Extensions between Cohen–Macaulay modules of Grassmannian cluster categories
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Abstract
In this paper we study extensions between Cohen–Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine \(\mathrm{Ext}^i(L_I, L_J)\) for arbitrary rank 1 modules \(L_I\) and \(L_J\). An explicit combinatorial algorithm is given for the computation of \(\mathrm{Ext}^i(L_I, L_J)\) when i is odd, and when i even, we show that \(\mathrm{Ext}^i(L_I, L_J)\) is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of \(\mathrm{Ext}^i(L_I, L_J)\) for any \(i>0\).
Keywords
Grassmannian cluster algebras Cohen–Macaulay modules Extension spacesMathematics Subject Classification
05E10 16A62 16G501 Introduction and preliminaries
In his study [7] of the total positivity of the Grassmannian Gr(k, n) of kplanes in \(\mathbb {C}^n\), Postnikov introduced alternating strand diagrams as collections of n curves in a disk satisfying certain axioms. Alternating strand diagrams associated with the permutation \(i\mapsto i+k\) of \({\mathbb {Z}}_n=\{1,2,\ldots , n\}\) (where \(i+k\) is taken modulo n if \(i+k>n\)) were used by Scott [8] to show that the homogeneous coordinate ring of Gr(k, n) has the structure of a cluster algebra, with each such diagram corresponding to a seed whose (extended) cluster consists of minors (i.e. of Plücker coordinates), where the minors are labelled by ksubsets of \(\{1,2,\ldots , n\}\). The diagram both gives the quiver of the cluster and the minors (cluster variables) contained in it: every alternating region of the diagram is obtained as a label the ksubset formed by the strands passing to the right of the region, and the quiver can be read off from the geometry of the strands. Oh et al. have proved in [6] that every cluster consisting of minors arises in this way, so there is a bijection between clusters of minors and strand diagrams for the Grassmann permutation. A categorification of this cluster algebra structure has been obtained by Geiss et al. [4] via (a subcategory of) the category of finitedimensional modules over the preprojective algebra of type \(A_{n1}\).
In [5], Jensen et al. gave a new and extended categorification of this cluster structure using the maximal Cohen–Macaulay modules [2] over the completion of an algebra B which is a quotient of the preprojective algebra of type \(\tilde{A}_{n1}\). In particular, a rank 1 Cohen–Macaulay Bmodule \(L_I\) is associated with each ksubset I of \(\{1,2,\ldots ,n\}\).
It was shown in [5] that every rigid indecomposable Cohen–Macaulay module for the above mentioned algebra B has a generic filtration by rank 1 modules. This enables a description of these modules in terms of the socalled profiles, given by collections of ksubsets that correspond to the rank 1 modules in this filtration. In particular, this profile determines the class of the module in the Grothendieck group of the category of Cohen–Macaulay modules. Therefore, rank 1 modules are the building blocks of the category of Cohen–Macaulay modules, and in order to understand representationtheoretic invariants for the category of all Cohen–Macaulay modules, we must first do so for rank 1 modules. Since the algebra B is infinite dimensional, most of the homological computations are difficult to conduct, but for some problems, it is possible to give complete answers. Such a problem is the computation of the extension spaces between rank 1 Cohen–Macaulay modules. To the Grassmannian Gr(k, n) we can associate a graph \(J_{k,n}\) by drawing a linear graph with nodes \(1,\ldots , n1\) and attaching an additional node to the kth node. The type of the cluster category associated to Gr(k, n) is finite if and only if this tree is a Dynkin graph. There is a degree function on the roots of the associated Kac–Moody algebra by taking the coefficient at the nth node [5, Sect. 2]. It is an open question raised by Zelevinsky whether the roots of height \(m>1\) correspond to Cohen–Macaulay modules obtained as extensions of m rank 1 modules whose ksubsets are cyclically equivalent. Knowing the extension spaces between rank 1 modules contributes to the understanding of this question. Also, extension spaces give us a lot of information about parts of the Auslander–Reiten quiver of the category of Cohen–Macaulay modules involving vertices that correspond to the rank 1 modules, and in this context, it is crucial to understand the structure of the syzygies appearing in the projective resolutions of rank 1 Cohen–Macaulay modules.
