A Multiplication Formula for the Modified Caldero Chapoton Map

A frieze in the modern sense is a map from the set of objects of a triangulated category $\mathsf{C}$ to some ring. A frieze $X$ is characterised by the property that if $\tau x\rightarrow y\rightarrow x$ is an Auslander-Reiten triangle in $\mathsf{C}$, then $X(\tau x)X(x)-X(y)=1$. The canonical example of a frieze is the (original) Caldero-Chapoton map, which send objects of cluster categories to elements of cluster algebras. \par In \cite{friezes1} and \cite{friezes2}, the notion of generalised friezes is introduced. A generalised frieze $X'$ has the more general property that $X'(\tau x)X'(x)-X'(y)\in\{0,1\}$. The canonical example of a generalised frieze is the modified Caldero-Chapoton map, also introduced in \cite{friezes1} and \cite{friezes2}. \par Here, we develop and add to the results in \cite{friezes2}. We define Condition F for two maps $\alpha$ and $\beta$ in the modified Calero-Chapoton map, and in the case when $\mathsf{C}$ is 2-Calabi-Yau, we show that it is sufficient to replace a more technical"frieze-like"condition from \cite{friezes2}. We also prove a multiplication formula for the modified Caldero-Chapoton map, which significantly simplifies its computation in practice.

0. Introduction 0.1. Summary. This paper focuses around two main topics: generalised friezes with integer values (see [14]) and generalised friezes taking values inside a Laurent polynomial ring (see [15]).
A frieze is a map X : obj C → A, where C is some triangulated category with Auslander-Reiten (AR) triangles and A is a ring, such that the following exponential conditions are satisfied: (0.1) X(0) = 1 and X(a ⊕ b) = X(a)X(b), and if τ x → y → x is an AR triangle, then (0.2) X(τ x)X(x) − X(y) = 1.
The canonical example of a frieze is the Caldero-Chapoton map, which we recall in Section 0.4. Generalised friezes are similarly defined maps X ′ : obj C → A, also satisfying the exponential conditions in (0.1), however we permit the more general property that The canonical example of a generalised frieze is the modified Caldero-Chapoton map, which we recall in Section 0.6. The arithmetic version π, with integer values, is defined in Equation (0.5), whilst the more general version ρ, taking values inside a Laurent polynomial ring, is defined in Equation (0.8).
The modified Caldero-Chapoton map was introduced in [15], and we improve and add to the results of that paper. When working with a 2-Calabi-Yau category, we manage to replace the technical "frieze-like" condition (see [15, def. 1.4]) for the maps α and β in the generalised Caldero-Chapoton map (Equation (0.8)), by our so-called Condition F (see Definition 2.1). This condition significantly simplifies the frieze-like condition and demonstrates the roles of α and β. We will see that α plays the role of a "generalised index", whilst β provides a correction term to α being exponential over a distinguised triangle.
We use this to establish a multiplication formula for the modified Caldero-Chapoton map ρ (see Theorem 5.2), allowing its computation in practice. In [15], the computation of ρ is not addressed. However, our multiplication formula does address the computation, and does so in a simpler manner than merely applying the definition. In particular, the formula allows us to compute values of ρ without calculating Euler characteristics of submodule Grassmannians which are otherwise part of the definition of ρ.

Cluster Categories. Cluster categories were first introduced in [4] by Buan-
Marsh-Reineke-Reiten-Todorov as a means of understanding the 'decorated quiver representations' introduced by Reineke-Marsh-Zelevinksy in [16]. Let Q be a finite quiver with no loops or cycles, and consider the category mod CQ of finitely generated modules over the path algebra CQ. Then, set the bounded derived category of mod CQ.
The cluster category of type Q, denoted C(Q), is defined to be the orbit category of D(Q) under the action of the cyclic group generated by the autoequivalence τ −1 Σ = S −1 Σ 2 , where τ is the Auslander-Reiten translation, Σ the suspension functor and S the Serre functor. That is, The objects in C(Q) are the same as those in D(Q), however, the morphism sets in C(Q) are given by Hom C(Q) (X, Y ) = n∈Z Hom D(Q) (X, (S −1 Σ 2 ) n Y ).
