TITLE: Is there a "correct" general setting for the principle:  "tensoring any object with a projective object yields another projective"? 
QUESTION [10 upvotes]: Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$.  (See Exercise 2 on p. 426 of Curtis & Reiner, Representation Theory of Finite Groups and Associative Algebras, 1962.)  Here the Hopf algebra structure of A yields a natural left module structure on the tensor product of two left modules over k.   
By the mid-1970s similar tensor product behavior was observed in other special cases for left A-modules and their tensor products, where A is a Hopf algebra over a commutative ring k:   (1)  the (finite dimensional) restricted enveloping algebra of a restricted Lie algebra $\mathfrak{g}$ over a field of prime characteristic; (2) more generally  the hyperalgebra of a higher Frobenius kernel  when $\mathfrak{g}$ is the Lie algebra of a reductive algebraic group;  (3) the universal enveloping algebra of a Kac-Moody algebra in characteristic 0;  (4) the full hyperalgebra of a reductive algebraic group in prime characteristic (with "projective" replaced by "injective" as in J.C. Jantzen's book Representations of Algebraic Groups, I.3).    Relevant references:
B. Pareigis, Kohomologie von p-Lie Algebren, Math. Z. 104 (1968); Lemma 2.5
J.E. Humphreys, Projective modules for SL(2,q), J. Algebra 25 (1973); Thms. 1, 2 (and note
added in proof referring to Pareigis)
J.E. Humphreys, Ordinary and modular representations of Chevalley groups, Springer
Lect. Notes in Math. 528 (1976); Appendix T (following Sweedler's suggestion)
H. Garland and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976); 1.7 and Remark
J.E. Humphreys, On the hyperalgebra of a semisimple algebraic group, in Contributions to Algebra, Academic Press, 1977; 3.1
The arguments here typically involve special cases of a general theorem suggested by Sweedler (and closely related to the "tensor identity" discussed in a recent MO post 37709 ):   Let $A$ be a Hopf algebra (with antipode) over a commutative ring $k$, with Hopf subalgebra $B$ (possibly k).   Given an $A$-module  $M$ and a $B$-module $N$,  there is a natural  $A$-module isomorphism:
$$(A \otimes_B N) \otimes_k M \cong A \otimes_B (N \otimes_k M)$$  On the left side, A acts via comultiplication, while on the right it acts on the first factor. 

Is this the optimal generality, and if so is there a textbook reference?

REPLY [5 votes]: Dear Jim,
Perhaps this is the correct statement, which is proven in Etingof's lectures on tensor categories (though its proof follows from the yoga of tensor categories, as I'll explain):
http://www-math.mit.edu/~etingof/tenscat1.pdf
Proposition:
Let $P$ be a projective object in a multiring category C. If $X\in C$ has a right dual, then the object $P \otimes X$ is projective. Similarly, if $X \in C$ has a left dual, then the object $X \otimes P$ is projective. 
Multi-ring categories are not so common, so let me specialize a little:
Let $P$ be a projective object in a tensor category $C$.  Then $P\otimes X$ is projective, for any $X\in C$.
The other things I mentioned are automatic in a tensor category.
The proof is that being projective means that the functor $Hom(P, -)$ is exact.  Since we have $Hom(P\otimes X,-) \cong Hom(P,-\otimes ^*X)$ (right dual), it means that $P\otimes X$ is exact whenever $P$ is (tensoring with an object is always exact).