After some introductory remarks, in the second section of this paper we prove that rank 1 Cohen–Macaulay modules over the above mentioned completion of the algebra B are periodic, with periods being even numbers in the case when I is a disjoint union of more than two intervals. We give an explicit combinatorial formula for computation of the period of a given rank 1 module \(L_I\) only in terms of the ksubset I, which is called the rim of the rank 1 module \(L_I\). In the last section of this paper, we give an explicit combinatorial description of the Extspaces between rank 1 Cohen–Macaulay modules. The description is in terms of a new combinatorial and geometric construction consisting of a sequence of trapezia given by the rims of rank 1 Cohen–Macaulay modules. An explicit algorithm is constructed for the computation of the Extspaces which turn out to be finite dimensional. Also, we prove directly that the Extfunctor is commutative for rank 1 modules and that \(\mathrm{Ext}^2(L_I,L_J)\), where \(L_I\) and \(L_J\) are rank 1 Cohen–Macaulay modules, is a cyclic module over the centre \({\mathbb {F}}[t]\) of B. By using the fact that rank 1 modules are periodic, it was proven that for any \(i>0, \mathrm{Ext}^i(L_I,L_J)\) is a finitedimensional vector space. At the end of the paper, we give a combinatorial criterion for vanishing of \(\mathrm{Ext}^i(L_I, L_J)\) for any \(i>0\).
The centre Z of B is the polynomial ring \(\mathbb {F}[t]\), where \(t=\sum _{i=1}^n x_iy_i\). The (maximal) Cohen–Macaulay Bmodules are precisely those which are free as Zmodules. Indeed, such a module M is given by a representation \(\{M_i\,:\,i\in C_0\}\) of the quiver with each \(M_i\) a free Zmodule of the same rank (which is the rank of M, cf. [5, Sect. 3]).
Definition 1.1
Definition 1.2

\(x_i:U_{i1}\rightarrow U_{i}\) to be multiplication by 1 if \(i\in I\), and by t if \(i\not \in I\),

\(y_i:U_{i}\rightarrow U_{i1}\) to be multiplication by t if \(i\in I\), and by 1 if \(i\not \in I\).
Remark 1.3
Note that we represent a rank 1 module \(L_I\) by drawing its rim in the plane and identifying the end points of the rim. Unless specified otherwise, we will assume that the leftmost vertex is the vertex labelled by n, and in this case, most of the time we will omit labels on the edges of the rim. If one looks at the rim from left to right, then the number of downward edges in the rim is equal to k (these are the edges labelled by the elements of I), and the number of upward edges of the rim is equal to \(nk\) (these are the edges labelled by the elements that do not belong to I).
Proposition 1.4
[5, Proposition 5.2] Every rank 1 Cohen–Macaulay Bmodule is isomorphic to \(L_I\) for some unique ksubset I of \(C_1\).
Every Bmodule has a canonical endomorphism given by multiplication by \(t\in Z\). For \({L}_I\) this corresponds to shifting \(\mathcal {L}_I\) one step downwards. Since Z is central, \(\mathrm{Hom}_B(M,N)\) is a Zmodule for arbitrary Bmodules M and N. If M, N are free Zmodules, then so is \(\mathrm{Hom}_B(M,N)\). In particular, for rank 1 Cohen–Macaulay Bmodules \(L_I\) and \(L_J, \mathrm{Hom}_B(L_I,L_J)\) is a free module of rank 1 over \(Z={\mathbb {F}}[t]\), generated by the canonical map given by placing the lattice of \(L_I\) inside the lattice of \(L_J\) as far up as possible so that no part of the rim of \(L_I\) is strictly above the rim of \(L_J\).
2 Periodicity of rank 1 modules
In this section we prove that all rank 1 Cohen–Macaulay Bmodules are periodic, and we give an explicit formula for the periods of these modules in terms of their rims.
