| Summary: | We begin by reviewing the theory of NC-schemes and NC-smoothness, as introduced by Kapranov in \cite{Kapranov} and developed further by Polishchuk and Tu in \cite{PT}.
For a smooth algebraic variety $X$ with a torsion-free connection $\nabla$, we study modules over the NC-smooth thickening $\tw \O_X$ of $X$ constructed in \cite{PT} via NC-connections. In particular we show that the NC-vector bundle $\tw E_{\bar\nabla}$ constructed via mNC-connections in \cite{PT} from a vector bundle $(E,\bar\nabla)$ with connection additionally admits a bimodule extension at least to nilpotency degree 3.
Next, in joint work with A. Polishchuk \cite{DP}, we show that the gap, as first noticed in \cite{PT}, in the proof from \cite{Kapranov} that certain functors are representable by NC-smooth thickenings of moduli spaces of vector bundles is unfixable. Although the functors do not represent NC-smooth thickenings, they lead to a weaker structure of \textit{NC-algebroid thickening}, which we define. We also consider a similar construction for families of quiver representations, in particular upgrading some of the quasi-NC-structures of \cite{Toda1} to NC-smooth algebroid thickenings.
This thesis includes unpublished co-authored material.
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