Here and throughout, K is some field. By a K-algebra A, we mean an associative and unital (but not necessarily commutative) ring with a compatible K-vector space structure. Any algebra A has an opposite algebra A^op.
Suppose Q is a finite quiver: that is, a directed graph with finitely many vertices and finitely many arrows, where loops and/or multiple edges are permitted. The paths of Q (also called Q-paths to emphasize their parent quiver), including the "stationary" paths at each vertex, form the basis of a vector space. multiplication can be defined on basis vectors p and q by "concatenation extended by zero"; more precisely, p ⋅ q = pq ("p then q") if pq is a path in Q, and p ⋅ q = 0 otherwise. This defines the path algebra KQ. Its multiplicative unit is the sum of stationary paths. It has finite K-dimension iff Q contains no (directed) cycles.
Let J ideal KQ denote the arrow ideal of KQ: the smallest two-sided ideal of KQ containing the arrows of Q. An ideal I ideal KQ is admissible iff there is an integer N ≥ 2 with J^N ⊆ I ⊆ J^2.
By a (bound) quiver algebra, we mean a quotient KQ/I of a path algebra KQ by an admissible ideal I. Quiver algebras are always finite-dimensional [ASS , Sec II.2]. Indeed, at least when K is algebraically closed, any finite-dimensional algebra is a direct product of connected ones (trivially), any connected finite-dimensional algebra is Morita equivalent to a basic one [ASS , Sec I.6] and any basic, connected algebra is isomorphic to a quiver algebra [ASS , Sec II.3].
In this document, we assume that K is algebraically closed and A is a quiver algebra KQ/I. By the above, this is no loss of generality. We also use the term A-path to mean a nonzero element p+I represented by a path of the quiver.
A representation of A is a homomorphism of algebras ϕ : A -> End X whose target is the endomorphism algebra End X of a K-vector space X; for convenience, write ϕ_a for the image of a in ϕ. In this case, we call X a (right) A-module, with associated action x ⋅ a = xϕ_a. The module is finite-dimensional iff X is.
Since A=KQ/I, we can work in terms of (bound) representations of quivers. These are assignments of a vector space X_i to each vertex i of Q and a linear map θ_α : X_i -> X_j to each arrow α : i -> j of Q such that for any ρ = ∑_k=1^m p_kλ_k ∈ I, the associated map ∑_k=1^m θ_p_k : ⨁_i X_i -> ⨁_i X_i is zero. Here, θ_p_k=θ_α_1⋯θ_α_r for a decomposition of a nonstationary p_k into a product α_1⋯α_r of arrows, and θ_i=id_X_i for any stationary path at i.
As is well-known [ASS , III.1], representations of quivers are equivalent to modules. More specifically, the categories Rep(Q,I) of bound quiver representations and Modcat A of A-modules are equivalent, and this equivalence restricts to their respective full subcategories rep(Q,I) and modcat A of finite-dimensional objects. In keeping with the quiver-minded approach from above, whenever we say module, we really mean the equivalent bound quiver representation.
We note in particular that all of the categories in the previous paragraph are abelian: thus, we can speak of the direct sum of modules (denoted with ⊕) . We call a module X indecomposable if X = Y⊕ Z implies Y or Z is zero. Further, if mathcalU is any set of A-modules, we define the additive closure add mathcalU of mathcalU as the full subcategory of modcat A whose objects are isomorphic to direct summands of finite direct sums of members of mathcalU.
Write [X] for the isomorphism type of X. One can seek to classify the isomorphism classes of indecomposable (finite-dimensional) modules of an algebra. A deep theorem of Drozd [Dro ] establishes that all finite-dimensional algebras fall into exactly one of three representation types. In increasing order of difficulty, the options are representation finite, tame or wild. The first simply means the algebra has only finitely many isoclasses of indecomposables. Speaking informally, tame algebras are those for which, in each dimension, almost all modules lie in one of finitely many classes each parameterized by the field. Speaking even more informally, wild algebras are those for which the classification problem is intractible in a very strong way. Discussion and formal definitions of representation type can be found in [Ben , Sec 4.4].