Again, because the rank is additive on short exact sequences, it follows that the kernel of an epimorphism from \(\oplus _{v\in V} P_v\) onto \({\varOmega }(L_I)\) is a rank 1 module. This means that there is a ksubset of \(\{1,2,\ldots ,n\}\), denoted by \(I^2\), such that this kernel, denoted by \({\varOmega }^2(L_I)\), is isomorphic to \(L_{I^2}\), i.e. \({\varOmega }^2(L_I)\cong L_{I^2}\). Note that \({\varOmega }^2(L_I)\) has no projective summands since it is an indecomposable module of rank 1. Also, since \({\varOmega }^2(L_I)\) is a module of rank 1, it is easy to show that \({\varOmega }^2(L_I)\) is a superfluous submodule of the module \(\oplus _{v\in V} P_v\), so \(\oplus _{v\in V} P_v\) is the projective cover of \({\varOmega }^1(L_I)\).
Using the same arguments as above, a projective module of the smallest possible rank that maps surjectively onto \({\varOmega }^2(L_I)\cong L_{I^2}\) is a module of rank \(r+1\), and the kernel of the corresponding epimorphism, denoted by \({\varOmega }^3(L_I)\), is a rank r module. Furthermore, the kernel of the epimorphism from a projective module of the smallest possible rank onto \({\varOmega }^3(L_I)\) is a rank 1 module, denoted by \({\varOmega }^4(L_I)\), and it is isomorphic to \(L_{I^4}\) for some ksubset \(I^4\) of \(\{1,2,\ldots ,n\}\). If we continue this construction of a projective resolution of \(L_I\), every other kernel will be a rank 1 module.
Since there are only finitely many rank 1 modules (they are in bijection with ksubsets of \(\{1,2,\ldots ,n\}\)), we must have that the above projective resolution of \(L_I\) is periodic. That is, for some indices a and \(b, a\ne b\), it holds that \({\varOmega }^a(L_{I})\cong {\varOmega }^b(L_{I})\), with \({\varOmega }^a(L_{I})\) denoting the ath syzygy of \(L_I\). In fact, we are going to prove a stronger statement that for some index t, we have that \({\varOmega }^{t}(L_I)\cong L_I\). The rest of this section is devoted to determining the minimal such index t.
Obviously, when \({\varOmega }^1(L_I)\) is of rank greater than 1, t must be an even number. Thus, we have to consider separately the case when \({\varOmega }^1(L_I)\) is a rank 1 module, because in this case in each step of the minimal projective resolution we get kernels that are rank 1 modules, so it can happen that in an odd number of steps we get a kernel that is isomorphic to \(L_I\), as we will see in the upcoming example.
Example 2.1
Let \(n=6, k=4,\) and \(I=\{1,2,4,5\}\). In this case, the number of peaks on the rim of \(L_I\) is equal to 2. For every \(i, {\varOmega }^i({L_I})\) is a rank 1 module.
Before moving on to the general case when \({\varOmega }(L_I)\) is a module of rank 1, let us introduce some of the notation used in this section.
If I is a ksubset of \(\{1,2,\ldots ,n\}\) that has \(r+1\) peaks when viewed as the rim of \(L_I\), then I can be written as a disjoint union of \(r+1\) segments \(A_1, A_2, \ldots , A_{r+1}\), where \(A_i=[a_i,b_i]\), and \(a_{i+1}b_i>1\), for all i. We can also assume without loss of generality that \(a_1=1\), because we can always assume that 0 is one of the peaks of the rim I, by renumbering if necessary. The size of the segment \(A_i\) is denoted by \(d_i\), and the difference \(a_{i+1}b_i1\) is denoted by \(l_i\). If one considers the rim of the module \(L_I\), it is clear that the numbers \(d_i\) (respectively \(l_i\)) represent the sizes of downward slopes (respectively upward slopes) of the rim, when looked at from left to right. Also, \(\sum d_i=k\), and \(\sum l_i=nk\).
Example 2.2
Continuing the previous example, we have that I is the union \(I=\{1,2\}\cup \{4,5\}\), and \(r+1=2\). There are two downward slopes, both of length 2, i.e. \(d_1=d_2=2\), and there are two upward slopes, both of length 1, i.e. \(l_1=l_2=1.\)
Theorem 2.3
Let \(L_I\) be a rank 1 module whose rim I has two peaks, and let \({\varOmega }^m(L_I)\) be as above. It holds that \(L_I\cong {\varOmega }^{2n/(n,k)}(L_I)\). The minimal projective resolution of \(L_I\) is periodic with period dividing 2n / (n, k).
