THE BOUNDED DERIVED CATEGORY OF A POSET

KOSMAS DIVERIS, MARJU PURIN, AND PETER WEBB

Abstract. We introduce a new combinatorial condition on a subinterval of a poset P (a clamped subinterval) that allows us to relate the Auslander-Reiten quiver of the bounded derived category of P to that of the subinterval. Applications include the determination of when a poset is fractionally Calabi-Yau and the computation of the Auslander-Reiten quivers of both the bounded derived category and of the module category.

1. Introduction

We study representations of a finite poset P over a field k. By this we mean representations of the incidence algebra kP; they may also be identified as functors from P to the category of vector spaces over k.

Such representations have been an object of study since at least the pioneering work of Mitchell [11, 12]. A notable result was the determination of the posets of finite representation type by Loupias [10]. More recent work has centered around the abstract properties of these representations, including the question of when the bounded derived category of an incidence algebra is fractionally Calabi-Yau [3, 7]. Examples show that such questions do not seem to have a particularly straightforward answer in terms of the combinatorics of the poset. Our main goal in this paper is to introduce a technique that allows us to describe the representation theory of a wide class of posets exactly by such combinatorial means.

We develop a new method, which we call clamping, to describe the Auslander-Reiten quiver components of kP in both the module category and the derived category. We show that the presence of a subposet of P of a kind that we call a clamped interval allows us to copy parts of the Auslander-Reiten quiver of the derived category of the clamped interval directly into the Auslander-Reiten quiver for kP. When the poset is built up entirely from clamped intervals it allows us to compute the shape of the Auslander-Reiten quiver of D^{b}(kP) very quickly, simply from the combinatorics of the poset. In this situation, it turns out that kP is derived equivalent to a hereditary algebra (i.e. kP is piecewise hereditary). In other cases, when we might not have such complete information, the method provides a way to show that certain bounded derived categories are not fractionally Calabi-Yau.

2010 Mathematics Subject Classification. Primary: 16G70; Secondary: 16G20, 18E30.

Key words and phrases. Auslander-Reiten quiver, poset incidence algebra, derived category, Calabi-Yau.

In the context of representations of finite dimensional algebras, we explore the connection between Auslander-Reiten triangles in the bounded derived category and Auslander-Reiten sequences in the module category, developing the fundamentals of this relationship. Applying this to posets built up out of clamped intervals, we use the approach to compute the shape of the Auslander-Reiten quiver of the module category. This allows us to construct posets with various properties: for each tree that is a barycentric subdivision, we are able to construct a lattice that is derived equivalent to the path algebra of any quiver that is an orientation of the given tree. In another example, we exhibit a lattice of finite representation type that is derived equivalent to a hereditary algebra of wild representation type; and we indicate other examples along these lines. The method provides a criterion for a poset constructed out of clamped intervals to have finite representation type, and we use this to list all such posets of finite representation type.

We present the definition and basic properties of clamping in Section 2. The main features are that restriction, induction, and coinduction between the clamped interval and the whole poset behave particularly well, allowing us to copy resolutions from the clamped interval to the whole poset, and also to copy regions of the Auslander-Reiten quiver in a similar way.

In Section 3 we use this in an approach that shows that many posets have bounded derived categories that are not fractionally Calabi-Yau.

In Section 4 we define the family of posets obtained iteratively out of clamping, denoting the class of posets obtained in this way IC. We use the results from Section 2 to describe the Auslander-Reiten quiver of the derived category for such posets, using the combinatorics of the poset. We prove that such algebras kP are piecewise hereditary and characterize the trees that arise as their ordinary quivers.

We start Section 5 by establishing a fundamental relationship between Auslander-Reiten triangles in the derived category and Auslander-Reiten sequences in the module category, in the context of representations of a finite-dimensional algebra. Although the results are straightforward, we have not so far found them in the literature, and they have broader application than this paper. This, along with clamping theory, allows us to compute Auslander-Reiten quiver components of the module categories of IC poset algebras.

We conclude in Section 6 by determining the Auslander-Reiten quivers of IC posets of finite representation type.

We assume familiarity with the methods of Auslander-Reiten theory, including the construction of Auslander-Reiten sequences and triangles. For the module category we refer to the books [1, 2] for this, and for triangulated categories we refer to [5]. For general aspects of representations of categories, and posets in particular, we refer to [15] for background.

2. Clamped intervals

We consider representations of a finite poset P over a field k and introduce the notation we will use. By a representation of P over k we mean a functor from P to the category of k-vector spaces, regarding P as a category. Such representations may be identified as modules for the category algebra (see [15]) or incidence algebra of P, which we denote by kP. Whenever Q is a subposet of P that is convex (meaning that a, b ∈ Q and a ≤ x ≤ b imply x ∈ Q) we let k_{Q} be the representation of P that takes the value k on every element of Q and that is 0 elsewhere, with identity maps corresponding to each comparison in Q. If M is a representation of P the support of M is the set of elements x ∈ P for which M(x) ≠ 0; thus the support of k_{Q} is Q. The fact that k_{Q} is a representation is part of the general observation that whenever U is a kQ-module we may regard it also as a kP-module with support in Q by letting morphisms of P that are not in Q act as zero. Thus k_{Q} is the constant functor on Q, regarded as a kP-module in this way.

For each element x ∈ P there is an indecomposable projective representation of P generated by its value at x, namely the representable functor at x. We will write it as P_{x}, or as P_{x}^{P} if we wish to emphasize the poset of which this is a representation. The corresponding indecomposable injective representation with socle at x is denoted I_{x} or I_{x}^{P}. Thus P_{x} = k_{[x,∞)} and I_{x} = k_{(−∞,x]}.

For each subposet Q of P we define functors

↑_{Q}^{P}: kQ-mod → kP-mod and ⇑_{Q}^{P}: kQ-mod → kP-mod

by U ↑_{Q}^{P}= kP ⊗_{kQ} U and U ⇑_{Q}^{P}= Hom_{kQ}(kP, U). These are the induction and coinduction to P of representations of Q. We will also make use of the corresponding functors between bounded derived categories

↑_{Q}^{P}: D^{b}(kQ) → D^{b}(kP) and ⇑_{Q}^{P}: D^{b}(kQ) → D^{b}(kP)

On an object M of D^{b}(kQ), namely a complex of kQ-modules with nonzero homology in only finitely many degrees, we define ↑_{Q}^{P} as the left derived functor

M ↑_{Q}^{P}= kP ⊗_{kQ}^{L} M

and ⇑_{Q}^{P} as the right derived functor

M ⇑_{Q}^{P}= R Hom_{kQ}(kP, M).

The way these functors are computed on a complex M is that we take a complex of projective modules isomorphic to M (a projective resolution) and apply ↑_{Q}^{P} term by term to get M ↑_{Q}^{P}. To get M ⇑_{Q}^{P} we take a complex of injective modules isomorphic to M (an injective resolution) and apply ⇑_{Q}^{P} term by term. In both the module and derived category cases, ↑_{Q}^{P} is the left adjoint of the restriction ↓_{Q}^{P}, and ⇑_{Q}^{P} is the right adjoint of ↓_{Q}^{P}. We say that a module (resp. complex) M is induced from Q if it has the form N ↑_{Q}^{P} for some module (resp. complex) N ∈ D^{b}(kQ) and coinduced from Q if it has the form N ⇑_{Q}^{P} for some module (resp. complex) N ∈ D^{b}(kQ).

We now come to an important new definition that specifies the way an interval may be positioned in a poset. Given elements a ≤ b in a finite poset P, we say that the closed interval [a, b] is clamped in P if a ≤ x implies either x ≤ b or b ≤ x, and y ≤ b implies either a ≤ y or y ≤ a. For example, in the following posets the interval [a, b] is clamped in the first poset, but no interval apart from single points or the whole poset is clamped in the second one.

\begin{matrix}a \\ b \end{matrix}

Our convention when drawing Hasse diagrams of posets is that smaller elements are positioned above larger elements, which is the opposite of the usual convention. We do this because when drawing diagrams of representations it is usual to put elements that generate a module at the top of the module.

We will show that the representation theory of a clamped interval reappears in the representation theory of P in a rather transparent way. We will start by considering the implications of this condition for the derived category, because in this setting induction and coinduction surprisingly coincide on the interior of a clamped interval. We write half-open intervals in P as [a, b) and (a, b]. We mention that some parts of the next proposition follow from Section IX.9 of Mitchell’s text [11], since clamped intervals are normal subsets in Mitchell’s terminology.

Proposition 2.1. Let the interval [a, b] be clamped in the poset P. Let M be a complex of kP-modules with homology supported on [a, b) and let N be a complex of kP-modules with homology supported on (a, b].

(1) The minimal projective resolution Π_{M} of M over kP only has terms P_{x} with x ∈ [a, b], and is induced from [a, b].

(2) The adjunction counit M ↓_{[a,b]}↑^{P}→ M is an isomorphism, so that M is induced from [a, b] in D^{b}(kP).

(3) The minimal injective resolution I_{N} of N over kP only has terms I_{x} with x ∈ [a, b], and is coinduced from [a, b].

(4) The adjunction unit N → N ↓_{[a,b]}⇑^{P} is an isomorphism, so that N is coinduced from [a, b] in D^{b}(kP).

