Cohomological Hall Algebras and Quiver Correspondences: Examples and Counterexamples

A one-loop representation of dimension one is a scalar, and change of basis by ${\mathrm{GL}}_1$ fixes it because conjugation by a scalar is trivial. A dimension-two representation is a $2\times 2$ matrix with conjugation action. Preserving a chosen line makes the matrix upper triangular in an adapted basis, as displayed.

For this quiver the off-diagonal rank is $e=1$ and the unipotent vertex-block rank is $u=1$; hence $q$ has relative stack dimension $e-u=0$, agreeing with $-{\chi }_Q(1,1)=0$. For the splitting change represented by $t\in \mathrm{Hom}(\mathbb{C},\mathbb{C})$, the change-of-basis matrix $g_t=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)$ acts by $A\mapsto g_{t}A{g}_t^{-1}$, sending $c$ to $c+(\mu -\lambda )t$. Thus when $\lambda \mathrm{\ne }\mu$ the extension coordinate is removed by a unique splitting change, whereas when $\lambda =\mu$ it is unchanged; this is consistent with a zero-dimensional stack morphism, not a claim that all extension fibers are points.

The fiber of $p$ at a scalar matrix $\lambda I$ is all of ${\mathbb{P}}^1$, so it is positive-dimensional. For a diagonal matrix with distinct eigenvalues, an invariant line must be an eigenspace, giving exactly two points. For a non-scalar Jordan matrix $\lambda I+N$ with $N^2=0$ and $\mathrm{rank}N=1$, an invariant line is $N$-stable; the nilpotent map induced on a one-dimensional line is zero, so the line lies in $\ker N$, which is the unique invariant line. These fibers agree with the closed incidence projection to ${\mathrm{Mat}}_2(\mathbb{C})\times {\mathbb{P}}^1$ and do not affect properness.

The incidence equations $A(L)\subseteq L$ are closed in ${\mathrm{Mat}}_2(\mathbb{C})\times {\mathbb{P}}^1$, so the forgetful map $p$ is projective and proper by [F2]. The example includes a loop and checks why stack-relative dimension and properness must be read from the specified correspondence, not from finiteness of its fibers. AC is inherited through [A1].

The trace definition in [F1] gives $W_1(z)=z^2$. Its differential is $d{W}_1=2zdz$; since the coefficient field has characteristic zero, this vanishes exactly at $z=0$, where $W_1(0)=0$. Thus the critical locus and critical value are as stated.

For any sufficiently small $ϵ\mathrm{\ne }0$, the local Milnor fiber $z^2=ϵ$ in a disk about zero consists of the two distinct points $\sqrt{ϵ}$ and $-\sqrt{ϵ}$. Its reduced zeroth cohomology is $\mathbb{Q}$, generated by the difference of the two point classes. When $ϵ$ goes once around zero, the two roots exchange, so this generator maps to its negative. The normalized vanishing-cycle stalk $\mathrm{\Phi }$ is therefore one-dimensional in degree one, with monodromy $-1$, by [F3]. The homological coefficient is its normalized equivariant Verdier dual as in [F1]; it must not be confused with an unshifted Milnor cohomology stalk.

This local calculation agrees with the degree-one normalization of [F3] and, under Thom–Sebastiani, with degree two for $x^2+y^2$. Trace additivity [F2] gives the extension value convention. The full operation package is proved by its separate supplier; this example verifies the specified local fibre, degree and monodromy.