Proof
Keeping the notation from the above discussion, if we set \(t=n/(n,k)\), then \(A_1=A_1^{2t},\, A_2=A_2^{2t}\), i.e. \(I=I^{2t}\). This means that \(L_I\cong {\varOmega }^{2n/(n,k)}(L_I)\). \(\square \)
We will now proceed by giving the explicit formula for the period of a rank 1 module whose rim has two peaks. We are looking for a minimal index m such that \(I^m=I\).
Theorem 2.4
Let \(L_I\) be a rank 1 Cohen–Macaulay module, and let \(d_1,d_2\) and \(l_1,l_2\) be as above. Depending on whether \(d_1=d_2\) or not, and \(l_1=l_2\) or not, the period of the module \(L_I\) is given by Eqs. (2.1), (2.2), (2.3) and (2.4).
This completes our determination of the periods for the rank 1 Cohen–Macaulay modules whose rims have only two peaks. For four different cases studied above, in general, we have four different formulas for computation of the period of a given rank 1 module.
Example 2.5
In Example 2.1 we had \(n=6, k=4\) and a rank 1 Cohen–Macaulay module \(L_I\) with the rim \( I=\{1,2,4,5\}\), and \(d_1=d_2=2\) and \(l_1=l_2=1\). In this case the period of \(L_I\) is given by Eq. (2.4). For \(t=1\), we have that \(d_1+kt\equiv 0 \mod 6\), meaning that the period of the module \(L_I\) is 3.
Example 2.6
We now assume that I is such that the rim of \(L_I\) has three or more peaks, and we set \(\mathrm{rk}\, {\varOmega }(L_I)=r>1.\)
From the above discussion we have that every other kernel in the above constructed projective resolution of \(L_I\) is a rank 1 module. If I is a disjoint union of segments \(A_1, A_2,\ldots , A_{r+1}\), then we assume that \(A_i\) has \(d_i\) elements and that the gap between \(A_{i}\) and \(A_{i+1}\) is of size \(l_i\). Also, we can assume without loss of generality that the smallest element in \(A_1\) is 1, i.e. \(A_1=\{1,2,\ldots , d_1\}, A_2=\{d_1+l_1+1,\ldots , d_1+l_1+d_2\}, \ldots , A_{r+1}=\{ \sum _{i=1}^r d_i+\sum _{i=1}^r l_i+1,\ldots , \sum _{i=1}^r d_i+\sum _{i=1}^r l_i+d_{r+1} \}\).
We proceed by computing the kernel of the above mentioned map D from the projective resolution of \(L_I\). We know that this kernel is a rank 1 module. If \(I=\{a_1,a_2,\ldots ,a_h\}\), then we set \(I+k=\{a_1+k,a_2+k,\ldots ,a_h+k\}\).
Proposition 2.7
The rim of the second syzygy of \(L_I\) is the rim I shifted by k, that is, the rim of \({\varOmega }^2(L_I)\) is \(I+k\).
Proof
If we fix a valley v of the rim I, then the elements of the module \(P_v\), where \(P_v\) is a summand of \(\bigoplus _{v\in V} P_v\), are mapped by the map D to two projective modules \(P_{u_{vl}}\) and \(P_{u_{vr}}\), where \(u_{vl}\) denotes the peak that is to the left of v and \(u_{vr}\) denotes the peak that is to the right of the valley v. So for a given peak u, only two \(P_vs\) are mapped to \(P_u\).
Remark 2.8
As a submodule of \(\bigoplus _{v\in V} P_v, {\varOmega }^2(L_I)\) is given as a diagonally embedded copy, with \({\varOmega }^2(L_I)\) seen as a submodule of each \(P_v\) by a canonical injective map given by placing the rim of \({\varOmega }^2(L_I)\) inside the \(P_v\) as high as possible.
