  Considered as commutative algebras, cluster algebras can be very unpleasant
objects. However, the first author introduced a condition known as "local
acyclicity" which implies that cluster algebras behave reasonably. One of the
earliest and most fundamental examples of a cluster algebra is the homogenous
coordinate ring of the Grassmannian. We show that the Grassmannian is locally
acyclic. Morally, we are in fact showing the stronger result that all positroid
varieties are locally acyclic. However, it has not been shown that all
positroid varieties have cluster structure, so what we actually prove is that
certain cluster varieties associated to Postnikov's alternating strand diagrams
are locally acylic. Moreover, we actually establish a slightly stronger
property than local acyclicity, which we term the Louise property, that is
designed to facilitate proofs involving the Mayer-Vietores sequence.
