# Is there a “correct” general setting for the principle: “tensoring any object with a projective object yields another projective”?

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?

-
add comment

## 1 Answer

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).

-
strange, I edited to correct a typo, and it created a duplicate. I'll check back in a second and delete the extra one, if it's still there. –  David Jordan Sep 7 '10 at 23:23
ok, deleted the extra. –  David Jordan Sep 7 '10 at 23:24
This is certainly the right way to approach some of the earlier examples and has become the standard proof of the tensoring-with-a-projective principle in a more limited case for the BGG category O (where it only makes sense in general to tensor with a finite dimensional module). I'll have to look closer at how other work like that of Garland-Lepowsky fits into the picture. Tensor categories clearly do provide a nice unified framework. –  Jim Humphreys Sep 8 '10 at 12:24
Yes, my answer was rather naive =]. –  David Jordan Sep 8 '10 at 14:29
It's not really naive, but I wonder about how restrictive tensor categories are for situations like Kac-Moody theory as well as for the somewhat dual but similar results on injectives for group schemes. For instance, I hope to minimize finiteness conditions. –  Jim Humphreys Sep 9 '10 at 20:30
add comment