I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image.
Towards the end, he has this representation $R_p\bigotimes\Lambda$ of the quiver $Q$. My questions are:
- is the tensor product taken over the field $\mathbb{K}$?
- what are the $\Lambda$-modules assigned at each vertex and what are the $\Lambda$-module morphisms assigned for each arrow for this representation $R_p\bigotimes\Lambda$?
(Here, $R_p$ is the usual representation of the quiver $Q$ where for each vertex you have a $\mathbb{K}$-vector space and for each arrow you have a $\mathbb{K}$-linear map between the corresponding vector spaces)
Best Answer
I think that $\Lambda$ should be a $k$-algebra and the tensor product should be over $k$. Then $R_p\otimes_k \Lambda$ makes sense as a representation of $Q$ over $\Lambda$, or equivalently as a $\Lambda Q$-module.
Namely, the Grassmannian $\mathcal{Gr}(R_p,\alpha)$ doesn't make sense as a scheme, but only as a $k$-scheme, determined by a functor from $k$-algebras to sets. In this case the functor sends a $k$-algebra $\Lambda$ to the set of summands of $R_p \otimes_k \Lambda$ of rank $\alpha$ (i.e., localizing at any maximal ideal of $\Lambda$, the projective $\Lambda$-module corresponding to a vertex $v$ should give a free module over the localization of rank $\alpha(v)$).