We note that C(Q) possesses a triangulated structure, it is C-linear, Hom-finite, Krull-Schmidt and 2-Calabi-Yau, meaning that its Serre functor is Σ 2 . It is also essentially small and has split idempotents. See [4, sec. 1] The cluster category of Dynkin type A n , denoted by C(A n ), has a very nice polygon model associated to it. This is due to Caldero, Chapoton and Schiffler in [7], who defined, for finite quivers of type A n , an equivalent category to the cluster category in [4] in a totally different manner. This is done using a triangulation of a regular (n + 3)-gon P , with objects and morphisms described in [7, sec. 2].
The category C(A n ) carries several nice properties. There is a bijection between the set of indecomposables of C(A n ), denoted indec C(A n ), and the set of diagonals of P . We also identify each edge of P with the zero object in C(A n ). Applying the suspension functor Σ to an indecomposable corresponds to rotating the endpoints of the associated diagonal one vertex clockwise. That is, if the vertices of P are Such coordinates should clearly be taken modulo n + 3.
Identifying indecomposables of C(A n ) with the diagonals of P carries the convenient property that for a, b ∈ indec C(A n ), , if a and b cross 0, if a and b do not cross.
0.3. The Auslander-Reiten Quiver. The Auslander-Reiten quiver for C(A n ) is ZA n modulo a glide reflection. A coordinate system may be put on the quiver, matching up with the diagonals of the (n + 3)-gon (see Figure 1 for an example of C(A 5 )). The Auslander-Reiten triangles in C(A n ) take the form where if {i − 1, j} or {i, j − 1} correspond to an edge of P , then they should be taken to be zero, see [5, lem. 3.15]. Notice that each AR triangle can be realised from a diamond in the AR quiver. That is, if is a diamond inside the AR quiver, then is an AR triangle. If a and d sit on the upper boundary of the AR quiver, then b should be taken as zero, whereas if a and d sit on the lower boundary, then c is taken to be zero. Note that a frieze X on C(A n ) satisfies:  [9] and [10]. An example of such a frieze pattern, known as a Conway-Coxeter frieze, is given in Figure 2. This frieze pattern is obtained from the original Caldero-Chapoton map γ by omitting the arrows from the AR quiver of C(A 7 ) and replacing each vertex by the value of γ applied to that indecomposable. In formal terms, for some positive integer n, a frieze pattern is an array of n offset rows of positive integers. Each diamond Observe that the Caldero-Chapoton map satisfies these equations by virtue of Equation (0.4), that is, because it is a frieze. Frieze patterns are known to be invariant under a glide reflection. A region of the frieze pattern, known as a fundamental domain, is enough to produce the whole frieze pattern by repeatedly performing a glide reflection. 0.6. A Modified Caldero-Chapoton Map. We assume in the rest of the paper that C is an essentially small, C-linear, Hom-finite, triangulated category, which is Krull-Schmidt and has AR triangles. In [14], Holm and Jørgensen introduce a modified version of the Caldero-Chapoton map, which we denote by π, that relies on a rigid object R ∈ obj C, a much weaker condition than that of being a cluster tilting object. We say that an object R is rigid if We also note that π does not require that the category is 2-Calabi-Yau, allowing us to work with a category C that is more general than a cluster category. Consider the endomorphism ring E = End C (R), and define mod E to be the category of finite dimensional right E-modules. Then, there is a functor For some object c ∈ C, the modified Caldero-Chapoton map is then defined by the formula: where Gr denotes the Grassmannian of submodules and χ is the Euler characteristic defined by cohomology with compact support (see [13, p. 93]). It is proved in [14] that π is a generalised frieze; that is, as well as the exponential properties given in Equation (0.1) it satisfies the property given in Equation (0.3).
Define R = add R, the full subcategory whose objects are finite direct sums of the summands of R (see Section 1 for details of this setup). This full subcategory, which is clearly closed under direct sums and summands, is rigid in the sense that Hom C (R, ΣR) = 0. A multiplication formula for computing π is also proved in [14]. Let m ∈ indec C and r ∈ indec R satisfy that Ext 1 C (m, r) and Ext 1 C (r, m) both have dimension one over C. Then, there are nonsplit triangles This formula can be applied iteratively to compute values of π.
