Representation stacks and exact-sequence correspondences of a finite quiver

Let $Q=(Q_0,Q_1,s,t)$ be a finite quiver, in the convention of Integral path ring of a finite quiver, and work over $\mathbb{C}$. A dimension vector is a function $\mathbf{d}:Q_0\to {\mathbb{Z}}_{\ge 0}$. Put $V_i^{\mathbf{d}}={\mathbb{C}}^{d_i},{\mathrm{Rep}}_Q(\mathbf{d})={⨁}_{h:i\to j}{\mathrm{Hom}}_{\mathbb{C}}(V_i^{\mathbf{d}},V_j^{\mathbf{d}}),G_{\mathbf{d}}={\prod }_{i\in Q_0}\mathrm{GL}(V_i^{\mathbf{d}})\mathrm{.}$ The group acts by change of basis: $(g\cdot X{)}_h=g_j{X}_h{g}_i^{-1}$. The representation stack ${\mathfrak{M}}_{\mathbf{d}}=[{\mathrm{Rep}}_Q(\mathbf{d})\mathrm{/}{G}_{\mathbf{d}}]$ is the quotient stack: over a scheme $T$, its objects are left $G_{\mathbf{d}}$-torsors $P\to T$ with equivariant maps $P\to {\mathrm{Rep}}_Q(\mathbf{d})$; arrows are equivariant isomorphisms commuting with these maps. The finite-dimensional atlas and the algebraic-stack conditions are verified in Affine quotients by products of general linear groups are algebraic stacks.

For two dimension vectors $\mathbf{a},\mathbf{b}$, write $\mathbf{d}=\mathbf{a}+\mathbf{b}$ and choose $V_i^{\mathbf{d}}=V_i^{\mathbf{a}}\oplus V_i^{\mathbf{b}}$. Let ${\mathrm{Rep}}_Q(\mathbf{a},\mathbf{b})$ be the vector space of representations preserving $V^{\mathbf{a}}$. For an arrow $h:i\to j$, its matrix has the form $X_h=\left(\begin{array}{cc}{X}_h^{\mathbf{a}}& C_h\\ 0& X_h^{\mathbf{b}}\end{array}\right),C_h\in \mathrm{Hom}(V_i^{\mathbf{b}},V_j^{\mathbf{a}})\mathrm{.}$ Let $P_{\mathbf{a},\mathbf{b}}\subseteq G_{\mathbf{d}}$ be the subgroup preserving the chosen graded subspaces. Define the exact-sequence stack ${\mathfrak{E}}_{\mathbf{a},\mathbf{b}}=[{\mathrm{Rep}}_Q(\mathbf{a},\mathbf{b})\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]\mathrm{.}$ Its maps are ${\mathfrak{M}}_{\mathbf{a}}\times {\mathfrak{M}}_{\mathbf{b}}\stackrel{q}{\leftarrow }{\mathfrak{E}}_{\mathbf{a},\mathbf{b}}\stackrel{p}{\to }{\mathfrak{M}}_{\mathbf{d}}\mathrm{.}$ The map $q$ sends an exact sequence to its subrepresentation and quotient; $p$ forgets the subrepresentation. Thus the order convention is $0\to M_{\mathbf{a}}\to M_{\mathbf{d}}\to M_{\mathbf{b}}\to 0$. The block form shows that the off-diagonal extension parameters are the coordinates $C_h$; it does not assert that extension classes are a vector space after quotienting by automorphisms.

\mathfrak M_{\mathbf a}\times\mathfrak M_{\mathbf b}
& \mathfrak E_{\mathbf a,\mathbf b}
  \arrow[l,"q"'] \arrow[r,"p"]
& \mathfrak M_{\mathbf a+\mathbf b}

The diagram records only the two maps just defined; it supplies no equality of maps or algebraic operation by itself.

The untwisted terminology below refers to the absence of Euler-form signs, automorphism denominators, and Tate twists in the product convention. The complex orientation in equivariant Borel–Moore homology remains part of the cohomology theory.