass \cite{blass}. This results give us an idea
of the conditions to be considered in an abstract study of the local Ramsey
property in the context of topological Ramsey spaces (see \cite{todo}).
