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, \(p \cdot q = pq\) ("\(p\) then \(q\)") if \(pq\) is a path in \(Q\), and \(p \cdot q = 0\) 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 \(N \geq 2\) with \(J^N \subseteq I \subseteq J^2\).
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 \( \phi \colon A \to \End X\) whose target is the endomorphism algebra \(\End X\) of a \(K\)-vector space \(X\); for convenience, write \(\phi_a\) for the image of \(a\) in \(\phi\). In this case, we call \(X\) a (right) \(A\)-module, with associated action \(x \cdot a = x\phi_a\). 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 \(\theta_\alpha \colon X_i \to X_j\) to each arrow \(\alpha \colon i \to j\) of \(Q\) such that for any \(\rho = \sum_{k=1}^m p_k\lambda_k \in I\), the associated map \(\sum_{k=1}^m \theta_{p_k} \colon \bigoplus_i X_i \to \bigoplus_i X_i\) is zero. Here, \(\theta_{p_k}=\theta_{\alpha_1}\cdots\theta_{\alpha_r}\) for a decomposition of a nonstationary \(p_k\) into a product \(\alpha_1\cdots\alpha_r\) of arrows, and \(\theta_{i}=\id_{X_i}\) 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 \(\oplus\)) . We call a module \(X\) indecomposable if \(X = Y\oplus Z\) implies \(Y\) or \(Z\) is zero. Further, if \(\mathcal{U}\) is any set of \(A\)-modules, we define the additive closure \(\add \mathcal{U}\) of \(\mathcal{U}\) as the full subcategory of \(\modcat A\) whose objects are isomorphic to direct summands of finite direct sums of members of \(\mathcal{U}\).
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 \(\Hom_A(X,-)\colon \Modcat A \to \Modcat Z\) is exact, or; injective if the functor \(\Hom_A(-,X) \colon \Modcat A \to \Modcat Z\) 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
\[0=X_0 < X_1 < X_2 < \cdots < X_{l-1} < X_l=X\]
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 \(\Hom_K(-, K) \colon \modcat A \to \modcat A^\op \) of modules (and its inverse) and we write \(*\) for the dualities \(\Hom_A(-, A) \colon \projcat A \to \projcat A^\op\) and \(\Hom_A(-, A) \colon \injcat A \to \injcat A^\op\).
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 \(P_1 \overset{f}{\to} P_0 \to X \to 0\). Applying \(*\) to \(f\) yields the map \(f^* \colon \P_0^* \to P_1^*\) in \(\projcat A^\op\). Its cokernel \(\cok f^*\) is called the transpose \(\Tr X\) of \(X\). (In \(\pstmodcat A\), transpose yields a duality \(\Tr \colon \pstmodcat A \to \pstmodcat A^\op \).)
The composition \(\D \Tr\) (transpose first, then dual) is called the Auslander-Reiten translation (and is an equivalence \(\pstmodcat A \to \istmodcat A\)). 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 \(\pi \colon \PP X \twoheadrightarrow X\). The kernel of this map is the (first) syzygy \(\syzygy^1 X\) of \(X\). We inductively define the \(k\)th syzygy as \(\syzygy^{k+1} M = \syzygy^1( \syzygy^k M )\) for \(k \geq 0\) 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 \(k\geq 0\) for which \(\syzygy^k X = 0\), or \(+\infty\) 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 \(k \geq 0\) 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 \(\mathcal{A}_k\) contains the isoclasses of indecomposables appearing at or \(\mathcal{A}\)fter the \(k\)th syzygy, while \(\mathcal{B}_k\) contains those appearing at or \(\mathcal{B}\)efore.) The \(\mathcal{A}_k\) and \(\mathcal{B}_k\) 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 \(\mathcal{B}_k \supseteq \mathcal{B}_{k+1}\) or \(\mathcal{A}_k \supseteq \mathcal{A}_{k+1}\), the appropriate set difference between them is finite. Note also that \(\cap_{k \geq 0}\mathcal{A}_k\) 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 \(\mathcal{A}_t=\mathcal{A}_{t+1}\)(\(=\mathcal{A}_{t+2}=\cdots=\cap_{k \geq 0} \mathcal{A}_k\)), then there is a minimal one \(t_\star\), in which case we say that the syzygy repetition index of \(X\) is \(t_\star\). This holds exactly when \(t_\star\) satisfies \(\mathcal{A}_{t_\star} = \cap_{k \geq 0} \mathcal{A}_k\) (and is the minimal index to enjoy this property). If no such \(t\) exists, the syzygy repetition of index of \(X\) is \(+\infty\).
If \(\mathcal{A}_0\) is finite, then we say \(X\) has syzygy type \(|\mathcal{A}_0|\) of index \(s_\star\), for \(s_\star\) the minimal index \(k\) such that \(\mathcal{B}_k=\mathcal{B}_{k+1}\)(\(=\mathcal{B}_{k+2}=\cdots=\mathcal{A}_0\)); the existence of \(s_\star\) in this case follows from an easy finiteness argument. Just as immediately, we see that if \(X\) has finite syzygy type \(|\mathcal{A}_0|\) then it has finite syzygy repetition index at most \(|\mathcal{A}_0|\).
If \([X] \in \cap_{k \geq 0} \mathcal{A}_k\), 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 \(k \geq 0\) for which it can be \(k\)-delooped, or \(+\infty\) if no such \(k\) exists. He subsequently defines the delooping level \(\delooping A\) of an algebra \(A\) as \(\delooping A = \max \{\delooping S \colon S \text{ is simple}\}\) 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 \(ab \notin I\) and
for every arrow \(a\) of \(Q\), there is at most one arrow \(c\) with \(ca \notin I\).
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 \(w \colon \Gamma \to Q\) from a quiver \(\Gamma\) such that: the domain is an orientation of a linear graph, \(w(\alpha) \neq w(\beta)\) whenever \(\alpha,\beta \in \Gamma_1\) have common source or target and where the image in \(w\) of any \(\Gamma\)-path \(p\) is linearly independent of all other \(A\)-paths. A string graph \(w\) is commonly depicted by labelling each vertex and arrow of \(\Gamma\) 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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