Cohomology of the Quot scheme of an infinite affine space

Abstract: We study the Quot scheme of points $$\mathrm{Quot}<em>d(\mathcal{O}</em>{\mathbb{A}^{{n}}^{\oplus} r})$$. We exhibit and compute the cohomology of explicit loci in $$\mathrm{Quot}<em>d(\mathcal{O}</em>{\mathbb{A}^{{n}}^{\oplus} r})$$, whose complement has codimension diverging to infinity as $n\rightarrow \infty$. In the case $1&lt;r&lt;\frac{d+1}{2}$ 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{Quot}d(\mathcal{O}{\mathbb{A}^{\infty}}^{\oplus r}) := \underset{n\rightarrow \infty}{colim} \mathrm{Quot}d(\mathcal{O}{\mathbb{A}^{n}}^{\oplus r}). ] Finally, we compute the cohomology (with integral coefficients) of the Quot scheme $$\mathrm{Quot}<em>2(\mathcal{O}</em>{\mathbb{A}^{n}^{\oplus} r})$$, confirming, in the case $d=2$, a conjecture of Pandharipande.