Canonical algebras had been introduced by C. M. Ringel [Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]. Let \(k\) be an algebraically closed field, \(t\geq 2\) a natural number, \(p=(p_1,\dots,p_t)\) a sequence of natural numbers with \(p_i\geq 2\) and \(\lambda=(\lambda_3,\dots,\lambda_t)\) a sequence of pairwise different nonzero elements in \(k\). Let \({\mathcal Q}\) be the quiver \[ \begin{matrix} & &\overset{\vec x_1}\circ &\overset {x_1}\longrightarrow &\overset {2\vec x_1}\circ &\longrightarrow &\cdots &\longrightarrow &\overset{(p_1-1)\vec x_1}\circ \\ &\overset {x_1}\nearrow &&&&&&&& \overset {x_1}\searrow \\ \overset {0}\circ &\underset {x_2}\longrightarrow &\overset{\vec x_2}\circ & \underset {x_2}\longrightarrow &\overset{2\vec x_2}\circ &\longrightarrow &\cdots &\longrightarrow &\overset{(p_2-1)\vec x_2}\circ &\underset {x_2}\longrightarrow &\overset\vec {c}\circ \\ &\underset {x_t}\searrow &&\vdots &&\vdots &&\vdots &&\underset {x_t}\nearrow \\ & &\underset {\vec x_t}\circ &\underset {x_t}\longrightarrow &\underset{2\vec x_t}\circ &\longrightarrow &\cdots &\longrightarrow &\underset {(p_t-1)\vec x_t}\circ\end{matrix} \] The canonical algebra \(\Lambda=\Lambda(p,\lambda)\) is the path-algebra of the quiver \({\mathcal Q}\), bound by the relations \(x^{p_i}_i-x_2^{p_2}+\lambda_i x_1^{p_1}\). If the tree \({\mathcal T}\), obtained from \({\mathcal Q}\) by deleting the vertex 0 is a Dynkin diagram, then \(\Lambda\) is a well known tame concealed algebra. If \({\mathcal T}\) is an extended Dynkin diagram, then \(\Lambda\) is a tubular algebra. Ringel [loc. cit.] gave a description of the category \(\text{mod }\Lambda\) of finite dimensional \(\Lambda\)-modules in this case. The paper under review studies the case, when \({\mathcal T}\) is a wild quiver and \(\Lambda\) then is called wild canonical.
In this case, the module category \(\text{mod }\Lambda\) decomposes into three parts, \(\text{mod }\Lambda=\text{mod}_+\Lambda\vee\text{mod}_0\Lambda\vee\text{mod}_-\Lambda\). The middle part \(\text{mod}_0\Lambda\) has as Auslander-Reiten quiver a separating family of pairwise orthogonal standard tubes. The category \(\text{mod}_+\Lambda\) has exactly one preprojective component \({\mathcal P}\), containing \(n-1\) indecomposable projectives, where \(n\) is the number of vertices of \({\mathcal Q}\). The remaining Auslander-Reiten components in \(\text{mod}_+\Lambda\) are either of type \(\mathbb{Z} A_\infty\) or have stable part \(\mathbb{Z} A_\infty\). All modules in \(\text{mod}_\geq\Lambda=\text{mod}_+\Lambda\vee\text{mod}_0\Lambda\) have projective dimension at most 1. The category \(\text{mod}_-\Lambda\) is dual to \(\text{mod}_+\Lambda \).
It was shown by W. Geigle and H. Lenzing [in Singularities, representations of algebras, and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)] that canonical algebras are quasitilted algebras. More precise, the hereditary category \(\text{coh} X\) of coherent sheaves over a certain weighted projective line \(X=X(p,\lambda)\) contains a tilting vector bundle \(T\) with \(\text{End}(T)=\Lambda\). The tilting object \(T\) has a decomposition \(T={\mathcal O}\oplus T'\), where \({\mathcal O}\) is a line bundle and \(T'\) is the minimal projective generator in the right perpendicular category \({\mathcal O}^\perp\). The endomorphism ring \(\text{End}(T')\) is \(\Lambda_0\), the path-algebra of the quiver \({\mathcal T}\), and \({\mathcal O}^\perp\) will be identified with \(\text{mod }\Lambda_0\).
The authors show that the subcategory \(\text{vect} X\) of \(\text{coh} X\), defined by the vector bundles over \(X\), consists of Auslander-Reiten components of type \(\mathbb{Z} A_\infty\), and they identify \(\text{mod}_\geq\Lambda\) with the torsion class \(\text{coh}_\geq X=\{{\mathcal F}\mid\text{Ext}({\mathcal O},{\mathcal F})=0\}\) in \(\text{coh} X\). Under this identification \(\text{mod}_0\Lambda\) becomes \(\text{coh}_0X\), the category of finite length sheaves, which implies the structure of \(\text{mod}_0\Lambda\).
For the study of \(\text{mod}_+\Lambda\subset\text{vect} X\), they compare the Auslander-Reiten translations \(\tau_X\) in \(\text{vect} X\) and the relative Auslander-Reiten translations \(\tau_\Lambda\) and \(\tau_{\Lambda_0}\) in \(\text{vect} X\). As result they get for a \(\Lambda\)-module \(M\) in \(\text{mod}_+\Lambda\), not in the \(\tau_\Lambda\)-orbit of a projective, that \(\tau_\Lambda^{-m-1} M=\tau^-_{\Lambda_0}\tau_\Lambda^{-m} M\) for \(m\gg 0\) and \(\tau_\Lambda^{m+1} M=\tau_X\tau_\Lambda^mM\) for \(m\gg 0\). This result not only implies the Auslander-Reiten structure of \(\text{mod}_+\Lambda\). It also shows that each Auslander-Reiten component in \(\text{mod}_+\Lambda\), different from the preprojective component, contains a right cone, consisting completely of \(\Lambda_0\)-modules and a left cone, which is a left cone of some Auslander-Reiten component in \(\text{vect} X\) simultaneously, in complete analogy to the Auslander-Reiten structure of wild tilted algebras [O. Kerner, J. Algebra 142, No. 1, 37-57 (1991; Zbl 0737.16007)]. Especially it implies bijections between the set of stable components of \(\text{mod}_+\Lambda\), the set of regular components of \(\text{mod}\Lambda_0\) and the set of Auslander-Reiten components in \(\text{vect} X\). But in contrast to wild tilted algebras, the dimension vectors \(\dim\tau_\Lambda^m M\) for \(M\in\text{mod}_+\Lambda\) grow linearly in \(m\).