# On the object-wise tensor product of functors to modules

## Marek Golasinski

We investigate preserving of projectivity and injectivity by the object-wise tensor product of $R\Bbb{C}$-modules, where $\Bbb{C}$ is a small category. In particular, let ${\cal O}(G,X)$ be the category of canonical orbits of a discrete group $G$, over a $G$-set $X$. We show that projectivity of $R{\cal O}(G,X)$-modules is preserved by this tensor product. Moreover, if $G$ is a finite group, $X$ a finite $G$-set and $R$ is an integral domain then such a tensor product of two injective $R{\cal O}(G,X)$-modules is again injective.

Keywords: category of canonical orbits, injective (projective) RC-module, linearly compact k-module, tensorpr oduct.

2000 MSC: Primary 18G05; secondary 16W50, 55P91.

Theory and Applications of Categories, Vol. 7, 2000, No. 11, pp 226-235.

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