In [15] Holm and Jørgensen redefine the modified Caldero-Chapoton map in a more general manner (the work in [14] is a special case of that in [15]). They define ρ by Here, the sum is taken over e ∈ K 0 (mod E), the Grothendieck group of the abelian category mod E, and Gr e (Gc) is the Grassmannian of E-submodules M ⊆ Gc with K 0 -class satisfying [M ] = e. The maps (0.9) α : obj C → A and β : K 0 (mod E) → A are both exponential maps in the sense that and A is some commutative ring, see [15, When the maps α and β satisfy a technical "frieze-like" condition, given in [15, def. 1.4], the map ρ becomes a generalised frieze, as in Equation (0.3). 0.7. This Paper. In this paper, we show a simpler condition on α and β than that in [15], which implies that ρ is a generalised frieze. We also show that a similar multiplication formula to that in Equation (0.7) holds with ρ instead of π. This permits a simpler iterative procedure for computing ρ than the one given in [15].
We give the following definition: We say that the maps α and β satisfy Condition F if for each triangle in C such that Gx, Gy and Gz have finite length in ModE, the following property holds: . The following main result shows that when C is 2-Calabi-Yau, this definition is sufficient to replace the frieze-like condition from [ In Section 5, we prove the multiplication formula for ρ, similar to the arithmetic case for π in (0.7). This multiplication formula is as follows: . This theorem can be applied inductively in order to simplify the computation of values of ρ. For some indecomposable m in C, one may find an a and b (which are the middle terms of the nonsplit extensions), such that ρ(m) = ρ(a) + ρ(b).
The theorem can then be reapplied to find each of ρ(a) and ρ(b). The process will eventually terminate at the stage where calculating ρ of some indecomposable reduces to calculating the index of that indecomposable. Substituting back into the equation allows a simple calculation of ρ(m).
We illustrate the procedure in Section 6 by computing ρ of an indecomposable in the Auslander-Reiten quiver for C(A 5 ). This retrieves one of the vertices of the AR quiver in Figure 3. Note that this example already appeared in [15, sec. 3], but Theorem 5.2 makes our computation much simpler. This paper is organised as follows. Section 1 gives an essential background to the modified Caldero-Chapoton map, and explains some important results from [15]. Section 2 introduces Condition F and proves Theorem 2.2, whilst Section 3 shows how one can construct maps α and β satisfying this condition. Section 4 demonstrates how to find the multiplication formula for π, Section 5 adapts this formula to ρ by proving the multiplication formula in Theorem 5.2, and Section 6 shows why this formula is useful.
It should be noted that Sections 1 and 4 contain no original work, however they provide an essential setup for subsequent sections in the paper.

A Modified Caldero-Chapoton Map: A Functorial Viewpoint
In this section, we redefine the modified Caldero-Chapoton map in detail, using an equivalent, functorial viewpoint, allowing us simpler calculations throughout the rest of the paper. To set up, we will follow the construction in [15, sec. 2]. We add the assumption that C is 2-Calabi-Yau. We let R be a functorially finite subcategory of C, which is closed under sums and summands and rigid; that is, Hom C (R, ΣR) = 0.
We also assume that C has a cluster tilting subcategory T, belonging to a cluster structure in the sense of [3, sec. II.1]. We additionally require that R ⊆ T. Note that the Auslander-Reiten translation for C is τ = Σ, and its Serre functor is S = Σ 2 .
There is a functor G, defined by: where Mod R denotes the category of C-linear contravariant functors R → Vect C. It is a C-linear abelian category, and we denote by fl R the full subcategory of Mod R, consisting of all finite length objects.
The modified Caldero-Chapoton map is then defined as in Equation (0.8), where the sum is now taken over e ∈ K 0 (fl R). Here, we have and these maps satisfy the exponential condition in Equation (0.10) For two objects a, b ∈ C such that Ga and Gb have finite length, it is known by [15, prop. 1.3] that ρ is also exponential; that is, It therefore suffices to calculate ρ for each indecomposable object in C. Note that the formula for ρ only makes sense when Gc has finite length in Mod R. That is, we require that Gc ∈ fl R.

