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The radical of functorially finite subcategories
- Publication Year :
- 2022
-
Abstract
- Let $\Lambda$ be an artin algebra and $\mathcal{C}$ be a functorially finite subcategory of mod$\Lambda$ which contains $\Lambda$ or $D\Lambda$. We use the concept of the infinite radical of $\mathcal{C}$ and show that $\mathcal{C}$ has an additive generator if and only if rad$^\infty_{\mathcal{C}}$ vanishes. In this case we describe the morphisms in powers of the radical of $\mathcal{C}$ in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that $\mathcal{C}$ is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in $\mathcal{C}$ is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left $\mathcal{C}$-approximations and right $\mathcal{C}$-approximations of simple $\Lambda$-modules, we give other criteria to describe whether $\mathcal{C}$ is of finite representation type. In addition, we give a nilpotency index of the radical of $\mathcal{C}$ which is independent from the maximal length of indecomposable $\Lambda$-modules in $\mathcal{C}$.<br />Comment: We changed the title and some results of the paper. We also added some new results
- Subjects :
- Mathematics - Representation Theory
16G10, 16E20, 18F30, 18A25
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2207.04225
- Document Type :
- Working Paper