Theorem 2.9
Proof
We notice here that if we take \(t=n/(n,k)\), and \(c=1\), then we have \(d_{c+i}=d_{1+i}\) and \(l_{c+i}=l_{1+i}\) for all i, and \(kt \equiv 0 \mod n\), i.e. \(A_{1+i}^{2n}=A_{1+i}\) for all i, and \(I=I^{2n/(n,k)}\). Thus, the upper bound for the period m is 2n / (n, k). It is clear that the period must be even, because every other syzygy is a rank 1 module. \(\square \)
As a corollary to the previous theorem, we immediately get a wellknown result.
Corollary 2.10
The algebra B has infinite global dimension.
Example 2.11
Example 2.12
Example 2.13
3 Extensions between rank 1 modules
In this section we compute all (higher) extensions \(\mathrm{Ext}^i(L_{I},L_{J})\), as a module over the centre \({\mathbb {F}}[t]\), for arbitrary rank 1 Cohen–Macaulay Bmodules \(L_I\) and \(L_J\). We give a combinatorial description and an algorithm for computation of extension spaces between rank 1 Cohen–Macaulay modules by using only combinatorics of the rims I and J.
Also, since \(\mathrm{Hom}({{\varOmega }(L_{I}}),{L_{J}})\) is a free module of rank r over the centre, and \(\mathrm{im}\, D^*\) is also a rank r submodule of a free \({\mathbb {F}}[t]\)module \(\displaystyle \bigoplus _{v\in V} \mathrm{Hom}({P_v},{L_{J}})\), we are left to compute invariant factors of \(D^*\) to determine generators of a free submodule \(\mathrm{im}\, D^*\) of \(\displaystyle \bigoplus _{v\in V} \mathrm{Hom}({P_v},{L_{J}})\).
Before proceeding, we note that the leading coefficient of the monomial \(d_{uv}^*\) is 1 if u is to the left of v in the cyclic ordering drawn in the plane; otherwise, it is \(1\).
Now we introduce a new combinatorial structure, consisting of a sequence of trapezia, that will enable us to describe extension spaces between rank 1 Cohen–Macaulay modules purely in terms of their rims.
Since \(I\ne J\), we can always assume, after cyclically permuting elements of \(\{1,2,\ldots ,n\}\) if necessary, that the first letter is L and that the last letter is R. The following step is to reduce the word \(w_{I,J}\) by replacing multiple consecutive occurrences of L (resp. R) by a single L (resp. R). What we are left with is a word of the form \({LRLRLRLR}\ldots {LR}=({LR})^s\). Let us call s the rank of the reduced word \(w_{I,J}\).
If in the above diagram we treat consecutive trapezia of the same orientation as a single trapezium, then we can see the above diagram as a collection of “boxes”, with box being a single pair consisting of one left trapezium and one right trapezium. The number s denotes the number of boxes for the rims I and J. If we ignore the parts of the rims of I and J that have the same tendency, then what we are left with is the two rims with always different tendencies (in other words, we have a sequence of boxes), and these rims are symmetric in the sense that one is a reflection of the other with respect to the horizontal line between them.
Our aim is to describe extension spaces between rank 1 Cohen–Macaulay modules by using combinatorics of the corresponding rims. As a module over the centre \({\mathbb {F}}[t]\), it turns out that the extension space between rank 1 Cohen–Macaulay modules \(L_I\) and \(L_J\) is a torsion module isomorphic to a direct sum of the cyclic modules which are computed directly from the rims I and J. Our main result states.
Theorem 3.1
In the coming proof of this result, we give an algorithm for the computation of the numbers \(h_i\) using only rims I and J.
We note here that there can only be left trapezia present between \(v_i\) and \(u_i\) since the rim of I has only upward tendency. Analogously, there can be only right trapezia between \(u_i\) and \(v_{i+1}\) since the tendency of I is downwards. If there are multiple left trapezia between \(v_i\) and \(u_i\), then we regard them as a single trapezium with an offset given by the sum of offsets of those left trapezia. The same goes for multiple right trapezia. So we regard every peak as having at most one left, and at most one right trapezium next to it. This corresponds to the previously mentioned reduction step of the word \(w_{I,J}\). Here, reduced letters come from the same peak.