Condition F on the maps α and β
We continue under the setup of Section 1. Consider the exponential maps α and β introduced earlier in Equation (1.2). The following definition describes a canonical condition on α and β, which will later be used in Section 5 to prove a multiplication formula for ρ.
is split short exact. It follows immediately that Gξ has trivial kernel. Applying Condition F to ∆, we obtain: where the final = is due to β being exponential. Case (ii), first part : Assume c / ∈ R ∪ Σ −1 R and G(∆) is a nonsplit short exact sequence. Then, by the same working as in Case (i), we see that: Now, consider the following triangle in C: Applying Condition F and using the fact that α is exponential, we obtain: We note that this manipulation works for any c ∈ obj C.
where P r is the indecomposable projective Hom R (−, r) in Mod R, described in [14, sec. 1.5]. Therefore, Gξ has zero kernel, and applyng Condition F shows: Case (iii): Let c = r ∈ R and again consider the triangle Applying Condition F, we see where the third = is since G(c) = G(r) = 0 and hence Ker Gν = 0. Now, by [14, lem. 1.12(ii)], applying G to ∆ gives the exact sequence: where the second = is since α(c ⊕ Σc) = 1.

Constructing Maps that Satisfy Condition F
We again continue under the setup of Section 1. We will show that there exist maps α and β satisfying Condition F, namely those given in [15, def. 2.8]. Let us first look at the necessary constructions behind the definitions of α and β in [15, def. 2.8].
Recall that T is some cluster tilting subcategory of C with R ⊆ T, and denote by indec T the set of indecomposable objects in T. For each t ∈ indec T, one may find a unique indecomposable t * ∈ indec C, called the mutation of t, such that replacing t with t * gives rise to another cluster tilting subcategory T * , see [3, sec. II.1]. Each such t and t * fit into two exchange triangles (see [3, sec. II.1]): where a, a ′ ∈ add (indec T) \ t . We denote by K split 0 (T) the split Grothendieck group of the additive category T which has a basis formed by the set of indecomposables in T. We note that K where Mod T is the abelian category of C-linear contravariant functors T → Vect C. Now, by [2, prop. 2.3(b)], for each t ∈ indec T, there is a simple object S t ∈ Mod T supported at t, and one may see that where S t denotes the simple object in Mod R supported at t. Due to i * being exact, we can restrict it to the subcategories fl T and fl R, made up of the finite length objects in Mod T and Mod R, respectively. Then, there is an induced (surjective) group homomorphism κ : K 0 (fl T) ։ K 0 (fl R), with the obvious property that For the category Mod T, there is a functor G similar to G from Equation (1.1). It is defined by: It is not hard to see that G has the property that i * G = G.
We define θ to be the group homomorphism making the following diagram commute: where a, a ′ ∈ add (indec T) \ t are from the exchange triangles for t in (3.1). Now, we may recall from [15, def. 2.8] that the maps α : obj C → A and β : K 0 (fl R) → A to a suitable ring A can be defined by: (3.5) α(c) = εQ(ind T (c)) and β(e) = εθ(e), where ε : K split 0 (T)/N → A is a suitably chosen exponential map, meaning that: Here, a and b denote two elements of K split 0 (T)/N .
In the above manipulation, the second = is due to [17, prop. 2.2] and the penultimate = occurs since ε is exponential, see Equation (3.6). In addition, where the first = is due to α being exponential and the penultimate = is just by the definition of β. Now, using the property that i * G = G, it follows that where (1) follows from i * being an exact functor, and (2) from the definition of κ.
We can now manipulate the expression ( * * ) further: where the second equality is due to the commutativity of Diagram (3.4).
Comparing ( * ) to ( * * * ), we see that the required equality for Condition F is satisfied if Making use of the "rolling" property on our triangle in (3.7), we obtain the following sequence: where any four consecutive terms form a triangle. Furthermore, since G is a homological functor, we may apply it to this sequence and produce the following long exact sequence in fl T: This shows Coker GΣ −1 ω = Ker Gϕ. Moreover, C is chosen such that GΣ −1 C = Coker GΣ −1 ω, and hence Ker Gϕ = GΣ −1 C. We can hence compute as follows: where the second = is due to [15, is an AR triangle in C such that Gc and G(Σc) have finite length in Mod T, the maps α and β from Equation (3.5) satisfy the frieze-like condition given in [15, def. 1.4]. By Lemma 3.1, we know that α and β as defined in Equation (3.5) satisfy Condition F. Theorem 2.2 proves that any α and β satisfying Condition F also satisfy the frieze-like condition for ∆ (recovering [15, thm. 2.11]). Hence by [15, thm. 1.6], these α and β turn ρ into a generalised frieze.

