Applications of Hochschild cohomology to perpendicular categories
Let \(k\) be an algebraically closed field. It is well known that any basic connected finite dimensional hereditary \(k\)-algebra is of the form \(kQ\) with \(Q\) a finite connected quiver without oriented cycles. We denote by \(\text{mod }kQ\) the category of finite dimensional right \(kQ\)-modules. If \(p=(p_1,\dots,p_t)\) with \(p_i>0\) and \(\lambda=(1=\lambda_3,\dots,\lambda_t)\) with \(\lambda_i\in k\setminus\{0\}\) we denote by \(\text{coh }\mathbb{X}(p,\lambda)\) the category of coherent sheaves defined by these data [\textit{W. Geigle, H. Lenzing}, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)]. For a sequence \(p\) as above we set \(\delta_p=t-2-\sum_{i=1}^t 1/p_i\).NEWLINENEWLINENEWLINEIf \(\mathcal H\) is \(\text{mod }kQ\) (with \(Q\) a wild quiver) or \(\text{coh }\mathbb{X}(p,\lambda)\) (with \(\delta_p>0\)), it is known that the majority of connected components in the Auslander-Reiten quiver of \(\mathcal H\) are of type \(\mathbb{Z}\mathbb{A}_\infty\). For an object \(X\) lying in such a component, we define the quasi-length of \(X\) to be the length of a sectional path from \(X\) to \(Y\) where \(Y\) has an indecomposable middle term in the Auslander-Reiten sequence ending at \(Y\), and denote it by \(\text{ql}(X)\). Let \(n=\text{rk }K_0(\mathcal H)\).NEWLINENEWLINENEWLINEThe author then proves the following results. Theorem: (a) If \(E\in\mathbb{Z}\mathbb{A}_\infty\) is indecomposable exceptional then \(\text{ql}(E)\leq n-2\). (b) Assume \(\mathcal H\) is either \(\text{mod }kQ\) where \(Q\) is a wild tree or \(\text{coh }\mathbb{X}(p,\lambda)\) , where \(\delta_p>0\) and \(t=3\). If \(E\in\mathbb{Z}\mathbb{A}_\infty\) is indecomposable exceptional, then \(\text{ql}(E)\leq n-5\). (c) If \({\mathcal H}=\text{coh }\mathbb{X}(p,\lambda)\) , with \(\delta_p>0\) and \(t\geq 4\), and if \(E\) is an exceptional object of \(\mathcal V\), where \(\mathcal V\) is the full subcategory of \(\text{coh }\mathbb{X}(p,\lambda)\) where each indecomposable object has infinite length, then \(\text{ql}(E)=1\).NEWLINENEWLINENEWLINEPart (c) of the theorem was shown with an alternative proof by \textit{H. Lenzing} and \textit{J. A. de la Peña} [Math. Z. 224, No. 3, 403-425 (1997; Zbl 0882.16011)]. The author obtains these results using Hochschild cohomology of algebras and perpendicular categories.
scientific article