There are certain canonical classes of module. A module X is: simple if it has no proper, nonzero submodules; projective if the covariant functor Hom_A(X,-): Modcat A -> Modcat Z is exact, or; injective if the functor Hom_A(-,X) : Modcat A -> Modcat Z is exact. The simple A-modules (necessarily indecomposable) are in one-to-one correspondence with the vertices of Q, as are the indecomposable projective and injective modules. We respectively write S_i, P_i and I_i for the simple, indecomposable projective and indecomposable injective module corresponding to the vertex i. We also write projcat A and injcat A to for the full subcategories of modcat A whose objects are respectively the (finite-dimensional) projective and injective modules.
A composition series for a module X is a strictly ascending chain of submodules
0=X_0 < X_1 < X_2 < ⋯ < X_l-1 < X_l=X
of X such that each consecutive quotient X_k+1/X_k is simple. A module is uniserial if it has a unique composition series; equivalently, if its submodules form a chain.
We write D for the vector-space duality Hom_K(-, K) : modcat A -> modcat A^op of modules (and its inverse) and we write * for the dualities Hom_A(-, A) : projcat A -> projcat A^op and Hom_A(-, A) : injcat A -> injcat A^op.
In this section, we describe certain module constructions which refer to projective presentations (described below). These constructions do not generally extend to functors on the module category because they depend on the presentation chosen. However, this dependence is usually only up to the adding or removing of projective direct summands. By working in the finite-dimensional universe modcat A where the Krull-Schmidt theorem applies, we can reduce to study of modules having no indecomposable projective direct summands. These objects admit minimal projective presentations. This approach suffices for SBStrips, since we will only be interested in constructions on modules and not their functorial extension to morphisms. Readers who care for the functorial approach should interpret the following in the (projectively) stable module category pstmodcat A: its objects are those of the usual module category modcat A and its hom-spaces are the quotients of their counterparts in modcat A by the subspaces of maps that factor through projective modules. They should also be aware of the injectively stable module category istmodcat A, similarly obtained by quotienting my maps factoring through injectives.
Any module M admits a projective presentation, which is to say an exact sequence P_1 oversetf-> P_0 -> X -> 0. Applying * to f yields the map f^* : P_0^* -> P_1^* in projcat A^op. Its cokernel cok f^* is called the transpose Tr X of X. (In pstmodcat A, transpose yields a duality Tr : pstmodcat A -> pstmodcat A^op.)
The composition D Tr (transpose first, then dual) is called the Auslander-Reiten translation (and is an equivalence pstmodcat A -> istmodcat A). Its inverse is the opposite composition Tr D.
For any finite-dimensional module X there is a smallest (in vector-space dimension) projective module PP X that maps onto it, say by the map π : PP X ↠ X. The kernel of this map is the (first) syzygy syzygy^1 X of X. We inductively define the kth syzygy as syzygy^k+1 M = syzygy^1( syzygy^k M ) for k ≥ 0 and, by convention, we set syzygy^0 M to be X/P, for P the largest projective direct summand of X.
The projective dimension projdim X of X is the smallest k≥ 0 for which syzygy^k X = 0, or +∞ if no such k exists. In particular, a module has projective dimension 0 iff it is projective.
We can define certain homological behavior with reference to syzygies. For k ≥ 0 and some fixed module X, let
\mathcal{A}_k = \big\{ [M] \colon M \in \add\{ \syzygy^t X \} \text{ for some } t \geq k \big\} \text{, } \mathcal{B}_k = \big\{ [M] \colon M \in \add\{ \syzygy^t X \} \text{ for some } t \leq k \big\}\text{.}
(These letters were chosen so that mathcalA_k contains the isoclasses of indecomposables appearing at or mathcalAfter the kth syzygy, while mathcalB_k contains those appearing at or mathcalBefore.) The mathcalA_k and mathcalB_k relate in the following way:
\mathcal{B}_0 \subseteq \mathcal{B}_1 \subseteq \mathcal{B}_2 \subseteq \cdots \subseteq \bigcup_{k \geq 0} \mathcal{B}_k = \mathcal{A}_0 \supseteq \mathcal{A}_1 \supseteq \mathcal{A}_2 \supseteq \cdots \supseteq \bigcap_{k \geq 0} \mathcal{A}_k \text{.}
We comment that, for each successive inclusion mathcalB_k ⊇ mathcalB_k+1 or mathcalA_k ⊇ mathcalA_k+1, the appropriate set difference between them is finite. Note also that ∩_k ≥ 0mathcalA_k contains exactly those isoclasses witnessed at syzygy^k X for infinitely many indices k. Below, we use this sequence of inclusions to define some terminology for patterns in syzygies. Our definitions are inspired by comparable work in [GH , Sec 2] and [Ric , Sec 7].