Remark 3.2
Let E be the \(s\times s\) submatrix of \(D_1\) consisting of the last s rows and columns of \(D_1\). Since \(\mathrm{im}\, D^*\) is a free submodule of corank 1 of the free module \(\bigoplus _{v\in V} \mathrm{Hom}{(P_v},{L_{J})}\), it follows that \(D^*\) is also a matrix of corank 1, and that E is a matrix of corank 1. There is a linear combination over \({\mathbb {F}}[t]\) of columns of \(D^*\) that is equal to zero. Moreover, at least one of the coefficients in this linear combination is equal to 1 (these are precisely the coefficients of the columns corresponding to the peaks that are placed on the rim of J when \(L_I\) is canonically mapped into \(L_J\), see Remark 3.6 below).
This proves Theorem 3.1!
Corollary 3.3
If \(I\ne J\), then \(\mathrm{Ext}^1(L_{I},L_{J})=0\) if and only if the number of LR boxes is equal to 1.
Remark 3.4
The case when the number of LR boxes is 1 is exactly the noncrossing case from [5, Proposition 5.6], because existence of exactly one box means that I and J are noncrossing.
Reenumerate indices of elements of \(H_i\), that is, for \(q>j, a_q\) becomes
\(a_{q1}\), and \(b_q\) becomes \(b_{q1}\).
Reenumerate indices of elements of \(H_i\), that is, for \(q>j, a_q\) becomes
\(a_{q1}\), and \(b_q\) becomes \(b_{q1}\).
UNTIL \(i=r+1\)
Example 3.5
Remark 3.6
Theorem 3.7
Proof
Let us draw the rims of \(L_I\) and \(L_J\) one below the other, with the rim of \(L_I\) above, and with an additional copy of the rim of \(L_I\) below the rim of \(L_J\). We also draw the trapezia we used to determine the extensions between the two rank 1 modules, with the upper trapezia used to compute \(\mathrm{Ext}^{1}(L_I,L_J)\) and lower trapezia to compute \(\mathrm{Ext}^{1}(L_J,L_I)\).
We now compute higher extensions for rank 1 Cohen–Macaulay modules. After showing how to compute higher extensions of odd degree, we prove that the even degree extensions are cyclic \({\mathbb {F}}[t]\)modules, and we show how to combinatorially compute generators of these cyclic modules. In the end we give a combinatorial criterion for vanishing of higher extension spaces between rank 1 modules.
From the first section we know that the rank 1 modules are periodic, and moreover, for a given rim I, every even syzygy in a minimal projective resolution of \(L_I\) is a rank 1 module. This immediately gives us the following statement.
Proposition 3.8
Proof
Theorem 3.9
Proof
From the above diagram we know that \(\mathrm{im}\, F^*\) is a free module isomorphic to a submodule of \(\mathrm{Hom}({\varOmega }^2(L_I),L_J)\). Hence, the matrix of \(F^*\) is a matrix of rank 1 over \({\mathbb {F}}[t]\). Since the map \(F^*\) is given by the maps from \(P_w\) to \(L_J\), which are given by multiplication by \(t^l\) for some exponent l, the matrix of \(F^*\) consists of the monomials. Because it is a matrix of rank 1, it follows that there is a column such that every other column is a multiple of that column. To find the invariant factor of this matrix, it remains to find a monomial with the smallest exponent from that column. This exponent gives us the integer a. \(\square \)
Corollary 3.10
Proof
Let us first note that \(W=I+k\) and label the matrix of \(F^*\) with pairs (u, v) rather than with pairs (w, v) with u corresponding to the element \(u+k=w\) (\(w=u(nk)\)) of W.
