arXiv subjects
Publications and source records attributed to Pengjie Jiao.
We characterize the generalized Auslander--Reiten duality on the category of finitely presented modules over some certain Hom-finite category. Examples include the category FI of finite sets with injections, and the one VI of finite dimensional vector spaces with linear injections over a finite field.
We prove Auslander's defect formula in an exact category, and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander-Reiten duality.
For an interval finite quiver $Q$, we introduce a class of flat representations. We classify the indecomposable projective objects in the category $\mathrm{rep}(Q)$ of pointwise finite dimensional representations. We show that an object in $\mathrm{rep}(Q)$ is projective if and only if it is a direct sum of countably generated flat representations.
We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.
Given a skeletally small category $\mathcal{C}$, we show that any locally finite endo-length $\mathcal{C}$-module is the direct sum of indecomposable $\mathcal{C}$-modules, whose endomorphism algebra is local.
We introduce a notion of generalized Auslander-Reiten duality on a Hom-finite Krull-Schmidt exact category $\mathcal{C}$. This duality induces the generalized Auslander-Reiten translation functors $τ$ and $τ^-$. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories $\mathcal{C}_r$ and $\mathcal{C}_l$ of $\mathcal{C}$. A non-projective indecomposable object lies in the domain of $τ$ if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of $τ^-$ if and only if it appears as the first term of an almost split conflation.
Given an exact category $\mathcal{C}$, we denote by $\mathcal{C}_l$ the smallest additive subcategory containing injectives and indecomposable objects which appear as the first term of an almost split conflation. We prove that a deflation is right determined by some object if and only if its intrinsic kernel lies in $\mathcal{C}_l$. We give characterizations for $\mathcal{C}$ having Auslander-Reiten duality.