Rank 1 modules are the building blocks of the category CM(B k,n ) of Cohen-Macaulay modules over a quotient B k,n of a preprojective algebra of affine type A. Jensen, King and Su showed in [8] that the category CM(B k,n ) provides an additive categorification of the cluster algebra structure on the coordinate ring C[Gr(k, n)] of the Grassmannian variety of k-dimensional subspaces in C n . Rank 1 modules are indecomposable, they are known to be in bijection with k-subsets of {1, 2, . . . , n}, and their explicit construction has been given in [8]. In this paper, we give necessary and sufficient conditions for indecomposability of an arbitrary rank 2 module in CM(B k,n ) whose filtration layers are tightly interlacing. We give an explicit construction of all rank 2 decomposable modules that appear as extensions between rank 1 modules corresponding to tightly interlacing k-subsets I and J.

In this paper we study indecomposable rank 2 modules in the Grassmannian cluster category CM(B5,10). This is the smallest wild case containing modules whose profile layers are 5-interlacing. We construct all rank 2 indecomposable modules with filtration {i, i + 2, i + 4, i + 6, i + 8} | {i + 1, i + 3, i + 5, i + 7, i + 9}, classify them up to isomorphism, and parameterize all infinite families of non-isomorphic rank 2 modules.