The quiver exact-sequence correspondence is smooth and proper

Put $Y={\mathrm{Rep}}_Q(\mathbf{a},\mathbf{b})$. The block description makes $Y$ an affine finite-type scheme. The block parabolic is $P_{\mathbf{a},\mathbf{b}}=U⋊L$ with underlying scheme ${\mathbb{A}}^u\times L$, where $L=G_{\mathbf{a}}\times G_{\mathbf{b}}$; hence $P_{\mathbf{a},\mathbf{b}}$ is affine and finite type, and it is smooth because $U$ is a vector group and each general linear factor of $L$ is smooth by [F5]. The quotient definition in [F1] makes $Y\to [Y\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]$ representable: after base change by a scheme $T\to [Y\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]$ corresponding to a $P_{\mathbf{a},\mathbf{b}}$-torsor $R\to T$, the pullback is $R$. Fppf-local triviality and [F8] make this torsor an affine scheme over $T$. To see that $R\to T$ is smooth, extend its structure group to $G_{\mathbf{a}+\mathbf{b}}$; the resulting $G$-torsor is a tuple of frame bundles of vector bundles $E_i$, while the reduction to $P_{\mathbf{a},\mathbf{b}}$ specifies subbundles $F_i\subset E_i$ of ranks $a_i$ with locally free quotients. Zariski-locally these sequences split and the bundles admit adapted frames, so $R\to T$ is locally $P_{\mathbf{a},\mathbf{b}}\times T\to T$, which is smooth by [F9] and [F10]; it is surjective as a torsor. For two underlying $P_{\mathbf{a},\mathbf{b}}$-torsors $R,R^{\mathrm{\prime }}$ over $T$, their equivariant isomorphism functor is fppf-locally represented by $P_{\mathbf{a},\mathbf{b}}\times T$, and these affine local representatives descend by [F8]. Fppf-locally trivialize both torsors, with their equivariant maps given by $y,y^{\mathrm{\prime }}:T^{\mathrm{\prime }}\to Y$. The compatible isomorphisms are cut out in $P_{\mathbf{a},\mathbf{b}}\times T^{\mathrm{\prime }}$ by $g\cdot y=y^{\mathrm{\prime }}$, the inverse image of ${\mathrm{\Delta }}_Y$ under $(g,t)\mapsto (g\cdot y(t),y^{\mathrm{\prime }}(t))$. This is closed because $Y$ is separated, and the closed subscheme descends effectively by [F8]. Thus the diagonal is affine representable. Effective descent of torsors with maps is obtained by descending their affine schemes, actions and compatible maps by the affine-descent argument in the proof of [F2]; that argument uses only affineness of the group and target, so also applies to this parabolic and to its vector subgroup $U$. By [F8], $[Y\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]$ is an algebraic stack. Now, for each arrow $h:i\to j$, the free off-diagonal block is $\mathrm{Hom}(V_i^{\mathbf{b}},V_j^{\mathbf{a}})$; hence $Y\to {\mathrm{Rep}}_Q(\mathbf{a})\times {\mathrm{Rep}}_Q(\mathbf{b})$ is a vector bundle of rank $e={\sum }_{h:i\to j}{b}_i{a}_j$. The kernel $U$ of $P_{\mathbf{a},\mathbf{b}}\to L$ consists of upper-right vertex blocks and has dimension $u={\sum }_i{a}_i{b}_i$.