(5) Induction and coinduction of complexes with homology supported on (a, b) are naturally isomorphic.

Note that for any convex subposet Q of P it is always the case that the unit L → L ↑^{P}↓_{Q} and the counit L ⇑^{P}↓_{Q}→ L are isomorphisms.

Proof. For the description and properties of projective kP-modules that we use here see [15], especially Proposition 4.4 in that exposition. Let Π_{M}^{[a,b]} → M ↓_{[a,b]}^{P} be a minimal projective resolution of M ↓_{[a,b]}^{P} over k[a, b]. The projective modules that appear in Π_{M}^{[a,b]} have the form P_{x}^{[a,b]} with x ∈ [a, b]. In the derived category,

M ↓_{[a,b]}^{P}↑_{[a,b]}^{P}≅ Π_{M}^{[a,b]} ↑_{[a,b]}^{P}= kP ⊗_{k[a,b]} Π_{M}^{[a,b]}

is isomorphic to a complex of projectives with summands of the form P_{x}^{P} with x ∈ [a, b], as representations of P, because P_{x}^{[a,b]} ↑_{[a,b]}^{P}≅ P_{x}^{P}. Furthermore P_{x}^{[a,b]}(z) ≅ P_{x}^{P}(z) for all z ∈ [a, b]. Hence on evaluation at b we have that Π_{M}^{[a,b]} ↑_{[a,b]}^{P} (b) = Π_{M}^{[a,b]}(b) is acyclic, because H_{∗}(M) is zero at b. Also, Π_{M}^{[a,b]} ↑_{[a,b]}^{P} is only non-zero on elements y ≥ a, in which case either y ≤ b, in which case Π_{M}^{[a,b]} ↑_{[a,b]}^{P} (y) = Π_{M}^{[a,b]}(y),or y ≥ b because [a, b] is clamped. In the latter case the unique morphism b → y induces an isomorphism

Π_{M}^{[a,b]} ↑_{[a,b]}^{P} (b) → Π_{M}^{[a,b]} ↑_{[a,b]}^{P} (y),

because the same is true for each P_{x}^{P} with x ∈ [a, b], and thus

Π_{M}^{[a,b]} ↑_{[a,b]}^{P} (y)

is acyclic. We conclude from all this that Π_{M}^{[a,b]} ↑_{[a,b]}^{P} is a projective resolution of M over kP, and the counit of the adjunction

Π_{M}^{[a,b]} ↑_{[a,b]}^{P}→ M

provides the isomorphism. The resolution is minimal because Π_{M}^{[a,b]} is. This proves (1) and (2). The proof of (3) and (4) is dual, and (5) follows by combining these results. □

We examine the effect of clamping on Auslander-Reiten triangles in the derived category. Since the incidence algebras of finite posets have finite global dimension, by [11, IX.10.3] for example, it follows that Auslander-Reiten triangles always exist in the bounded derived category, by [5], and they are constructed using the Serre functor ν on D^{b}(kP) which is the left derived functor of the Nakayama functor, also written ν.

Proposition 2.2. Let the interval [a, b] be clamped in the poset P and let M be a complex of kP-modules. Let ν be the Serre functor on D^{b}(kP). Then M has homology supported on [a, b) if and only if νM has homology supported on (a, b]. Furthermore, if this is the case and M is assumed to have homology supported on [a, b), then (νM) ↓_{[a,b]}^{P}≅ ν(M ↓_{[a,b]}^{P}). Dually, if N is a complex whose homology is supported on (a, b], then (ν^{−1}N) ↓_{[a,b]}^{P}≅ ν^{−1}(N ↓_{[a,b]}^{P}).

Proof. Suppose that M has non-zero homology only at elements of [a, b). The Serre functor ν is obtained by applying the Nakayama functor term by term to the projective resolution Π_{M} of M, replacing each indecomposable projective by the indecomposable injective with the same socle. Assume that the resolution is minimal, so that by Proposition 2.1 the projective modules that appear have the form P_{x} with x ∈ [a, b]. We obtain the isomorphism (νΠ_{M})(a) ≅ Π_{M}(b) because these projectives have the property (νP_{x})(a) = I_{x}(a) ≅ k ≅ P_{x}(b), and the maps that appear in the two complexes are the same when transported via this isomorphism. Furthermore, all indecomposable injectives that appear in νΠ_{M} have the form I_{x} with x ∈ [a, b]. Since Π_{M}(b) is acyclic, so is ν(Π_{M})(a) and hence so is ν(Π_{M})(y) if y ≤ a, because the morphism y → a gives an isomorphism ν(Π_{M})(y) → ν(Π_{M})(a). This shows that ν(Π_{M}) has non-zero homology only at elements of (a, b]. The converse statement, assuming that ν(Π_{M}) has non-zero homology only at elements of (a, b], is proved dually.

By Proposition 2.1 the projectives that appear in the minimal projective resolution of M as a kP-module have exactly the same labels as those that appear in the minimal projective resolution over k[a, b], because Π_{M} is induced, and so applying the Serre functor commutes with restriction. The dual argument for N is similar. □

We write Auslander-Reiten triangles in the bounded derived category as τN → M → N → τN[1], where τ = ν[−1] is the Auslander-Reiten translate.

Corollary 2.3. Let the interval [a, b] be clamped in the poset P and let M be a complex of kP-modules with homology supported on [a, b). Let ν be the Serre functor on D^{b}(kP).

(1) For any N ∈ D^{b}(kP) we have

Hom_{D^{b}(kP)}(M, N) ≅ Hom_{D^{b}(k[a,b])}(M ↓_{[a,b]}^{P}, N ↓_{[a,b]}^{P})

(2) There is a ring isomorphism

End_{D^{b}(kP)}(M) ≅ End_{D^{b}(k[a,b])}(M ↓_{[a,b]}^{P}).

Identifying ring actions via this isomorphism we have

Hom_{D^{b}(kP)}(M, νM) ≅ Hom_{D^{b}(k[a,b])}(M ↓_{[a,b]}^{P}, νM ↓_{[a,b]}^{P})

as End_{D^{b}(kP)}(M)-modules.

(3) On M, the Auslander-Reiten translate commutes with restriction: τ(M ↓_{[a,b]}^{P}) ≅ (τM) ↓_{[a,b]}^{P}.

(4) The Auslander-Reiten triangle in D^{b}(k[a, b]) terminating at M ↓_{[a,b]}^{P} is the restriction of the Auslander-Reiten triangle in D^{b}(kP) terminating at M, and it is also the restriction of the Auslander-Reiten triangle in D^{b}(kP) starting at τM.

Proof. (1) We have

Hom_{D^{b}(kP)}(M, N) ≅ Hom_{D^{b}(kP)}(M ↓_{[a,b]}^{P}↑_{[a,b]}^{P}, N)
≅ Hom_{D^{b}(k[a,b])}(M ↓_{[a,b]}^{P}, N ↓_{[a,b]}^{P})

by the adjoint property.

(2) These both follow from (1).

(3) This follows because τ(U) = ν(U)[−1] for any object U, and both ν and the shift commute with restriction on M.

(4) The Auslander-Reiten triangles terminating at M and M ↓_{[a,b]}^{P} are related by restriction because they are both computed by taking the mapping cone of a homomorphism M → νM in the first case and M ↓_{[a,b]}^{P}→ νM ↓_{[a,b]}^{P} in the second case. This homomorphism must lie in the socle for the action of End(M), and such a homomorphism for M restricts to one for M ↓_{[a,b]}^{P} by (2). Since the mapping cone construction also commutes with restriction we obtain that the Auslander-Reiten triangle ending at M restricts as claimed. □

We now put together these technicalities. The next corollary describes the basic relationship between the Auslander-Reiten quivers of the bounded derived categories of a poset and of a clamped subinterval, showing that part of one is copied into the other.

Corollary 2.4. Let the interval [a, b] be clamped in the poset P and let Θ be the region of the Auslander-Reiten quiver of D^{b}(kP) consisting of the meshes where either the right hand term has homology supported on [a, b) or the left hand term has homology supported on (a, b]. Then all complexes in Θ have homology supported on [a, b] and restriction ↓_{[a,b]}^{P} gives an isomorphism between Θ and the corresponding region of the Auslander-Reiten quiver of D^{b}([a, b]).

Example 2.5. We illustrate Corollary 2.4 with a straightforward example. We take the poset P to be the chain

\begin{matrix}4 \\ 3 \\ 2 \\ 1 \\ P = \end{matrix}

whose representations are the same as those of the Dynkin quiver A_{4} with all arrows oriented in the same direction. The Auslander-Reiten quiver of D^{b}(kP) is familiar from [5] and part of it looks as follows.

1 2 3 4 [−1] 4 3 2 1 2 3 4 [−1] 1 2 3 [−1] 3 4 2 3 =M 1 2 2 3 [−1] 1 2 [−1] 2 3 4 =Rad P_{1} 1 2 3 =P_{1}/ Soc P_{1} 3 [−1] 2 [−1] 1 [−1] 1 2 3 4 =P_{1} 4 [1]

We consider the clamped interval [2, 3]. The region Θ described in Corollary 2.4 consists of infinitely many triangles, of which two are shown shaded. These triangles make up the quiver of D^{b}(k[2, 3]) and are copied into the quiver of D^{b}(kP). We will see in Proposition 4.1 exactly how these triangles are glued in to the bigger quiver. For future reference we have also labeled terms in the Auslander-Reiten triangle containing the indecomposable projective P_{1}.