Remark 3.11
Let \(r+1\) be the number of peaks of the rim I, i.e. assume that \(F^*\) is a matrix of the format \((r+1)\times (r+1)\). From the proof of the previous corollary we see that in order to compute the smallest exponent a for the entries of \(F^*\), it is sufficient to compute entries of one column and one row, which means that we have to compute at most \(2r+1\) entries of the matrix \(F^*\) determined by (3.9). We pick an arbitrary row, compute its entries, and choose the minimal one. Then we compute entries of the column that contains that minimal entry. Then the exponent a is the minimal entry from that column. \(\square \)
Theorem 3.12
Proof
From the proof of the previous theorem and corollary we know that \(\mathrm{Ext}^{2}(L_I,L_J)=0\) if and only if there is an element of the matrix of \(F^*\) that is equal to 1. This happens only if for some \(u_i\in U\) and \(v_j\in V\) the number \(a_{u_j,v_i}\) is zero. For a given \(u_j\) and \(v_i\), recalling the picture from the proof of the previous corollary, \(a_{u_j,v_i}=0\) if and only if both vertical distances at a given node \(u_j(nk)\) between the rim of \(P_{v_i}\) and the rim of \(L_J\), and between the rim of \(P_{v_i}\) and the rim of \({\varOmega }^2(L_I)\) inside the rim of \(P_{v_i}\) are equal to 0. Obviously, this can not happen if \(u_j(nk)\) is a node on the rim of \({\varOmega }^2(L_I)\) that is not on the rim of \(P_{v_i}\) at the same time, as in the case of the pictured node w in the picture from the proof of the previous corollary. In this case, the vertical distance between w on the rim of \({\varOmega }^2(L_I)\) and \(u(nk)=w\) on the rim of \(P_{v_i}\) is strictly positive, so \(a_{u_j,v_i}>0\) in this case. We conclude that if \(a_{u_j,v_i}=0\), it must be that \(u_j(nk)\) is on both the rim of \({\varOmega }^2(L_I)\) and the rim of \(P_{v_i}\). So for a given \(v_i\), the only candidates \(u_j\) for \(a_{u_j,v_i}\) to be 0 are \(u_{i}\), which is to the right of \(v_i\), with the corresponding node \(u_i(nk)\) on both rims of \({\varOmega }^2(L_I)\) and \(P_{v_i}\), and \(u_{i1}\), which is to the left of \(v_i\), with the corresponding node \(u_{i1}+k\) on both rims of \({\varOmega }^2(L_I)\) and \(P_{v_i}\).
For these two nodes \(u_i(nk)\) and \(u_{i1}+k\), in order for the vertical distance between the rim of \(P_v\) and the rim of \(L_J\) to be equal to zero at the node \(u_i(nk)\) (resp. \(u_{i1}+k\)), it has to be that the rim J has the same tendency between \(u_i(nk)\) and \(v_i\) (resp. between \(v_i\) and \(u_{i1}+k\)) as the rim of \(P_{v_i}\). This means that it must be that \(\# J\cap (u_i(nk),v_i]=0\) (resp. \(\# J\cap (v_i,u_{i1}+k]=k(v_iu_{i1}\)). \(\square \)
Remark 3.13
Combined with Corollary 3.3 and periodicity of rank 1 modules, the previous theorem gives us a combinatorial criterion for vanishing of \(\mathrm{Ext}^{i}(L_I,L_J)\) for arbitrary \(i>0\), and for any rank 1 modules \(L_I\) and \(L_J\). This criterion is given purely in terms of the rims I and J.
Example 3.14
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). We would like to thank Alastair King for all his help with this project. The authors were supported by the Austrian Science Fund Project Number P25647N26, the first author was also supported by the Project FWF W1230.
References
 1.Baur, K., King, A., Marsh, R.J.: Dimer models and cluster categories of Grassmannians (2013). arXiv:1309.6524
 2.Buchweitz, R.O.: Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings. Unpublished notes (1985)Google Scholar
 3.Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations. I. Mutations. Sel. Math. (N.S.) 14, 59–119 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Geiss, C., Leclerc, B., Schroer, J.: Partial flag varieties and preprojective algebras. Ann. Inst. Fourier (Grenoble) 58(3), 825–876 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Jensen, B., King, A., Su, X.: A categorification of Grassmannian Cluster Algebras (2013). arXiv:1309.7301
 6.Oh, S., Postnikov, A., Speyer, D.E.: Weak Separation and Plabic Graphs (2011). arXiv:1109.4434
 7.Postnikov, A.: Total positivity, Grassmannians, and networks (2006). arXiv:math/0609764
 8.Scott, J.S.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. (3) 92(2), 345–380 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
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