Cohomology of the Quot scheme of an infinite affine space

We study the Quot scheme of points ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_d({\mathcal{O}}_{{\mathbb{A}}^n}^{\oplus r})$. We exhibit and compute the cohomology of explicit loci in ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_d({\mathcal{O}}_{{\mathbb{A}}^n}^{\oplus r})$, whose complement has codimension diverging to infinity as $n\to \mathrm{\infty }$. In the case $1<r<\frac{d+1}{2}$ this loci is an irreducible component. The main ingredient in our proof are classification results on maximal-dimensional spaces of commutative matrices satisfying certain generating conditions. Our primary motivation is the study of the ind-scheme $${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_d({\mathcal{O}}_{{\mathbb{A}}^{\mathrm{\infty }}}^{\oplus r}):={\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_d({\mathcal{O}}_{{\mathbb{A}}^n}^{\oplus r})\mathrm{.}$$ Finally, we compute the cohomology (with integral coefficients) of the Quot scheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_2({\mathcal{O}}_{{\mathbb{A}}^n}^{\oplus r})$, confirming, in the case $d=2$, a conjecture of Pandharipande.