In the next sections we will use this isomorphism between parts of the Auslander-Reiten quiver for a poset and a clamped interval in two different ways. Our first application will be to give a criterion in the next section for the bounded derived category of a poset to fail to be fractionally Calabi-Yau. After that we will show how this approach can be used to construct the Auslander-Reiten quivers of both the bounded derived categories and of the module categories of posets.

3. Posets that are fractionally Calabi-Yau

We say that a triangulated category is Calabi-Yau if some power of the shift functor is a Serre functor ν. More generally, we say that the category is fractionally Calabi-Yau if it has a Serre functor ν and ν^{n} = [m] for some powers m, n of the shift functor and the Serre functor. Algebras whose bounded derived categories are fractionally Calabi-Yau have received attention recently in the work of several authors, and we mention [3], [7], [8] and [13] as examples.

In the context of the bounded derived category of a finite dimensional algebra Λ the Auslander-Reiten translate on D^{b}(Λ) has the form τ(M) = ν(M)[−1] so that, when Auslander-Reiten triangles exist, D^{b}(Λ) is fractionally Calabi-Yau if and only if the Auslander-Reiten quiver is periodic up to some power of the shift. This possibility was considered by Scherotzke in [13, Theorem 4.13], and she showed that if a stable Auslander-Reiten component of D^{b}(Λ) contains a shiftperiodic complex, then its tree class is either a finite Dynkin diagram or A_{∞}. Furthermore, in the Dynkin case this is the only Auslander-Reiten component of D^{b}(Λ), and D^{b}(Λ) has ‘finite representation type’. This means that there are only finitely many indecomposable objects in D^{b}(Λ) up to shift, and it implies that Λ itself has finite representation type. It is interesting to understand which algebras of infinite representation type have the fractional Calabi-Yau property.

Using Scherotzke’s result, we present the following as a corollary of Corollary 2.4.

Corollary 3.1. Let P be a poset for which kP has infinite representation type and which contains a clamped subinterval Q = [a, b]. Suppose there is an object of D^{b}(kQ) represented by a complex M with homology supported only in [a, b) such that the Auslander-Reiten triangle terminating in M has more than two middle terms. Then D^{b}(kP) is not fractionally Calabi-Yau.

The same conclusion holds if there is a complex M with homology supported only in (a, b] if the Auslander-Reiten triangle starting in M has more than two middle terms.

Proof. Suppose to the contrary that D^{b}(kP) is fractionally Calabi-Yau and consider the Auslander-Reiten component of D^{b}(kP) containing M. By Scherotzke’s theorem [13, 4.13] we conclude that the tree class of the component must be A_{∞}, since kP has infinite representation type. Take Q = [a, b] in Proposition 2.2. Since the Auslander-Reiten triangle terminating in M has more than two indecomposable middle terms, part (4) of Proposition 2.2 says that D^{b}(kP) also has an Auslander-Reiten triangle with more than two indecomposable middle terms and so the tree class cannot by A_{∞}. This contradiction allows us to conclude that D^{b}(kP) is not fractionally Calabi-Yau. □

Example 3.2. Let P be a poset for which kP has infinite representation type. Suppose that one of the following three intervals Q = [a, b] is clamped in P:

\begin{matrix}b \\ a \\ 1 \\ 2 \\ Poset 1 \end{matrix}  \begin{matrix}b \\ a \\ 1 \\ 2 \\ 3 \\ 4 \\ Poset 2 \end{matrix}  \begin{matrix}b \\ a \\ 5 \\ 6 \\ 3 \\ 4 \\ 1 \\ 2 \\ Poset 3 \end{matrix}

We will show in each case that D^{b}(kP) is not fractionally Calabi-Yau. We apply Corollary 3.1 using a certain kP-complex M with homology supported in [a, b), and in each case the complex will be a module in degree 0. As before, let P_{x} denote the indecomposable projective representation of kP generated at vertex x. When P contains the clamped interval described by the first diagram, take M = P_{a}/P_{b}. For the second diagram take M = Rad P_{a}/P_{b}. In the third case we take M to be the indecomposable middle term of the Auslander-Reiten triangle terminating in the indecomposable complex P_{6} → P_{3}. Then M

is a kP-module of the form \begin{aligned}& 0 \\ k & & k^{2} \\ & k^{2} & & k^{2} \\ & & k & & k \\ & & & 0 \end{aligned} which we will denote as [ \begin{matrix}1 & 0 \\ 2 & 2 \\ 1 & 2 \\ 0 & 1 \end{matrix} ].

By Proposition 2.2, and in each case, the Auslander-Reiten triangles over kP and kQ terminating at M are the same. Furthermore, one calculates over kQ that in each case the Auslander-Reiten triangle terminating at M has three indecomposable middle terms. In particular the following are Auslander-Reiten triangles:

(Poset 1)  Rad P_{a} → P_{a} ⊕ k_{1} ⊕ k_{2} → P_{a}/P_{b} → Rad P_{a}[1]

(Poset 2)

( \begin{matrix}P_{b} \\ ↓ \\ P_{1}⊕P_{2} \end{matrix} ) → Rad P_{a} ⊕ P_{1}/P_{b} ⊕ P_{2}/P_{b} → (Rad P_{a})/P_{b} → ( \begin{matrix}P_{b} \\ ↓ \\ P_{1}⊕P_{2} \end{matrix} ) [1].

In the above triangle the two middle terms are modules in degree 0, and the two end complexes (which are also isomorphic to shifts of modules) have their lowest shown terms in degree 0.

For the third poset, the calculations become more involved but we again find an Auslander-Reiten triangle consisting of modules in degree 0. We will use dimension vectors of representations to simplify the notation. The following describes the first three terms of an Auslander-Reiten triangle in D^{b}(kQ) with three indecomposable middle summands:

(Poset 3) [ \begin{matrix}1 & 0 \\ 2 & 1 \\ 1 & 2 \\ 0 & 1 \end{matrix} ] → [ \begin{matrix}1 & 0 \\ 1 & 1 \\ 1 & 2 \\ 0 & 1 \end{matrix} ] ⊕ [ \begin{matrix}0 & 0 \\ 1 & 1 \\ 0 & 1 \\ 0 & 1 \end{matrix} ] ⊕ [ \begin{matrix}1 & 0 \\ 2 & 1 \\ 1 & 1 \\ 0 & 0 \end{matrix} ] → [ \begin{matrix}1 & 0 \\ 2 & 2 \\ 1 & 2 \\ 0 & 1 \end{matrix} ] → [ \begin{matrix}1 & 0 \\ 2 & 1 \\ 1 & 2 \\ 0 & 1 \end{matrix} ] [1].

In fact, for each of the three posets in our example the Auslander-Reiten quiver of D^{b}(kQ) is a single component of Dynkin type D_{4}, E_{6}, and E_{8}, respectively. An application of Corollary 3.1 then shows that kP is not fractionally Calabi-Yau.

It is interesting to note that continuing this pattern and taking four stacked boxes for the poset Q, does not yield a restriction on whether D^{b}(kP) is fractionally Calabi-Yau. This is because the Auslander-Reiten triangles for kQ have at most two middle terms.

Example 3.3. As another example, we use clamping theory to show that incidence algebras of posets P that have the following posets Q as clamped intervals are not fractionally Calabi-Yau.

\begin{matrix}ω \\ β_{1} \\ β_{2} \\ β_{r−1} \\ α \end{matrix}  \begin{matrix}ω \\ β_{1} \\ β_{2} \\ β_{r−1} \\ α \end{matrix}

For the posets Q corresponding to these diagrams, the incidence algebra kQ is derived equivalent to a canonical algebra of Euclidean type [9, 1.1]. Canonical algebras of Euclidean type are a class of finite-dimensional algebras of global dimension at most 2 that have an Auslander-Reiten quiver component whose section is a Euclidean diagram [14]. They are of special interest as they are of minimal infinite representation type in the sense of Happel-Vossieck [6]. More precisely, the posets Q corresponding to the diagram on the left have incidence algebras that are derived equivalent to canonical algebras that are denoted in the literature as C(2, 2, r) where 2 ≤ r. These are said to be of Euclidean type D̃_{r+2}. The posets corresponding to the diagram on the right on the other hand have incidence algebras that are derived equivalent to canonical algebras C(2, 3, r) where r = 3, 4, 5. These are of Euclidean type Ẽ_{i} for i = 6, 7, 8 respectively. We refer the reader to [14] for further background on these algebras.

Whenever one of the above posets Q is clamped in a poset P, an application of Corollary 3.1 shows that the incidence algebra kP is not fractionally Calabi-Yau.

4. Iterated clamping

In this section and the next we turn our attention to posets constructed by repeatedly clamping intervals in an inductive process. It turns out that the posets constructed in this way are all piecewise hereditary, meaning that they are derived equivalent to hereditary algebras. We are able to specify combinatorially the tree of the corresponding hereditary algebra and hence determine the entire Auslander-Reiten quiver of D^{b}(kP). We are also able to reverse-engineer this process and construct posets that are piecewise hereditary with a given tree class for many trees, including all trees that are barycentric subdivisions.

