5
$\begingroup$

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 $A^{\mathrm{op}}\otimes_k A$.

$\endgroup$
5
$\begingroup$

[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.

$\endgroup$
  • $\begingroup$ ¡Gracias Mariano! Es un ejemplo estupendo $\endgroup$ – Fernando Muro Jun 19 '11 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 Jun 19 '11 at 21:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.