Hereditary noetherian categories of positive Euler characteristic. (English) Zbl 1105.18010
Consider a noetherian abelian hereditary Ext-finite category \(\mathcal{H}\) over a field \(k\), and assume that \(\mathcal{H}\) is equipped with an equivalence \(\tau\) which satisfies Serre duality.
In the paper under review, two major invariants are attached to \(\mathcal{H}\), namely its function field \(k(\mathcal{H})\) and its Euler characteristic \(\chi_{\mathcal{H}}\), and they are proved to determine the shape of \(\mathcal{H}\) under many aspects. Several examples of these categories are outlined in the paper, the most relevant being that of coherent sheaves on a smooth projective curve \(C\) over \(k\), in which case the invariants of \(\mathcal{H}\) equal those of \(C\).
Consider the full subcategory \(\mathcal{H}_{0}\) consisting of objects of finite length. This is the union over a set \(C(\mathcal{H})\) (called the set of points of \(\mathcal{H}\)) of categories whose Auslander-Reiten quiver can be of type \(\mathbb{Z\,A}/\tau^{p}\) (a finite point) or of type \(\mathbb{Z\,A}_{\infty}\) (an infinite point). The authors first prove that these alternatives cannot coexist, and that in the latter \(\mathcal{H}\) can have at most two points.
On the full subcategory \(\mathcal{H}_{+}\) of objects (called bundles) with no indecomposable subobject, a convenient notion of rank takes positive value. Once chosen a line bundle on \(\mathcal{H}\) the authors provide a definition of \(\chi_{\mathcal{H}}\), a degree function, and a definition of semistable bundles in analogy with the case of smooth projective curves [see C. S. Seshadri, Astérisque 96 (1982; Zbl 0517.14008)].
The case of \(\chi_{\mathcal{H}}\) being positive (\(\mathcal{H}\) is then called domestic) is studied in great detail. The condition \(\chi_{\mathcal{H}} > 0\) takes place, for instance, if and only if the rank function is bounded on some component of \(\mathcal{H}_{+}\), or if and only if each indecomposable is stable (or exceptional). The case of \(\mathcal{H}\) having an infinite point is proved to be domestic too. Moreover, in analogy with the case of \(\mathbb{P}^{1}\), it is proved that \(\mathcal{H}\) is domestic if and only if \(\mathcal{H}\) has a hereditary tilting class, which in turn amounts to \(\mathcal{H}\) being derived equivalent to \(\roman{mod}(\Lambda)\), where \(\Lambda\) is a hereditary locally bounded category.
This allows to complete the classification of domestic categories over an algebraically closed field, which turn out to be associated to path algebras of extended simply laced Dynkin graphs, or of an infinite quiver of type \(\mathbb{A}_{\infty}^{\infty}\) or \(\mathbb{D}_{\infty}\).
This complements the classification appearing by I. Reiten and M. Van den Bergh [J. Am. Math. Soc. 15, 295–366 (2002; Zbl 0991.18009)]. The tubular case (that is, when \(\chi_{\mathcal{H}}=0\)) is also studied, and the Auslander-Reiten quiver of \(\mathcal{H}\) must consist of infinitely many points of finite period. A number of open questions is listed at the end of the paper.
In the paper under review, two major invariants are attached to \(\mathcal{H}\), namely its function field \(k(\mathcal{H})\) and its Euler characteristic \(\chi_{\mathcal{H}}\), and they are proved to determine the shape of \(\mathcal{H}\) under many aspects. Several examples of these categories are outlined in the paper, the most relevant being that of coherent sheaves on a smooth projective curve \(C\) over \(k\), in which case the invariants of \(\mathcal{H}\) equal those of \(C\).
Consider the full subcategory \(\mathcal{H}_{0}\) consisting of objects of finite length. This is the union over a set \(C(\mathcal{H})\) (called the set of points of \(\mathcal{H}\)) of categories whose Auslander-Reiten quiver can be of type \(\mathbb{Z\,A}/\tau^{p}\) (a finite point) or of type \(\mathbb{Z\,A}_{\infty}\) (an infinite point). The authors first prove that these alternatives cannot coexist, and that in the latter \(\mathcal{H}\) can have at most two points.
On the full subcategory \(\mathcal{H}_{+}\) of objects (called bundles) with no indecomposable subobject, a convenient notion of rank takes positive value. Once chosen a line bundle on \(\mathcal{H}\) the authors provide a definition of \(\chi_{\mathcal{H}}\), a degree function, and a definition of semistable bundles in analogy with the case of smooth projective curves [see C. S. Seshadri, Astérisque 96 (1982; Zbl 0517.14008)].
The case of \(\chi_{\mathcal{H}}\) being positive (\(\mathcal{H}\) is then called domestic) is studied in great detail. The condition \(\chi_{\mathcal{H}} > 0\) takes place, for instance, if and only if the rank function is bounded on some component of \(\mathcal{H}_{+}\), or if and only if each indecomposable is stable (or exceptional). The case of \(\mathcal{H}\) having an infinite point is proved to be domestic too. Moreover, in analogy with the case of \(\mathbb{P}^{1}\), it is proved that \(\mathcal{H}\) is domestic if and only if \(\mathcal{H}\) has a hereditary tilting class, which in turn amounts to \(\mathcal{H}\) being derived equivalent to \(\roman{mod}(\Lambda)\), where \(\Lambda\) is a hereditary locally bounded category.
This allows to complete the classification of domestic categories over an algebraically closed field, which turn out to be associated to path algebras of extended simply laced Dynkin graphs, or of an infinite quiver of type \(\mathbb{A}_{\infty}^{\infty}\) or \(\mathbb{D}_{\infty}\).
This complements the classification appearing by I. Reiten and M. Van den Bergh [J. Am. Math. Soc. 15, 295–366 (2002; Zbl 0991.18009)]. The tubular case (that is, when \(\chi_{\mathcal{H}}=0\)) is also studied, and the Auslander-Reiten quiver of \(\mathcal{H}\) must consist of infinitely many points of finite period. A number of open questions is listed at the end of the paper.
Reviewer: Daniele Faenzi (Firenze)
MSC:
| 18E20 | Categorical embedding theorems |
| 16D90 | Module categories in associative algebras |
| 16E60 | Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 14H10 | Families, moduli of curves (algebraic) |
Keywords:
hereditary noetherian categories; Euler characteristic; domestic categories; path algebras; infinite points; non-commutative curvesReferences:
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