To this end we define a class of posets IC recursively as follows. All of the posets will have a unique minimal element α and a unique maximal element ω. We define IC_{0} to contain only the poset with a single element: • α = ω. Now supposing that n ≥ 1 and that IC_{n−1} has been defined, we define IC_{n} to consist of the posets that have a unique minimal element α and a unique maximal element ω so that the open interval (α, ω) is a disjoint union of posets from IC_{n−1}:

(α, ω) = P_{1} ⊔ ⋯ ⊔ P_{s}, s ≥ 0, P_{i} ∈ IC_{n−1}.

The interpretation of s = 0 is that the disjoint union could be empty, in which case the poset consists either of a single point, or of two points that are comparable. We easily see that IC_{n−1} ⊂ IC_{n} always, arguing by induction and starting with IC_{0} ⊂ IC_{1}. We define

IC = ⋃_{i=0}^{∞} IC_{n}.

The posets in IC are obtained by Iterative Clamping. Thus, for example, posets such as

\begin{matrix}ω \\ z \\ a \\ α \end{matrix}

lie in IC but posets 2 and 3 in Example 3.2 and the posets in Example 3.3 do not. The labels in this picture are relevant for Proposition 4.1. We observe that the posets in IC are all lattices.

The next result applies to posets in IC, and also more generally. It identifies the form of certain Auslander-Reiten triangles and shows how the part of the Auslander-Reiten quiver of D^{b}(kP) identified in Corollary 2.4 is glued on to the rest of the quiver. In our notation we will identify a kP-module M with the complex of kP-modules whose only term is M, concentrated in degree 0. An example illustrating this glueing process was given as Example 2.5.

Proposition 4.1. Let P be a poset with a unique minimal element α and a unique maximal element ω, with α ≠ ω.

(1) There are Auslander-Reiten triangles

I_{α}[−1] → Rad P_{α} → P_{α} → I_{α},
P_{α} → P_{α}/ Soc P_{α} → Soc P_{α}[1] → P_{α}[1]

and

Rad P_{α} → P_{α} ⊕ (Rad P_{α}/ Soc P_{α}) → P_{α}/ Soc P_{α} → Rad P_{α}[1].

(2) Suppose, further, that there is a clamped interval [a, z] in P where a covers α and ω covers z. Let M = P_{a}/P_{ω} = k_{[a,z]} in the notation used previously. Then there are Auslander-Reiten triangles

k_{{P−{α,a}}} → Rad P_{α} ⊕ Rad M → M → k_{{P−{α,a}}}[1]

and

M → P_{α}/ Soc(P_{α}) ⊕ M/ Soc(M) → k_{{P−{ω,z}}} → M[1].

Proof. (1) We establish that these are Auslander-Reiten triangles by direct calculation. The AR triangle ending at P_{α} is obtained by completing to a triangle the map P_{α} → I_{α} that sends the simple top to the simple socle, and rotating. The mapping cone of this map is isomorphic to Rad P_{α}[1]. The AR triangle starting at P_{α} is obtained similarly, noting that P_{α} = I_{ω}. Note also that Soc P_{α} = P_{ω}.

For the second triangle, the module P_{α}/ Soc P_{α} is isomorphic to a complex P_{ω} → P_{α}, and applying the Nakayama functor we get I_{ω} → I_{α} ≅ (Rad P_{α})[1]. Thus there is an Auslander-Reiten triangle with end terms Rad P_{α} and P_{α}/ Soc P_{α} as claimed, and whose middle term is the mapping cone of a map (P_{ω} → P_{α})[−1] → (I_{ω} → I_{α})[−1]. This mapping cone is a complex P_{ω} → P_{α} ⊕ I_{ω} → I_{α}, which is seen to be non-zero only in degree 0. Note that P_{α} ≅ I_{ω} is both projective and injective and that P_{ω} and I_{α} are both simple. Because of this the mapping cone is the direct sum of a diagonally embedded copy of P_{α} ≅ {(x, −x) | x ∈ P_{α}} and a complex (P_{ω} → P_{α} → I_{α}) ≅ Rad P_{α}/ Soc P_{α}, with both modules in degree 0.

(2) Suppose that there is a clamped interval [a, z] in P as in the statement of the proposition and write N = k_{{P−{α,a}}}. Let M = P_{a}/P_{ω} = k_{[a,z]}. We know that M is isomorphic to the complex P_{ω} → P_{a} so that ν(M) = I_{ω} → I_{a} = N[1] is the shift of N. We also know that there is an Auslander-Reiten triangle Rad P_{α} → P_{α} ⊕ M ⊕ U → P_{α}/ Soc P_{α} → Rad P_{α}[1] for some module U by what we already proved, since M is a summand of Rad P_{α}/ Soc P_{α}. Thus there is an irreducible morphism Rad P_{α} → M. There is also an irreducible morphism Rad M → M because the Auslander-Reiten quiver of [a, b] is copied in the Auslander-Reiten quiver of P by Corollary 2.4, and by part (1) of this proposition applied to [a, b].

The mapping cone C of M[−1] → νM[−1] has non-zero homology only in degree 0 by an argument already used in part (1), so it is equivalent to a module, and the long exact sequence in homology of the triangle νM[−1] → C → M → νM becomes a short exact sequence 0 → N → H_{0}(C) → M → 0 which is the start of the Auslander-Reiten triangle with M on the right. We have seen that Rad P_{α} and Rad M are summands of H_{0}(C). Since

dim M + dim N = dim Rad P_{α} + dim Rad M

we have H_{0}(C) = Rad P_{α} ⊕ Rad M.

The argument for the Auslander-Reiten triangle starting at M is similar. □

By a slice of a component of an Auslander-Reiten quiver we mean a maximal connected subgraph with the property that, for each vertex x, at most one element of {x, τx} belongs to the subgraph. So that we can apply an inductive argument to posets obtained by iterative clamping, we consider posets satisfying the following technical condition:

Hypothesis 4.2. The poset P has a unique minimal element a and a unique maximal element z. Furthermore, the component of the Auslander-Reiten quiver of D^{b}(kP) that contains the largest projective module P_{a} has a slice that consists of P_{a} and some other modules (regarded as complexes concentrated in degree 0). Apart from P_{a}, none of these modules have z in their support.

The next result contains the inductive step that shows that posets in IC satisfy Hypothesis 4.2.

Proposition 4.3. Let P be a poset with a unique minimal element α and a unique maximal element ω such that the open interval

(α, ω) = [a_{1}, z_{1}] ⊔ ⋯ ⊔ [a_{n}, z_{n}]

is a disjoint union of closed intervals. Suppose that, for each i, the Auslander-Reiten quiver component of D^{b}(k[a_{i}, z_{i}]) that contains the projective P_{a_{i}} has a slice S_{i} with tree class T_{i} satisfying Hypothesis 4.2.

(1) Then P also satisfies Hypothesis 4.2 with the slice

S = S_{1} ∪ ⋯ ∪ S_{n} ∪ {P_{α}, P_{α}/ Soc P_{α}}.

(2) The tree class T of the Auslander-Reiten quiver component of D^{b}(kP) that contains the largest projective P_{α} is obtained by gluing the T_{i} to the ends of a graph that is a star consisting of n + 1 edges joined at a single point. One of these edges has nothing glued to it, and the remaining edges have the T_{i} glued to them at the point where the projective P_{a_{i}} was fastened.

(3) If the direct sum of modules in each slice S_{i} is a tilting complex U_{i} for the interval [a_{i}, z_{i}], then the direct sum of modules in the slice S is a tilting complex U for kP.

(4) If each ring End_{D^{b}(k[a_{i},z_{i}])}(U_{i}) is hereditary then so is End_{D^{b}(kP)}(U), and so kP is piecewise hereditary.

Proof. By Proposition 4.1 there is an Auslander-Reiten triangle in D^{b}(kP) with terms

Rad P_{α} → P_{α} ⊕ k_{[a_{1},z_{1}]} ⊕ ⋯ ⊕ k_{[a_{n},z_{n}]} → P_{α}/P_{ω} → Rad P_{α}[1].

By Hypothesis 4.2, for each i we have a slice S_{i} that includes the module k_{[a_{i},z_{i}]} and whose remaining terms do not have z_{i} in their support. By Proposition 2.2 the Auslander-Reiten triangles in D^{b}(k_{[a_{i},z_{i}]}) that have these remaining terms on the right remain Auslander-Reiten triangles for kP. This means that the slice S_{i} remains part of a slice for kP.

By Proposition 4.1 it is glued to the rest of the quiver in the manner described in (2), where we include the modules P_{α} and P_{α}/ Soc(P_{α}) in the slice. Again by Proposition 4.1, the slice is complete. The only term in this slice that has ω in its support is P_{α}, so that P satisfies Hypothesis 4.2. This proves parts (1) and (2).

