Category equivalences involving graded modules over path algebras of quivers.(English)Zbl 1264.16042
Consider a quiver \(Q\) and a field \(k\). Let \(I\) denote the vertex set of \(Q\) and \(Q_1\) the arrow set. The path algebra \(kQ\) can be decomposed in the form \(kQ=KI\oplus kQ_1\oplus kQ_2\oplus\cdots\) and can be seen as the tensor algebra of the \(kI\)-bimodule \(kQ_1\). Next define \(S_n:=\text{End}_{kI}(kQ_n)\). Since \(kQ_{n+1}\cong kQ_1\otimes kQ_n\) (as \(kI\)-modules), we can consider the \(k\)-algebra homomorphism \(\theta_n\colon S_n\to S_{n+1}\) such that for any \(f\in S_n\) and \(x\otimes z\in S_{n+1}\), we have \(\theta(f)(x_1\otimes z)=x_1\otimes f(z)\). Thus, we have a direct system \(S_0\to S_1\to\cdots\) and we can define \(S(Q):=\varinjlim S_n\).
The main result of the paper is the chain of category equivalences: \[\text{QGr}(kQ)\equiv\text{Mod}S(Q)\equiv\text{Gr}L_k(Q^\circ)\equiv\text{Mod}L_k(Q^\circ)_0\equiv\text{QGr}(kQ^{(n)})\] where: (1) \(\text{QGr}(kQ)\) is the quotient category \(\text{QGr}(kQ):=\text{Gr}(kQ)/\text{Fdim}(kQ)\), with \(\text{Gr}(kQ)\) the category of \(\mathbb Z\)-graded left \(kQ\)-modules; \(\text{Fdim}(kQ)\) the localizing (full) subcategory of modules which agree with the sum of their finite-dimensional submodules.
(2) \(Q^\circ\) is the quiver without sinks or sources that is obtained by repeatedly removing all sinks and sources from \(Q\).
(3) \(L_k(Q^\circ)\) is the Leavitt path algebra of \(Q^\circ\) and \(L_k(Q^\circ)_0\) is its zero-homogeneous component.
(4) \(Q^{(n)}\) is the quiver whose incidence matrix is the \(n\)-th power of that of \(Q\).
The main result of the paper is the chain of category equivalences: \[\text{QGr}(kQ)\equiv\text{Mod}S(Q)\equiv\text{Gr}L_k(Q^\circ)\equiv\text{Mod}L_k(Q^\circ)_0\equiv\text{QGr}(kQ^{(n)})\] where: (1) \(\text{QGr}(kQ)\) is the quotient category \(\text{QGr}(kQ):=\text{Gr}(kQ)/\text{Fdim}(kQ)\), with \(\text{Gr}(kQ)\) the category of \(\mathbb Z\)-graded left \(kQ\)-modules; \(\text{Fdim}(kQ)\) the localizing (full) subcategory of modules which agree with the sum of their finite-dimensional submodules.
(2) \(Q^\circ\) is the quiver without sinks or sources that is obtained by repeatedly removing all sinks and sources from \(Q\).
(3) \(L_k(Q^\circ)\) is the Leavitt path algebra of \(Q^\circ\) and \(L_k(Q^\circ)_0\) is its zero-homogeneous component.
(4) \(Q^{(n)}\) is the quiver whose incidence matrix is the \(n\)-th power of that of \(Q\).
Reviewer: Candido Martín González (Málaga)
MSC:
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16G20 | Representations of quivers and partially ordered sets |
| 16D90 | Module categories in associative algebras |
Cite
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