Almost Split Sequences for Comodules

✍ Scribed by William Chin; Mark Kleiner; Declan Quinn

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 147 KB
• Volume: 249
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.9086

✦ Synopsis

Let be a coalgebra over a field k. We introduce an operator Tr that takes a right quasi-finitely copresented -comodule M to a left quasi-finitely copresentedcomodule Tr M. If M is indecomposable not injective and Tr M is finite-dimensional over k, we prove the existence of an almost split sequence 0 → M → E → DTr M → 0 in the category of all right -comodules, where D = Hom k k . If is right semiperfect and the embedding of each simple right comodule S into its injective envelope I S has the property that the socle of I S /S is finite-dimensional, the above almost split sequence exists for each finite-dimensional M, and DTr M is also finite-dimensional.