For the proof of (3) we first show that the slice just constructed has the property that, for every pair of modules M, N in it, Hom_{D^{b}(kP)}(M, N[i]) = 0 if i ≠ 0. We consider several cases. Suppose first that M ∈ S_{r} and N ∈ S_{s} with M supported on [a_{r}, z_{r}). Then by Proposition 2.1 M is induced from k[a_{r}, z_{r}] and

Hom_{D^{b}(kP)}(M, N[i]) ≅ Hom_{D^{b}(k[a_{r},z_{r}])}(M ↓_{[a_{r},z_{r}]}^{P}, N[i] ↓_{[a_{r},z_{r}]}^{P}).

If r ≠ s then N[i] ↓_{[a_{r},z_{r}]}= 0, so this Hom group is 0. If r = s then the Hom group is 0 if i ≠ 0 from the hypothesis on S_{r}.

We next consider the case when M = k_{[a_{r},z_{r}]} is the largest projective module for k[a_{r}, z_{r}]. As an object in D^{b}(kP) it is isomorphic to the complex P_{ω} → P_{a_{r}}. Because this projective resolution has terms only in degrees 0 and 1, if N is any kP-module then Hom_{D^{b}(kP)}(M, N[i]) = 0 unless i is 0 or 1, and if i = 1 and there is a non-zero homomorphism then the simple module P_{ω} lies in Soc N. When N lies in the slice under consideration this means N = P_{α}, which is injective as well as projective, so Hom_{D^{b}(kP)}(M, P_{α}[1]) = 0 in this case. This shows that for this M we have Hom_{D^{b}(kP)}(M, N[i]) = 0 when N lies in the slice and i ≠ 0.

The argument is similar when M = P_{α}/ Soc P_{α}. This module is isomorphic to the complex P_{ω} → P_{α} so that for any kP-module N we have Hom_{D^{b}(kP)}(M, N[i]) = 0 unless i is 0 or 1, and if i = 1 and there is a non-zero homomorphism then the simple module P_{ω} lies in Soc N. For N in the slice this again implies that N = P_{α}, which is injective, so Hom_{D^{b}(kP)}(M, P_{α}[1]) = 0.

We have now accounted for Hom_{D^{b}(kP)}(M, N[i]) in all cases except when one of M or N is P_{α}. As already observed, this module is both projective and injective, so in these remaining cases the Hom group is 0 if i ≠ 0.

We next show that D^{b}(kP) is generated by the modules in S. For this it is sufficient to show that the injective kP-modules lie in the subcategory generated by S. Evidently P_{α} is one of these. If I is any other injective kP-module there is a surjection P_{α}/ Soc P_{α} → I whose kernel is a direct sum of modules, each of which is a k[a_{i}, z_{i}]-module for some i. We know that the modules in the S_{i} generate such modules by hypothesis. This shows that the modules in S generate D^{b}(kP).

Finally we observe that the direct sum of modules in S is a perfect complex since kP has finite global dimension.

(4) The only non-zero homomorphisms between P_{α}, P_{α}/ Soc P_{α} and members of S are the endomorphisms of these modules, a 1-dimensional space of homomorphisms P_{α} → P_{α}/ Soc P_{α}, and a 1-dimensional space of homomorphisms k_{[a_{i},z_{i}]} → P_{α}/ Soc P_{α} for each i. There are no other homomorphisms than these because P_{α} has ω as its socle and no other module in S has ω as a composition factor, P_{α} and P_{α}/ Soc P_{α} have α as their top and no other modules have α as a composition factor, and P_{α}/ Soc P_{α} has z_{1}, …, z_{n} forming its socle, while none of the modules in S except P_{α}, P_{α}/ Soc P_{α} and k_{[a_{i},z_{i}]} have such a composition factor. The same observations show that dim End_{D^{b}(kP)}(P_{α}) = dim End_{D^{b}(kP)}(P_{α}/ Soc P_{α}) = 1. This analysis of the homomorphisms shows, by induction, that End_{D^{b}(kP)}(U) is the path algebra of the quiver obtained by joining a star with center corresponding to P_{α} and arms • → • ← • to the quivers coming by induction from S_{i}. □

Proposition 4.3 was an inductive step that we use to describe part of the Auslander-Reiten quiver of posets obtained by iterative clamping. We already see it illustrated in Example 2.5, where the quiver of the poset with tree class A_{4} is obtained from the clamped interval with tree class A_{2} by joining on two more edges.

It is convenient to refer to the wing of a quiver component determined by a vertex x in that component. By this we mean the region of the quiver component whose vertices are those that can appear in some slice together with x. It is bounded on one side by the vertices y for which there is a chain of irreducible morphisms y = y_{0} → y_{1} → y_{2} → ⋯ → y_{n} = x with y_{i} ≠ τy_{i+2} always, and on the other side by vertices y for which there is a chain of irreducible morphisms x = y_{0} → y_{1} → y_{2} → ⋯ → y_{n} = y satisfying the same condition.

Corollary 4.4. Let P ∈ IC be a poset obtained by iterative clamping. Then

(1) P satisfies Hypothesis 4.2.

(2) kP is piecewise hereditary.

(3) The component of the Auslander-Reiten quiver of D^{b}(kP) that contains the largest projective P_{α} has the form ℤT where the tree T is obtained inductively by taking a star with an edge for each interval clamped at the top level, plus an extra edge whose free vertex is labelled with P_{α}, and joining to each of the remaining free vertices the trees for the clamped intervals, joining at the vertices corresponding to the largest projectives.

(4) The algebra kP is derived equivalent to the path algebra of any quiver whose underlying graph is the tree T.

(5) The wing of the quiver component determined by P_{α} consists entirely of modules.

Proof. The proof is by induction on n, where P ∈ IC_{n}. When n = 0 the poset consists of just a single point, and the result is immediate because kP = k is a field. Parts (1), (2), (3) and (4) now follow using Proposition 4.3 as the induction step. The argument is that we obtain a slice of the quiver component containing P_{α} with underlying tree T, the direct sum of whose terms is a tilting complex with endomorphism ring a hereditary algebra, necessarily of tree class T. Thus the Auslander-Reiten quivers of the bounded derived categories of kP and this hereditary algebra are isomorphic. It follows that the quiver component for kP containing P_{α} is ℤT (rather than some quotient of this) because of the description of the quiver component containing projectives of a hereditary algebra given in [5].

The argument that proves (5) also proceeds by induction. We merely sketch how it goes, to avoid some of the technicalities. The wings of all of the largest projectives of the clamped intervals, taken in the quivers of those intervals, consist entirely of modules, by induction; and by an application of Corollary 2.4 (after the conditions on the support of these modules are checked) are copied into the quiver component of P_{α}. Note that none of these modules have α or ω in their support, and so cannot be projective or injective. The wing of P_{α} consists of all the modules just considered, together with complexes M for which there is a directed path from M to P_{α} or from P_{α} to M. The arguments in these two cases are similar, and we show by induction on the distance between M and P_{α} that M is a module. The argument is that M appears as the first or third term in an Auslander-Reiten triangle where the other two remaining terms are modules. We use the long exact sequence in homology to show that M has homology only in degree 0, using also a result we have not yet come to: Lemma 5.1. To apply that lemma we must verify that the third term of the Auslander-Reiten triangle is not projective. When M is the first term, the third term is one of the copied modules that does not have ω in its support, so the third term is not projective. If M were the third term and projective, the first term is the −1 shift of an injective and also one of the copied modules, which is not possible. □

More generally, the inductive argument just indicated shows that, in the wing of P_{α}, projectives can only appear on the ‘left boundary’ of the wing, and injectives only on the ‘right boundary’ of the wing. Thus the ‘interior’ of the wing of P_{α} contains no projectives or injectives.

We will present various applications of Corollary 4.4. The first is a determination of when kP is fractionally Calabi-Yau if P is obtained by iterative clamping.

Corollary 4.5. Let P ∈ IC be a poset obtained by iterative clamping. Then kP is fractionally Calabi-Yau if and only if the tree constructed in Corollary 4.4 is a Dynkin diagram.

Proof. This is immediate from Scherotzke’s result [13, 4.13] since for kP to be fractionally Calabi-Yau the tree class of each Auslander-Reiten component must be a Dynkin diagram or A_{∞}, and the tree constructed in Corollary 4.4 is always finite. □

Example 4.6. Let P(q_{1}, q_{2}, …, q_{t}) be the poset that consists of t chains of lengths q_{1}, q_{2}, …, q_{t}, all of which are clamped intervals lying between an initial vertex α and a terminal vertex ω. For example, the following poset has a chain of length 1 and a chain of length 2 that are both clamped intervals, lying between an initial vertex α and a terminal vertex ω.

\begin{matrix}ω \\ α \\ P(1, 2) = \end{matrix}

Then kP(q_{1}, q_{2}, …, q_{t})) is derived equivalent to the path algebra of any quiver whose underlying graph is a star, consisting of t + 1 arms of lengths 2, q_{1} + 1, q_{2} + 1, …, q_{t} + 1 joined at a single vertex. It is clear that we can obtain any star in this way provided one of the arms has length 2. (We will see in a moment how to obtain an arbitrary star by a modification of the above construction.)

