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The category of Cohen-Macaulay modules of an algebra B k,n is used in [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. In this paper, we find canonical Auslander-Reiten sequences and study the Auslander-Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra in the tame cases.Proposition 5.2. Let I and J be r-interlacing. Then there exist 0 ≤ a 1 ≤ a 2 ≤ . . . a r−1 such that, as Z-modules,Proof. We assume that we have drawn the rims I and J one above the other, say I above J, as in the proof of Theorem 3.1 in [BB16, Section 3]. For every i 2s ∈ J \ I we have that rims I and J are not parallel between points i 2s−1 and i 2s , yielding a left trapezium. Similarly, for every i 2s+1 ∈ I \ J we have that rims I and J are not parallel between points i 2s and i 2s+1 , yielding a right trapezium. Since I and J are r-interlacing, we have, in alternating order r-left and r-right trapezia, giving us in total r boxes. The statement now follows from the proof of Theorem 3.1 in [BB16] which says that Ext 1 (L I , L J ) is a product of r − 1 cyclic Z-modules. Corollary 5.3. Let k = 3 and I and J be rims. Then Ext 1 (L I , L J ) ∼ = C × C if and only if I and J are 3-interlacing.Proof. If I and J are 3-interlacing, then they are both unions of three one-element sets. Hence, all the lateral sides in the boxes from the above proof are of length 1 and the statement follows since a i from the previous proposition are strictly positive, but at most equal to the lengths of the boxes involved. Corollary 5.4. Let k = 3. If I and J are crossing but not 3-interlacing, then Ext 1 (L I , L J ) ∼ = C.Note that if in Corollary 5.4 we have L J = τ (L I ), then we are in the situation of Theorem 3.12.Proposition 5.5. Assume that (k, n) = (3, 9) or (k, n) = (4, 8). Let M ∈ CM(B k,n ) be a rigid indecomposable rank 2 module. Then M ∼ = L I | L J where I and J are 3-interlacing.

Abstract. We characterize the syzygies and co-syzygies over 2-Calabi-Yau tilted algebras in terms of the Auslander-Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen-Macaulay modules, the representation dimension of algebras and the Igusa-Todorov functions. In particular, we prove that the Igusa-Todorov dimensions of d-Gorenstein algebras are equal to d.For cluster-tilted algebras of Dynkin type D, we give a geometric description of the stable Cohen-Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander-Reiten quiver of the stable Cohen-Macaulay category using tagged arcs and geometric moves.

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-1 prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss-Leclerc-Schröer.