The Multiplication Formula from [14]
In this section we demonstrate some of the technicalities behind the proof of the multiplication formula for π from Equation (0.7), proved in [14, prop. 4.4]. This is done with a view of proving a similar formula for ρ in Section 5. Following the setup of [14, sec. 4], for this section we do not require a cluster tilting subcategory T, as the theory in [14] uses only the rigid subcategory R.
Let m ∈ indec C and r ∈ indec R be indecomposable objects such that Ext 1 C (r, m) and Ext 1 C (m, r) both have dimension one over C. As in [14, rem. 4.2], this allows us to construct the following nonsplit triangles in C: with δ and ζ nonzero. Note that "rolling" the first triangle gives: which is also a triangle in C. Applying the functor G to both the "rolled" triangle and the triangle in (4.2) gives the following exact sequences in Mod R, obtained in [14]: Remark 4.1.
(1) The zeros arise in each exact sequence due to G(r) = Hom(−, Σr)| R being the zero functor. Indeed, since R is rigid, evaluating G(r) at any x in R will make the corresponding Hom-space zero.
(2) The exact sequences are in Mod R; that is, each term is a C-linear contravariant functor R → Vect C.
Letting Gr denote the Grassmannian of submodules, we have morphisms of algebraic varietes, It was proved in [14,Lemma 4.3] that if M in Mod R is some subfunctor of Gm, then either M ⊆ Im Gη or Ker Gµ ⊆ M , but not both. This means that for M ⊆ Gm we can find either a subfunctor N ⊆ Gb such that (Gη)(N ) = M or we can find P ⊆ Ga such that (Gµ) −1 (P ) = M . Hence, the subfunctor M is either of the form (Gη)(N ) or (Gµ) −1 (P ).
It is hence clear that Gr(Gm) is isomorphic to the disjoint union of the images of ξ and ν. That is, Gr(Gm) ∼ = Gr(Gb) Gr(Ga).
(1) We should note that for M in Mod R, the Grassmannian Gr(M ) is an algebraic variety. Therefore, it makes sense to calculate the Euler characteristic of Gr(Gm), Gr(Ga) and Gr(Gb).
(2) In addition, we note that ξ and ν are both constructible maps, hence the images of ξ and ν form constructible subsets in Gr(Gm). See [17, Section 2.1] for the definitions of a constructible map and a constructible set.
The following statement then follows in [14]: where χ again denotes the Euler characteristic defined by cohomology with compact support (see [13, p. 93]). Using Remark 4.2, since the images of ξ and ν are constructible sets inside Gr(Gm), we know that χ is additive (see [13, p. 92, item (3)]), which gives the above equality in (4.3). That is π(m) = π(a) + π(b). [15] This section builds on the material covered in the previous section and makes necessary adjustments and additions in order to obtain the multiplication formula for ρ, given in Theorem 5.2. Clearly, now that we are back working with ρ, we again require the setup of Section 1; that is, we need a cluster tilting subcategory T, with R ⊆ T.

Adaptation of the Multiplication Formula to
As with π, we look to understand how to evaluate ρ for some m ∈ indec C. In the definition of ρ, we take a sum over e ∈ K 0 (fl R). In order to do this, we will require knowledge of the Grothendieck group K 0 (fl R) and the K 0 -classes of some of its key elements.