For example, the Auslander-Reiten quiver component of D^{b}(P(1, 2)) that contains the largest projective module P_{α} has tree class that is a star with arms of length 2, 2 and 3. This is the Dynkin diagram D_{5}. Furthermore, kP(q_{1}, q_{2}, …, q_{t}) is fractionally Calabi-Yau if and only if (q_{1}, …, q_{t}) = (q_{1}), (1, q_{2}) or (2, 2), (2, 3), (2, 4), by Corollary 4.5. In these cases the Auslander-Reiten quiver of D^{b}(P(q_{1}, q_{2}, …, q_{t})) consists of a single component that has tree class A_{q_{1}+2}, D_{q_{2}+3}, E_{6}, E_{7}, E_{8}.

So as to obtain a greater variety of tree classes we now describe a modification of the construction of the posets in IC. The modification is to adjoin only a new minimal element α to a poset [a, z] that already has unique minimal element a and unique maximal element z. (We could equally adjoin a new maximal element ω, leaving a as the minimal element.) The interval [a, z] is clamped in the new poset and we can apply Proposition 4.1 again.

Proposition 4.7. Let P = {α}∪[a, z] be a poset that is the union of an interval [a, z] and another element α with α < a. Thus α is the unique minimal element and z is the unique maximal element. Suppose that the Auslander-Reiten quiver component of D^{b}(k[a, z]) that contains the projective P_{a} has a slice S with tree class T satisfying Hypothesis 4.2. Assume, furthermore, that P_{a}/P_{z} lies in S.

(1) Then P also satisfies Hypothesis 4.2 with the slice

S^{+} = (S − {P_{a}}) ∪ {P_{α}, P_{α}/P_{z}}.

(2) The tree class T^{+} of the Auslander-Reiten quiver component of D^{b}(kP) that contains the largest projective P_{α} is obtained from T by gluing an extra edge to the point corresponding to P_{a}.

(3) If the direct sum of modules in S is a tilting complex U for the interval [a, z], then the direct sum of modules in the slice S^{+} is a tilting complex V for kP.

(4) If End_{D^{b}(k[a,z])}(U) is hereditary then so is End_{D^{b}(kP)}(V), and so kP is piecewise hereditary.

Proof. The proof is very similar to that of Proposition 4.3. We start differently in that, by Proposition 4.1, we now have an Auslander-Reiten triangle in D^{b}(kP) with terms

P_{a} → P_{α} ⊕ (P_{a}/P_{z}) → P_{α}/P_{z} → P_{a}[1],

noting that P_{a} = k_{[a,z]}, P_{z} = Soc P_{a} = Soc P_{α}, P_{a}/P_{z} = k_{[a,z)} and P_{α}/P_{z} = k_{[α,z)}. We wish to obtain a slice consisting of modules and containing P_{α}, so that the only module in the slice with z in its support is P_{α}. Since S contains P_{a}, which has z in its support, we get rid of it and replace it by τ^{−1}P_{a} = P_{α}/P_{z}. It then suffices to adjoin one new module P_{α} to obtain a slice S^{+}. From the construction we see it satisfies Hypothesis 4.2 and condition (2).

We must now establish (3) and (4), and we leave the details of this to the reader. The techniques used are exactly the same as in the proof of Proposition 4.3 with some different modules appearing that come from the slightly different situation. □

As an example of the situation described in Proposition 4.7, the particular case of Example 2.5 shows explicitly how the tree class of the quiver when P is a chain of length 4 is obtained from the tree class of the quiver of a chain of length 3 by adding a single new vertex (as well as in the manner described in Proposition 4.3).

The construction of the class of posets IC was done so as to facilitate the inductive argument of Proposition 4.3. Evidently we can enlarge IC by allowing the further operation of Proposition 4.7, giving a class of posets IC^{+} that are also piecewise hereditary. We now show how to construct lattices in IC^{+} with certain specified tree classes. We will refer to the largest indecomposable projective of a lattice P: this is the module k_{P}.

Theorem 4.8. Let T be a finite tree with the property that whenever x, y are vertices with valence ≥ 3 then d(x, y) ≥ 2. Then there is a lattice P ∈ IC^{+} with the properties:

(1) the component of the Auslander-Reiten quiver of D^{b}(kP) containing the largest indecomposable projective kP-module has tree class T,

(2) the largest indecomposable projective kP-module may lie in the τ-orbit corresponding to any specified extreme vertex of T, and

(3) the wing of the quiver component of D^{b}(kP) determined by the largest indecomposable projective kP-module consists entirely of modules.

Proof. We proceed by induction on the number v of vertices of T of valence ≥ 3. When this number is 0, T is the Dynkin diagram A_{n} for some n. Letting P be a chain with n elements produces the desired result.

When v = 1, T is a star and we have seen in Example 4.6 lattices with the required properties.

Now let v > 1 and suppose the result is proved for smaller values of v. Let p be the extreme vertex of T that is to correspond to the largest projective k_{P} and let x be the vertex of T of valence ≥ 3 closest to p. Let the edges of T incident with x be e_{0}, e_{1}, …, e_{r}, where e_{0} lies on the path between p and x, and label the vertices of e_{i} by x and y_{i}. Removing from T the star consisting of the path from x to p and the edges e_{1}, …, e_{r}, yields a disjoint union of r trees T_{1}, …, T_{r}, and y_{i} is an extreme vertex of T_{i} for each i. By induction we can find a lattice P_{i} satisfying the condition of the theorem with tree T_{i} and largest projective k_{P_{i}} corresponding to y_{i}. Clamping P_{1}, …, P_{r} between elements α′ and ω produces a lattice P′ so that the component of the Auslander-Reiten quiver of D^{b}(kP′) containing the largest projective has tree class the subtree T′ of T that consists of T minus the edges from y_{0} to p, and with the largest projective associated to y_{0}. Finally, by adjoining a chain of elements in bijection with the vertices of T −T′ to the smallest element α′ of P′ produces a poset P with tree T as required, by Propositions 4.3 and 4.7. This completes the inductive step. The final statement about the wing of the largest projective is proved in the same way as part (4) of Corollary 4.4 □

5. The connection with the Auslander-Reiten quiver of kP-mod

We explore the connection between Auslander-Reiten triangles in the bounded derived category and in the module category. It is tempting to suppose that if we take an Auslander-Reiten triangle in D^{b}(Λ), where Λ is a finite dimensional algebra, and evaluate its homology in some dimension, we will get either a split short exact sequence, or an Auslander-Reiten sequence direct sum a split sequence. This is not true in general: the Auslander-Reiten triangle ending in a projective module has zero homology that does not produce a short exact sequence.

We start by providing a sufficient condition for the homology of an Auslander-Reiten triangle in some dimension to be an Auslander-Reiten sequence direct sum a split sequence, or split. This approach will provide us with a method to compute the Auslander-Reiten quiver of kP-mod, starting from the Auslander-Reiten quiver of D^{b}(kP). Our first results hold in the generality of representations of a finite dimensional algebra and have wider interest than just representations of posets.

We denote the Auslander-Reiten translate by τ, and for any object M we refer to the τ-orbit of M as the set of objects of the form τ^{i}(M) for some i ∈ ℤ. This is an abuse of terminology because τ may only be partially defined on a τ-orbit, so that the operation of applying τ might not be invertible.

The first lemma follows from [16, 2.1]; apart from this we have been unable to find references for it in the literature.

Lemma 5.1. Let Λ be a finite dimensional algebra and let

L → M → N → L[1]

be an Auslander-Reiten triangle in D^{b}(Λ). Assume that N is not the shift of a projective module (a complex with only one nonzero term). Then the associated long exact homology sequence is the splice of short exact sequences

0 → H_{i}(L) → H_{i}(M) → H_{i}(N) → 0

with zero connecting homomorphisms.

Proof. The homology H_{i} is a representable functor:

H_{i}(N) = Hom_{D^{b}(Λ)}(Λ[i], N).

The connecting homomorphisms in the long exact sequence of homology arising from the triangle are all zero, except when N is the shift a projective module, because of the lifting property of the Auslander-Reiten triangle, since N is never a summand of Λ[i]. □

A similar analysis shows that, when N is the shift of a projective module, the long exact sequence in homology of the Auslander-Reiten triangle is not a splice of short exact sequences. We now describe circumstances in which we obtain short homology sequences that are Auslander-Reiten sequences, up to split summands.

Proposition 5.2. Let Λ be a finite dimensional algebra and let

L → M → N → L[1]

be an Auslander-Reiten triangle in D^{b}(Λ). Suppose that N is a truncated projective resolution of some non-projective Λ-module. Thus N has at most two non-zero homology groups, one in degree 0 and possibly one more in a higher degree at the other end of the complex, and is not the shift of a projective module. Then

0 → H_{0}(L) → H_{0}(M) → H_{0}(N) → 0

is a short exact sequence of Λ-modules that is the direct sum of a split exact sequence and (possibly) an Auslander-Reiten sequence.