If there is an index t for which mathcalA_t=mathcalA_t+1(=mathcalA_t+2=⋯=∩_k ≥ 0 mathcalA_k), then there is a minimal one t_⋆, in which case we say that the syzygy repetition index of X is t_⋆. This holds exactly when t_⋆ satisfies mathcalA_t_⋆ = ∩_k ≥ 0 mathcalA_k (and is the minimal index to enjoy this property). If no such t exists, the syzygy repetition of index of X is +∞.
If mathcalA_0 is finite, then we say X has syzygy type |mathcalA_0| of index s_⋆, for s_⋆ the minimal index k such that mathcalB_k=mathcalB_k+1(=mathcalB_k+2=⋯=mathcalA_0); the existence of s_⋆ in this case follows from an easy finiteness argument. Just as immediately, we see that if X has finite syzygy type |mathcalA_0| then it has finite syzygy repetition index at most |mathcalA_0|.
If [X] ∈ ∩_k ≥ 0 mathcalA_k, then we call X weakly periodic.
We say a module X can be k-delooped if there is some module Y for which syzygy^k X is a direct summand syzygy^k+1 Y. Here, we either discard projective direct summands of both modules or, formally, work in pstmodcat A. Gélinas [Gél , Thm 1.10] showed that it suffices to check Y = suspension^k+1 syzygy^k X, where suspension = Tr syzygy Tr is called suspension.
Gélinas defines the delooping level delooping X of a module X to be the smallest k ≥ 0 for which it can be k-delooped, or +∞ if no such k exists. He subsequently defines the delooping level delooping A of an algebra A as delooping A = max {delooping S : S is simple} and relates this invariant to the finitistic dimension of A.
A special biserial (SB) algebra is a quiver algebra KQ/I such that
every vertex of Q is the source of at most 2 arrows,
every vertex of Q is the target of at most 2 arrows,
for every arrow a of Q, there is at most one arrow b with ab ∉ I and
for every arrow a of Q, there is at most one arrow c with ca ∉ I.
These algebras emerged from the modular representation theory of finite groups. A key text on them is [WW ], which establishes in particular that they are tame algebras. Their indecomposable modules fall into three classes: band modules, string modules and a finite class of projective-injective nonuniserial ("pin") modules.
String modules earn their name from the string graphs that describe them so well. A string graph for A=KQ/I is a quiver homomorphism w : Γ -> Q from a quiver Γ such that: the domain is an orientation of a linear graph, w(α) ≠ w(β) whenever α,β ∈ Γ_1 have common source or target and where the image in w of any Γ-path p is linearly independent of all other A-paths. A string graph w is commonly depicted by labelling each vertex and arrow of Γ by its respective images in Q. Then vertices of w provide a basis of the associated string module, and the labels describe the A action. We can identify a string graph with the string module it represents.
One subtle point: here, we do not require string graphs to be connected; accordingly we do require string modules to be indecomposable.
The dual D X, the transpose Tr X and the syzygy syzygy^1 X of a string module X are all string modules, albeit for the opposite algebra in the first two cases [WW , Sec 3] [LM , Sec 2]. In as-yet-unpublished work of Galstad [Gal] (and publicized without proof by Huisgen-Zimmermann [Hui ]) the syzygy of a band module X is also a string module provided that at least one indecomposable direct summand of PP X is a string module.
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