Firstly, we know In the above calculation, the third = is due to χ(Gr e (0)) being zero for all nonzero e ∈ K 0 (fl R) and one when e = 0. The last = is since β is exponential. Now consider ρ(r)ρ(m) for m ∈ indec C: ρ(r)ρ(m) = α(r)α(m) e χ(Gr e (Gm))β(e) = α(r)α(m) e χ(Im ξ ∩ Gr e (Gm)) + χ(Im ν ∩ Gr e (Gm)) β(e), where the second equality arises from Gr(Gm) being the disjoint union of the images of ξ and ν. We now make an important remark about the two intersections in the second equality above.    Applying Condition F to our two triangles in the theorem, whilst remembering that α is exponential, gives Returning these equalities into Equation (5.3), the expression for ρ(r)ρ(m) becomes: Gr e (Gb) β(e) = ρ(a) + ρ(b).

Example for C(A 5 )
In this section, we will demonstrate the multiplication formula for ρ in Theorem 5.2 by recomputing a vertex in the AR quiver in Figure 3. We first give some brief background on the polygon model for C = C(A n ), the cluster category of Dynkin type A n . Here, n ≥ 2 is an integer. By [7], the indecomposables of C can be identified with the diagonals of a regular (n + 3)-gon P with the set of vertices {0, ..., n+2}. By [14, thm. 5.4], the indecomposables of the rigid subcategory R ⊆ T give a polygon dissection of P , and by [4] the cluster tilting subcategory T gives a full triangluation of the (n + 3)-gon. Indeed, recall that there is a full subcategory S, which is closed under direct sums and summands, such that indec T = indec R ∪ indec S.
Hence, the indecomposables in S correspond to a triangulation of each of the cells of P given by the polygon dissection from indec R. We should note in addition that each edge of the (n + 3)-gon is identified with the zero object inside C.
This model also comes with the convenient property that for two indecomposables a and b in C dim C Ext 1 C (a, b) = 1, if a and b cross 0, otherwise.
It is known by Theorem 5.2 that for m ∈ indec C and r ∈ indec R such that where a and b are the middle terms of the nonsplit extensions in (4.1) and (4.2).
In the case of C = C(A n ), a and b can be obtained as seen in the polygon in Figure  4, where we have a = a 1 ⊕ a 2 and b = b 1 ⊕ b 2 . See [15, sec. 5] for details. Now, to compute our example, we refer to the setup of [15, sec. 3]; that is, we set C = C(A 5 ), the cluster category of Dynkin type A 5 . Thus, the indecomposables of C can be identified with the diagonals on a regular 8-gon. As in [15] we will denote by {a, b} the indecomposable corresponding to the diagonal connecting the vertices a and b. We use the same polygon triangulation as in [15, sec. 3]; that is, indec R corresponds to the red diagonals in Figure 5 and indec S corresponds to the blue diagonals. Hence, T contains the following indecomposable objects = v + z −1 where (1) is by substituting the values from Equations (6.5) and (6.6), and (2) is due to the definition of ε from Equation (6.1). We notice here that this is indeed the same result for ρ({4, 6}) obtained in [15, ex. 3.5].
Remark 6.1. In general, the formula from Theorem 5.2 can be applied iteratively to calculate ρ(m). Indeed, ρ(r)ρ(m) = ρ(a) + ρ(b) is an iterative formula on m, and hence, calculating ρ of each indecomposable in C can be reduced to calculating ρ of the indecomposables in C whose corresponding diagonals in the (n + 3)-gon do not cross any of the diagonals in R. Namely, it is clear from Figure 4 that each of a 1 , a 2 , b 1 and b 2 sit inside "smaller" polygons than m. Here, the smaller polygons are those obtained from r dissecting the (n+3)-gon.
Since R consists of only non-crossing diagonals, the remaining diagonals in R sit inside these smaller polygons. Reapplying Theorem 5.2 to each of a 1 , a 2 , b 1 and b 2 will again create a series of even smaller polygons, containing a new a or b. After repeated iterations, this process will eventually terminate at the stage when the new a or b does not cross any of the diagonals in R. Now, in the case when a diagonal, say m ′ , does not cross a diagonal in R, calculating ρ(m ′ ) is acheived by calculating α(m ′ ). This is clear since G(m ′ ) = 0. Computing α(m ′ ) is done by calculating the index of m ′ , and then applying the maps Q and ε. Hence, finding ρ(m) for each m ∈ indec C can be reduced by Theorem 5.2 to computing the index of each of the indecomposables in C whose corresponding diagonals do not cross any diagonals in R.