Proof. Exactness is proved in Lemma 5.1. Note that N is indecomposable, by the Auslander-Reiten triangle hypothesis. It follows that H_{0}(N) is indecomposable, since N is a truncated projective resolution of H_{0}(N), and this would decompose if H_{0}(N) did. We show that if f : U → H_{0}(N) is a module homomorphism that is not a split epimorphism then f lifts to a homomorphism U → H_{0}(M). Let P_{U} be the start of a projective resolution of U with the same number of non-zero terms as the complex N. Then f lifts to a homomorphism of complexes ϕ : P_{U} → N, so that H_{0}(ϕ) = f, and ϕ is not a split epimorphism because f is not a split epimorphism. Thus ϕ lifts to a homomorphism of complexes P_{U} → M by the Auslander-Reiten property. Taking zero homology we get that f lifts through H_{0}(M). From this it follows that 0 → H_{0}(L) → H_{0}(M) → H_{0}(N) → 0 is a direct sum of a split short exact sequence and (possibly) an Auslander-Reiten sequence of modules. The argument is standard: either H_{0}(M) → H_{0}(N) is split epi, or else it is right almost split, in which case it breaks up as a direct sum of a minimal right almost split map and a zero map (see [2]). In the first case the sequence is split and in the second case it is the direct sum of a split sequence and Auslander-Reiten sequence. □

Corollary 5.3. Let Λ be a finite dimensional algebra.

(1) Suppose that L → M → N → L[1] is an Auslander-Reiten triangle in D^{b}(Λ) where L, M and N are all isomorphic to modules (i.e. their only non-zero homology is in degree 0). Then the sequence of Λ-modules 0 → H_{0}(L) → H_{0}(M) → H_{0}(N) → 0 is an Auslander-Reiten sequence.

(2) If part of the Auslander-Reiten quiver of D^{b}(Λ) consists of modules, forming a union of meshes, then this part is copied into the Auslander-Reiten quiver of Λ-mod.

Proof. (1) Note that N cannot be a projective module (or the shift of a projective module) because if it were, then L ≅ νN[−1] would fail to be a module, contrary to the assumption on L. Thus, by Proposition 5.2, 0 → H_{0}(L) → H_{0}(M) → H_{0}(N) → 0 is a short exact sequence of modules that is a direct sum of a split sequence and possibly an Auslander-Reiten sequence. Since L and N are indecomposable, so are the modules H_{0}(L) and H_{0}(N), because these are the only non-zero homology modules. For the same reason, the sequence 0 → H_{0}(L) → H_{0}(M) → H_{0}(N) → 0 is not split. We deduce that the sequence is an Auslander-Reiten sequence.

(2) is an immediate consequence. □

Corollary 5.4. Let P ∈ IC be a poset obtained by iterative clamping. The wing of the quiver component of D^{b}(kP) containing the largest projective P_{α} consists entirely of modules, and is a union of meshes. A copy of this wing appears in the Auslander-Reiten quiver of kP-mod.

Proof. We have seen in Corollary 4.4 that the wing of the quiver of D^{b}(kP) determined by the largest projective consists of modules, and from the definition of the wing it is a union of meshes, except in the case when P is a single point. In that case kP = k and the result is true. It follows from Corollary 5.3 the wing is copied into the Auslander-Reiten quiver of kP-mod. □

We now show how to use the information from Corollary 5.4 to help compute the Auslander-Reiten quiver of kP-mod for a poset P obtained by iterative clamping. Note that something more general can also be done for an extended class of posets using Proposition 4.7, but we leave this to the reader. We have seen how to obtain a part of a quiver component of kP-mod, but this component will not be stable, because it contains the projective module P_{α}. To describe the whole component we consider the stable part as well as P_{α} and the τ-orbits of the finitely many other projectives and injectives that may happen to be there.

By slight extension of a notation used previously, if θ is an element of P we denote by k_{θ} the simple kP-module supported at θ, with dimension 1 at θ and 0 at other elements of P. We will make use of additive functions on the Auslander-Reiten quiver of kP-mod. Writing the largest element of P as ω, as before, let f_{ω} be the function defined on kP-modules given by f_{ω}(M) := dim_{k} Hom(P_{ω}, M), the composition factor multiplicity of k_{ω} in M. Since composition factor multiplicities are additive on short exact sequences, this function is additive on every mesh in the Auslander-Reiten quiver of kP-mod. Given a translation quiver with Auslander-Reiten translate τ, we term a τ-orbit interval a set of vertices of the form τ^{i}M with r ≤ i ≤ s for some integers r and s.

Proposition 5.5. Let P ∈ IC be a poset obtained by iterated clamping. Thus P has a smallest element α, a largest element ω, and satisfies the hypotheses of Proposition 4.3, with a slice S and underlying tree T as specified there. Then the component of the Auslander-Reiten quiver of kP-mod that contains P_{α} is a subset of ℤT. Identifying the modules in S with a copy of T in ℤT, the subset of ℤT that is the Auslander-Reiten quiver of kP-mod is a union of τ-orbit intervals, each interval containing an element of S and either infinite or terminating in a projective module to the left, or an injective module to the right.

Proof. The function f_{ω} is 0 on S except on P_{α}, where it takes the value 1. This is because of the hypothesis that the only module in S with support on ω is P_{α}. There is no mesh starting at P_{α}, because it is injective. If a new projective module P_{x} appears in this quiver component to the right of S, then it must arise via the only irreducible morphism terminating in P_{x}, namely Rad P_{x} → P_{x}. Notice that f_{ω}(Rad P_{x}) = f_{ω}(P_{x}) = 1, so that f_{ω} must already have taken the value 1 to the right of S before the projective P_{x} appears. If S has at least three elements (the case of one or two is left to the reader), there is a mesh starting at an element of S with its middle terms in S, perhaps apart from a projective. There can, in fact, be no projective, because f_{ω} is zero on the left of the mesh. Thus f_{ω} is zero at the right term of this mesh. Repeating in this way we deduce that f_{ω} is zero everywhere to the right of S and that no new projective modules P_{x} appear there. From this we see that the quiver to the right of S consists entirely of negative powers of τ applied the non-injective terms in S, with a finite τ-orbit interval to the right if it is terminated by an injective module. Because the hypothesis that P ∈ IC implies that the opposite poset P^{op} ∈ IC, we see dually that no injective can appear to the left of S and that every module in this quiver component has the form τ^{n}M for some n ∈ ℤ and M ∈ S in an interval that may be terminated on the left by a projective, and on the right by an injective. □

The calculation of Auslander-Reiten quivers of kP-mod in our examples is facilitated by the following proposition.

Proposition 5.6. (1) Suppose the poset P consists of two clamped intervals [β, δ] and [γ, ϵ] lying between an initial element α and a terminal element ω. Then the Auslander-Reiten translate on kP-mod has values τk_{ϵ} = P_{β} and τk_{δ} = P_{γ}.

(2) Whenever a chain a < b < c < ⋯ < z is clamped we have τk_{a} = k_{b}, τk_{b} = k_{c}, and so on.

Proof. (1) We calculate the effect of τ on k_{ϵ} by taking the first two terms of a projective resolution P_{ω} → P_{ϵ} → k_{ϵ} → 0 (in fact this is the complete resolution) and applying the Nakayama functor to get I_{ω} → I_{ϵ}. The kernel of this map is P_{β}, so τk_{ϵ} = P_{β}. The calculation of τk_{δ} is similar.

(2) We calculate the resolution P_{b} → P_{a} → k_{a} → 0, apply the Nakayama functor to get I_{b} → I_{a} and observe that the kernel is k_{b}. □

Example 5.7. We are ready to give a fairly elaborate example of the application of these methods. We calculate the Auslander-Reiten quivers of both D^{b}(kP) and kP-mod when P is the poset with Hasse diagram:

\begin{matrix}ω \\ θ \\ η \\ ϵ \\ δ \\ γ \\ β \\ α \\ ι \\ ζ \end{matrix}

It is remarkable that we will compute the quiver of kP-mod without computing the structure of every module in detail. We see that P is obtained by iterative clamping, starting from the intervals [δ, ϵ], [ζ, ζ] and [ι, ι] and so, according to Proposition 4.3, kP piecewise hereditary, being derived equivalent to the path algebra of a quiver with underlying tree T: which is of wild representation type. On the other hand we claim that kP is of finite representation type, and that its Auslander-Reiten quiver has shape indicated by the following diagram:

0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 0 0 0 0 1 1 1 1 1 1 2 2 0 0 0 0 0 1 1 1 1 1 1 2 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 0

In this diagram the vertical dashed line indicates that the picture continues to the right as a reflection of the part to the left of the vertical line. The numbers attached to vertices are the values of the function f_{ω}, which is not invariant under reflection in the dashed line.

To show that this is the Auslander-Reiten quiver of kP-mod, we observe that the quiver component containing P_{α} is a subset of ℤT by Proposition 5.5. By Proposition 5.6 and Corollary 5.3 the following is part of this quiver component, because it is part of the quiver of D^{b}(kP) and all the terms are modules:

k_{ϵ}  k_{δ}
↘  ↗
k_{[δ,ϵ]}
↗  ↘
Rad k_{[γ,η]} → k_{ζ} → k_{[γ,η]}/ Soc k_{[γ,η]}
↘  ↗
k_{[γ,η]}
↗  ↘
k_{[γ,θ]}  k_{[β,η]}
↘  ↗
Rad P_{γ}  ?  k_{[β,θ]}
↘  ↗  ↘  ↗  ↘
P_{γ}  Rad P_{α}→ P_{α} → P_{α}/ Soc P_{α}
↘ ↗ ↘ ↗
P_{β}  k_{ι}

We write ? to denote a module whose precise structure has not been determined. Notice that the remaining indecomposable projectives not listed in the above diagram have the property that their radical is also projective, and so the following is part of the Auslander-Reiten quiver of kP-mod:

P_{δ}
↗
P_{ϵ}
↗
P_{η} → P_{ζ}
↗
P_{θ}
↗
P_{ω}
↘
P_{ι}

From this we deduce that these projectives all lie in the same quiver component, so either they are all in the same component as P_{α} or none of them is. We conclude that the τ-orbits in the component of P_{α} that do not terminate in P_{α}, P_{β} or P_{γ} are either all infinite, or else they all terminate to the left, since there are 7 of these orbits and 7 projectives that might terminate them.