First construct the relative Grassmannian used here. For a rank-$n$ bundle $E$ on any scheme $T$ and $0\le a\le n$, impose the homogeneous Plucker relations of [F3] in ${\mathbb{P}}_T(({\bigwedge }^{a}E{)}^{\vee })$. On a frame chart with $p_I\mathrm{\ne }0$, normalize $p_I=1$. The one-replacement coordinates give the entries of a matrix with identity pivot rows; the Plucker relation expressing a coordinate with $s$ nonpivot indices in terms of coordinates with fewer such indices proves inductively that every coordinate is its corresponding maximal minor. This is the calculation in [F3], using polynomial identities and only inversion of $p_I$, hence valid over every $\mathbb{C}$-algebra, including nonreduced ones. Conversely those minors satisfy all relations by alternating multilinearity. Thus each chart has polynomial coordinate ring in $a(n-a)$ entries and represents graph subbundles whose projection onto the pivot summand is invertible. Any locally direct-summand rank-$a$ subbundle is covered by such charts, since its maximal minors generate the unit ideal. The minor-ratio transitions identify the same subbundle, so the charts represent the subbundle functor, glue under change of frame, and give a closed relative Plucker embedding. Ranks $0,n$ give $T$. Now let an arbitrary scheme $T$ map to ${\mathfrak{M}}_{\mathbf{a}+\mathbf{b}}$. By the quotient-stack definition this is a family of vector bundles $E_i$ of ranks $a_i+b_i$, with arrow maps. The base change of $p$ parametrizes subbundles $F_i\subset E_i$ of ranks $a_i$ preserved by every arrow. It is the closed subscheme of the relative Grassmannian ${\prod }_i{\mathrm{Gr}}_T(a_i,E_i)$ cut out by the vanishing of the bundle maps $F_i\to E_j\to E_j\mathrm{/}{F}_j$ for each arrow $h:i\to j$. The relative Plucker map sends $F_i$ to the line subbundle $\det (F_i)\subseteq {\bigwedge }^{a_i}{E}_i$. Under the quotient convention of Projective bundle in the quotient convention, this is represented by the quotient line of $({\bigwedge }^{a_i}{E}_i{)}^{\vee }$ obtained by dualizing the inclusion. Thus each relative Grassmannian is a closed subscheme of ${\mathbb{P}}_T(({\bigwedge }^{a_i}{E}_i{)}^{\vee })$; on local frames this is the closed embedding just constructed from [F3], and the equations glue under changes of frame. Each such projective bundle is proper: after any base change $T^{\mathrm{\prime }}\to T$, trivialize $E_i$ over an open cover of $T^{\mathrm{\prime }}$; on each open the projection is a projective-space projection, hence proper by [F4]. The image of a closed set is closed on each member of the cover and therefore closed globally; separatedness and finite type are also local on the base. Thus every base change is proper by [F7]. The product of these proper maps is proper by base change and composition [F4]. The arrow conditions are closed because they are zero loci of regular bundle maps, so the incidence scheme is proper over $T$ as a closed subscheme of that product [F4]. In the standard frame atlas this scheme is exactly $I_{\mathbf{a},\mathbf{b}}=\{(X,(F_i)):\dim F_i=a_i,X_h(F_i)\subseteq F_j\text{ for every }h:i\to j\}\mathrm{.}$, and its defining equations are the same zero-locus equations just described; in an ordinary Grassmannian chart they are the vanishing of the complementary matrix block. Hence every scheme base change of $p$ is a proper scheme.

Pull back $q$ along the atlas $Z={\mathrm{Rep}}_Q(\mathbf{a})\times {\mathrm{Rep}}_Q(\mathbf{b})\to {\mathfrak{M}}_{\mathbf{a}}\times {\mathfrak{M}}_{\mathbf{b}}$. This fixes the endpoint representations. Every middle representation is a block matrix from step 1.1, and every isomorphism inducing the identity on the fixed endpoints is a unique upper-right vertex block. Hence the pullback stack is $[Y\mathrm{/}U]\to Z$, where $Y\to Z$ is the rank-$e$ vector bundle of off-diagonal blocks and $U$ is the rank-$u$ vector group of changes of splitting. For any scheme $T\to [Y\mathrm{/}U]$, its base change is a $U\cong {\mathbb{G}}_a^u$-torsor, represented by an affine scheme using [F8]. Cover $T$ by affine opens. On each affine base, choose an affine fppf cover of the torsor as in [F11]; the unique group element between the two pulled-back torsor points is a vector $\delta \in (B{\otimes }_{A}B{)}^u$ satisfying the additive Amitsur $1$-cocycle equation coordinatewise. Exactness in [F11] gives $b\in B^u$ with $\delta =b\otimes 1-1\otimes b$, and translation by $b$ descends a section, trivializing the torsor there. Thus $Y\to [Y\mathrm{/}U]$ is Zariski-locally $U\times T\to T$, which is smooth by [F9] and [F10], and it is surjective as a torsor. The same inverse-image-of-${\mathrm{\Delta }}_Y$ argument as in step 1.1, with $U$ in place of $P_{\mathbf{a},\mathbf{b}}$, gives $[Y\mathrm{/}U]$ an affine representable diagonal, so it is an algebraic stack. Its composite $Y\to Z$ is a vector bundle projection of relative dimension $e$. By the atlas convention just stated, $[Y\mathrm{/}U]\to Z$ is smooth of relative stack dimension $e-u$. Smoothness and relative dimension are checked after the displayed smooth target atlas, so the same holds for $q$. Substitution of the definition of ${\chi }_Q$ gives $e-u=-{\chi }_Q(\mathbf{b},\mathbf{a})$, including arrows that are loops or occur with multiplicity.

The construction in step 1.2 identifies every scheme base change of $p$ with the incidence scheme, so [F7] proves that $p$ is representable by schemes, hence by algebraic spaces; the same step proves every such base-changed map is proper. If some $a_i=0$ or $a_i=a_i+b_i$, the corresponding Grassmannian is a point and the same closed-incidence argument applies. Step 2.1 proves the smoothness of $q$, so both claims of the Statement follow. AC is used only through [A1].