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Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product $ \cal{E} \otimes_A \cal{E} $ is again finitely generated, projective, and of finite rank. My question is when does it hold that $$ \text{ rank}({\cal E} \otimes_A {\cal E}) \leq n^2 ~ ? $$ To be more specific, I am looking for a set of sufficient (and hopefully natural) conditions for this to hold.

I will volunteer the condition of locality as a sufficient condition. Since in this case we have a well defined notion of minimal number of generators, which coincides with the rank, and so, the tensor product should (?) have a set of generators of size less than or equal to that of the square of the original generating set. I may be totally wrong here though!

[Edit] With regard to the definition of the rank of a projective module, I am using the one found here. The algebras I am interested in are sub-algebras of (noncommutative) Hopf algebras over the complex numbers, and so, the homomorphism $\phi$ can just be taken to be the counit $\epsilon$ of the Hopf algebra.

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I think you must be missing some conditions if this is to make sense. For a general non-commutative algebra $A$, $\cal{E}\otimes_A\cal{E}$ only makes sense if $\cal{E}$ is both a left and right $A$-module structure. Also, what do you mean by "rank"? –  Jeremy Rickard Apr 28 '13 at 11:58
    
Yes, quite true, I am thinking here of bimodules, I will change it now. –  Milan Bernolak Apr 28 '13 at 12:05
    
What definition of rank are you thiking of? –  Fernando Muro Apr 28 '13 at 12:08
    
P.S. For the definition of the rank of a projective module I am using the K-theoretic version found here encyclopediaofmath.org/index.php/Rank_of_a_module –  Milan Bernolak Apr 28 '13 at 12:11
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If $A$ is a Hopf algebra over $\mathbb{C}$, and you take $\cal{E}=A\otimes A$, then $\cal{E}$ is a finitely generated projective bimodule that has rank 1 by this definition, but $\cal{E}\otimes_A\cal{E}$ is isomorphic to a direct sum of $d=\dim_{\mathbb{C}}(A)$ copies of $\cal{E}$, and so will have rank $d$ (and won't even be finitely generated unless $A$ is finite-dimensional). Are you sure this is what you mean? –  Jeremy Rickard Apr 28 '13 at 19:02

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