Adjoints to tensor for graded algebras and coalgebras
type: Item
instance of: scholarly article
title: Adjoints to tensor for graded algebras and coalgebras
zbMATH Open document ID: 0592.16001
DOI: 10.1016/0022-4049(86)90112-X
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let K be a commutative ring. It is well-known when tensoring with a K- module, K-algebra, commutative K-algebra, K-coalgebra, or cocommutative K-coalgebra has a left or a right adjoint in the same category. The author generalizes these results to the graded case, which includes N- graded, Z-graded and differential Z-graded. For modules, an N-graded module \(M=\sum \oplus M_ n\), tensoring with M has a left adjoint iff each \(M_ n\) if finitely-generated projective. For the other two kinds of grading, the same result holds but also requires \(M_ n\) to be zero for almost all n. In the 3 kinds of grading for algebras and commutative algebras, tensoring has a left adjoint iff it does in the underlying graded module category. In the 3 kinds of grading for coalgebras and cocommutative coalgebras, tensoring with a fixed object always has a right adjoint.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 3950741
zbMATH Keywords: tensoring
zbMATH Keywords: right adjoint
zbMATH Keywords: N-graded
zbMATH Keywords: Z-graded
zbMATH Keywords: differential Z-graded
zbMATH Keywords: graded module category
zbMATH Keywords: cocommutative coalgebras
author: Susan Niefield
reviewed by: Earl J. Taft
MaRDI profile type: Publication
cites work: Coalgebras over a commutative ring
cites work: Q5586461
cites work: Categorical algebra
cites work: Q5639839
cites work: On the structure of Hopf algebras
cites work: Cartesianness: topological spaces, uniform spaces, and affine schemes
cites work: Q3961636
Recommended article: Q4337905
Recommended article: Q4761528
Recommended article: Associated Graded Algebras and Coalgebras
Recommended article: Tensor product of N-complexes and generalization of graded differential algebras
Recommended article: Tensor product algebras, Grassmannians and Khovanov homology
Recommended article: A criterion for graded coherence of tensor algebras and applications to higher dimensional AR-theory
Recommended article: Q5392463
Recommended article: Tensor products of matrix algebras over the Grassmann algebra
Recommended article: Tensor Algebras in Finite Tensor Categories
Recommended article: Tensor products of strongly graded vertex algebras and their modules
description: scientific article; zbMATH DE number 3950741