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Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective?

The algebra $A$ cannot be finite-dimensional, since then $A$ is Frobenius and so is $A^{\mathrm{op}}\otimes_k A$.

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[Nagata, Masayoshi. A conjecture of O'Carroll and Qureshi on tensor products of fields. Japan. J. Math. (N.S.) 10 (1984), no. 2, 375--377. MR0884425] proved that the Krull dimension of the tensor product $\mathbb C(x,y)\otimes_{\mathbb C}\mathbb C(x,y)$ is $2$. If this tensor product is reduced (I think it is, but it is late...) then it cannot be self-injective. In that case, $\mathbb C(x,y)$ is an example.

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  • $\begingroup$ ¡Gracias Mariano! Es un ejemplo estupendo $\endgroup$
    – Fernando Muro
    Commented Jun 19, 2011 at 21:05
  • $\begingroup$ In fact it seems that $k(x)\otimes_k k(x)$ is also valid since it has Krull dimension 1 and is also reduced, being $k\subset k(x)$ separable. $\endgroup$
    – Fernando Muro
    Commented Jun 19, 2011 at 21:08

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