Cotorsion pairs in categories of quiver representations
Cotorsion pairs in categories of quiver representations
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DOI：
10.1215/21562261-2018-0018
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发表时间：
2016-04
影响因子：
0.6
通讯作者：
Henrik Holm;Peter Jørgensen
中科院分区：
文献类型：
--
作者：
Henrik Holm;Peter Jørgensen
我们研究了值在阿贝尔范畴$\mathcal{M}$中的颤振$Q$的表示的范畴$\mathrm{Rep}(Q,\mathcal{M})$。在一定的假设下，我们证明了$\mathcal{M}$中的每个扭对$(\mathcal{A},\mathcal{B})$都会诱导$\mathrm{Rep}(Q,\mathcal{M})$中的两个（明确描述的）扭对$(\Phi(\mathcal{A}),\mathrm{Rep}(Q,\mathcal{B}))$和$(\mathrm{Rep}(Q,\mathcal{A}),\Psi(\mathcal{B}))$。这与Gillespie的结论类似，他断言$\mathcal{M}$中的扭对$(\mathcal{A},\mathcal{B})$会诱导$\mathcal{M}$中链配合物$\mathrm{Ch}(\mathcal{M})$类别中的扭对$(\widetilde{\mathcal{A}}, \mathrm{dg}\,\widetilde{\mathcal{B}})$和$(\mathrm{dg}\,\widetilde{\mathcal{A}}, \widetilde{\mathcal{B}})$。我们的结果的特殊情况恢复了在$\mathrm{Rep}(Q,\mathcal{M})$中由Enochs， Estrada和Garcia Rozas证明的投射物和射物的描述。
We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces two (explicitly described) cotorsion pairs $(\Phi(\mathcal{A}),\mathrm{Rep}(Q,\mathcal{B}))$ and $(\mathrm{Rep}(Q,\mathcal{A}),\Psi(\mathcal{B}))$ in $\mathrm{Rep}(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\widetilde{\mathcal{A}}, \mathrm{dg}\,\widetilde{\mathcal{B}})$ and $(\mathrm{dg}\,\widetilde{\mathcal{A}}, \widetilde{\mathcal{B}})$ in the category $\mathrm{Ch}(\mathcal{M})$ of chain complexes in $\mathcal{M}$. Special cases of our results recover descriptions of the projective and injective objects in $\mathrm{Rep}(Q,\mathcal{M})$ proved by Enochs, Estrada, and Garcia Rozas.