To show that the orbits are finite we consider the function f_{ω} defined before as f_{ω}(M) := dim_{k} Hom(P_{ω}, M), which is additive on every mesh. The values of f_{ω} on the part of the quiver already computed are sufficient to compute f_{ω} everywhere, by additivity. These values are shown on the quiver diagram. If τ-orbits were infinite the values shown would continue with negative values, which is not possible. We deduce that the quiver component is finite, and hence by a theorem of Auslander that kP has finite representation type and this is the entire quiver. Furthermore f_{ω} takes the value 1 on all indecomposable projectives, so the quiver must terminate to the left where there is a pattern of 1s the same as the pattern of the projectives. The only time this happens before f_{ω} becomes negative is at the left of what is shown of the quiver, and hence the quiver must terminate at the left as shown. Because P is isomorphic to its opposite poset, the quiver must terminate to the right with injectives in the same fashion.

Example 5.8. We may compute the Auslander-Reiten quivers of both D^{b}(kP) and of kP-mod by these means for many other posets P. For instance, the poset

\begin{matrix}ω \\ ϵ \\ δ \\ γ \\ β \\ α \\ ζ \end{matrix}

has finite representation type but is derived equivalent to the path algebra of a tree with underlying diagram

which is Euclidean and of tame representation type The poset

\begin{matrix}ω \\ η \\ ζ \\ ϵ \\ δ \\ γ \\ β \\ α \\ θ \end{matrix}

is of infinite representation type and has a stable component of the Auslander-Reiten quiver of kP-mod that has Euclidean tree class of type Ẽ_{6}. We leave the details to the reader.

We can extract from the above methods a criterion for a poset to be of finite representation type.

Corollary 5.9. Let P be a poset satisfying the conditions of Proposition 4.3, so that the component of the Auslander-Reiten quiver of D^{b}(kP) containing P_{α} has tree class T. If, after removing the vertex of T corresponding to P_{α}, the resulting tree is a Dynkin diagram then kP has finite representation type.

Proof. We know from Proposition 5.5 that the component of the Auslander-Reiten quiver of kP-mod that contains P_{α} is a subset of ℤT, and the infinite τ-orbits to the left of the slice S correspond to a subdiagram of T that does not include the vertex for P_{α}. This subdiagram must be a union of Dynkin trees. The function f_{ω} must be non-zero on these infinite orbits, since any adjacent finite τ-orbit must connect to an infinite orbit via an irreducible morphism of the form Rad P_{σ} → P_{σ} and f_{ω} is 1 on both these modules. We now conclude that f_{ω} must take negative values on some infinite τ-orbit, if it exists (because there is no positive additive function on ℤΔ when Δ is Dynkin), which is not possible. From this we conclude that the Auslander-Reiten quiver is finite to the left, and by a dual argument it is finite to the right. Hence the Auslander-Reiten quiver of kP has a finite component, and by a theorem of Auslander kP has finite representation type. □

We note that the converse of Corollary 5.9 is false, as is shown by Example 5.7.

6. IC posets of finite representation type

The posets of finite representation type were determined by Loupias [10], and enumerated by Drozdowski and Simson [4]. Here we illustrate how the techniques from the previous section can be used to determine the Auslander-Reiten quivers of IC posets of finite representation type. In [4] the authors organized the posets of finite representation type into families where the IC posets of finite representation type are named Rys 28, 29, and 30. We remark that while the families Rys 1, 2, and 12 are not necessarily IC posets, clamping theory can also be used to examine their Auslander-Reiten quivers.

In what follows we divide the IC posets of finite representation type into eight classes. For each class we give a slice of the Auslander-Reiten quiver of kP-mod containing the projective module P_{α} that we denote by an open circle in our diagram. While our focus is on the module category, we remark that, by Corollary 5.4, this diagram is the tree class of an Auslander-Reiten component in the derived category D^{b}(kP).

For all of these algebras, the subquiver of their Auslander-Reiten quiver consisting of only the projective modules is a disjoint union of Dynkin diagrams. For the classes (1)–(4), after removing the vertex corresponding to P_{α}, the slice of the Auslander-Reiten quivers is a Dynkin diagram. We remark that these algebras can be seen to have finite representation type by a quick application of Corollary 5.9. The remaining classes (5)–(8) behave similarly to Examples 5.7 and 5.8. That is, the Auslander-Reiten quiver has some τ-orbits that terminate quickly. These orbits correspond to the boxed vertices in our diagrams. After removing these vertices and the vertex corresponding to P_{α} the remaining τ-orbits, given as filled circles, form a Dynkin diagram.

We illustrate our notation on Example 5.7. There the slice for the Auslander-Reiten quiver of kP-mod would be the tree

THE BOUNDED DERIVED CATEGORY OF A POSET 31

We now give the promised list of IC posets of finite representation type.

Poset  Slice of Auslander-Reiten Quiver  Rys Label

p  p  \begin{matrix}28 \\ p ≥ 1 \end{matrix}

p  p  \begin{matrix}28 \\ 2 ≤ p ≤ 4 \end{matrix}

\begin{matrix}n \\ p \end{matrix}  \begin{matrix}n \\ p \end{matrix}  \begin{matrix}29 \\ n ≥ 0 \\ p ≥ 0 \end{matrix}

\begin{matrix}p \\ r \\ n \end{matrix}  \begin{matrix}p \\ n + r \end{matrix}  \begin{matrix}30 \\ n ≥ 1 \\ r ≥ 1 \\ p ≥ 1 \end{matrix}

q  q  \begin{matrix}30 \\ 1 ≤ q ≤ 4 \end{matrix}

THE BOUNDED DERIVED CATEGORY OF A POSET 33

Poset  Slice of Auslander-Reiten Quiver  Rys Label
p  p  \begin{matrix}30 \\ 1 ≤ p ≤ 3 \end{matrix}

30 30 30

References

[1] I. Assem, D. Simson and A. Skowronski, Elements of the representation theory of associative algebras Vol. 1: Techniques of representation theory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.

[2] M. Auslander, I. Reiten and S.O. Smalø, Representation theory of Artin algebras, Cambridge studies in advanced mathematics 36, Cambridge University Press, Cambridge, 1995.

[3] F. Chapoton, On the Categories of Modules Over the Tamari Posets pp. 269– 280 in: F. Müller-Hoissen, J. M. Pallo, and J. Stasheff, Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift, Springer Basel, (2012).

[4] G. Drozdowski and D. Simson, Remarks on posets of finite representation type. (unpublished?) Accessed at: http://www-users.mat.umk.pl/ simson/DrozdowskiSimson1978.pdf

[5] D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge University Press, Cambridge, 1988.

[6] D. Happel, D. Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42, (1983), 221–243.

[7] M. Herschend and O. Iyama, n-representation-finite algebras and twisted fractionally Calabi-Yau algebras, Bull. Lond. Math. Soc. 43 (2011), 449–466.

[8] D. Kussin, H. Lenzing, and H. Meltzer, Triangle singularities, ADE-chains, and weighted projective lines, Advances in Mathematics, 237, (2013), 194–251.

[9] S. Ladkani, Which canonical algebras are derived equivalent to incidence algebras of posets Comm. Algebra 36 (2008), 4599–4606.

[10] M. Loupias, Indecomposable representations of finite ordered sets. Representations of algebras (Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974), pp. 201-209. Lecture Notes in Math., Vol. 488, Springer, Berlin, (1975).

[11] B. Mitchell, Theory of categories. Pure and Applied Mathematics, Vol. XVII Academic Press, New York-London, (1965).

[12] B. Mitchell, On the dimension of objects and categories. II. Finite ordered sets, J. Algebra 9 (1968), 341–368.

[13] S. Scherotzke, Finite and bounded Auslander-Reiten components in the derived category, J. Pure Appl. Algebra 215 (2011), 232–241.

[14] D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras 2: Tubes and Concealed Algebras of Euclidean Type, Cambridge University Press, (2007).

[15] P.J. Webb, An introduction to the representations and cohomology of categories, pp. 149-173 in: M. Geck, D. Testerman and J. Thévenaz (eds.), Group Representation Theory, EPFL Press (Lausanne) (2007).

[16] P.J. Webb, Bilinear forms on Grothendieck groups of triangulated categories, preprint.

E-mail address: diveris@stolaf.edu

Department of Mathematics, St. Olaf College, Northfield, MN 55057, USA

E-mail address: purin@stolaf.edu

Department of Mathematics, St. Olaf College, Northfield, MN 55057, USA

E-mail address: webb@math.umn.edu

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA