Adjoints to tensor for graded algebras and coalgebras (Q1075407)

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.