Picard group of the tensor product $\bar K \otimes_K \bar K$ (product of schemes) Let $K$ be a field. What is the Picard group of the ring $R := \bar K \otimes_K \bar K$?

I have seen the question Picard group of product of spaces, but it only deals with varieties over an algebraically closed field.
If $K$ is perfect, then $R$ is a direct product of copies of $\bar K$ indexed by the absolute Galois group of $K$. I guess in that case that $\operatorname{Pic}(R)=0$, but I am not sure. Moreover, in general, $R$ may not be reduced.
 A: Indeed, $\operatorname{Pic} R=0$. This follows from a few facts:

*

*If $S=\lim S_i$ is a limit of a directed system of qcqs schemes $S_i$ with affine transition maps, then $\operatorname{Pic} S = \operatorname{colim} \operatorname{Pic} S_i$ (ref)

*Every finite-dimensional algebra over a field is a finite direct product of Artinian local rings (ref 1 + ref 2)

*The Picard group of a finite direct product of rings is the product of the Picard groups of the factors (ref)

*Line bundles over affine schemes correspond to projective modules (ref)

*Any projective module over a local ring is free (ref)

By 1, we may reduce to considering the case of $\overline{K}\otimes_K L$ for finite $L/K$, each of which are a finite-dimensional algebra over $\overline{K}$. By 2, such an algebra splits as a finite direct product of Artinian local rings. Then by 3 we can reduce to considering each of the factors. Next, by 4, we're looking to classify all rank one projective modules over each of these local rings, but by 5 they're all free, and so the Picard group vanishes. (Thanks to Aphelli for pointing out part 1.)
