Derived Hall Algebras: Examples and Counterexamples

Let $S=k$ concentrated in degree $0$, let $x=S$, $y=S[1]$, and $z=S\oplus S[1]$. Since $\mathrm{Hom}(S,S[1])=0$ and $\mathrm{Hom}(S,S)=k$, every map $f:S\to z$ is $f_a=(a,0)$ for a unique $a\in k$. Its cone is isomorphic to $S[1]$ exactly when $a\mathrm{\ne }0$: for $a\mathrm{\ne }0$ the first component is an isomorphism and cancels in the cone, while for $a=0$ the cone has nonzero degree-zero cohomology. Hence $\mathrm{\mid }[x,z{]}_y\mathrm{\mid }=q-1$ and $\mathrm{\mid }\mathrm{Aut}(x)\mathrm{\mid }=q-1$, so $n_{x,y}^z=1$.

In this category ${\mathrm{Ext}}^{-1}(x,z)=\mathrm{Hom}(S,z[-1])\cong k$, while ${\mathrm{Ext}}^{-i}(x,z)=0$ for $i\ge 2$ and ${\mathrm{Ext}}^{-i}(x,x)=0$ for all $i>0$. Thus the higher-Ext factor in [F1] is $q^{-1}$ and the Toën coefficient is $g_{x,y}^z=q^{-1}\mathrm{\ne }1=n_{x,y}^z$. The claimed equality is false.

Kernels and cokernels of representation morphisms are formed vertexwise and inherit the arrow map; the coimage-to-image map is an isomorphism at both vertices, so $\mathcal{A}$ is abelian. Replacing each vertex space by $k^m$ gives a set-sized equivalent category of matrices $k^m\to k^n$. A morphism is a pair of matrices satisfying the arrow-commuting equation, hence each Hom set is finite.

The representations $P_1=(k\stackrel{\mathrm{i}\mathrm{d}}{\to }k)$ and $P_2=(0\to k)$ are projective because $\mathrm{Hom}(P_1,V)\cong V_1$ and $\mathrm{Hom}(P_2,V)\cong V_2$, and vertex evaluation preserves epimorphisms. For $V=(V_1\stackrel{a}{\to }{V}_2)$, the sequence $0\to P_2{\otimes }_k{V}_1\stackrel{\iota }{\to }(P_1{\otimes }_k{V}_1)\oplus (P_2{\otimes }_k{V}_2)\stackrel{\pi }{\to }V\to 0$ has ${\pi }_1={\mathrm{i}\mathrm{d}}_{V_1}$, ${\pi }_2(u,v)=a(u)+v$, and ${\iota }_2(u)=(u,-a(u))$ with ${\iota }_1=0$. Its vertexwise kernel equals the image of $\iota$, so this is a length-one projective resolution. The resulting two-term projective complex $P$ is homotopically projective: for any acyclic cochain complex $C$ and chain map $f:P\to C$, projectivity lifts $f^0:P^0\to Z^0(C)$ through $d_C^{-1}:C^{-1}\twoheadrightarrow Z^0(C)$ to $h^0$; then $u^{-1}=f^{-1}-h^0{d}_P^{-1}$ lands in $Z^{-1}(C)$, and projectivity lifts it through $d_C^{-2}$ to $h^{-1}$. The equations $f^0=d_C^{-1}{h}^0$ and $f^{-1}=d_C^{-2}{h}^{-1}+h^0{d}_P^{-1}$ give a null-homotopy; only two lifts are used. By [F3], ${\mathrm{Hom}}_{D^b(\mathcal{A})}(V,W[i])$ is computed by $\mathrm{Hom}(P,W[i])$, which vanishes for $i>1$. Thus the canonical heart is hereditary in the cohomological-dimension sense of [F4].

A middle representation has dimension vector $(1,1)$ and is a scalar map $\lambda :k\to k$. Isomorphisms act by $(g_1,g_2)\cdot \lambda =g_2\lambda g_1^{-1}$ with $g_1,g_2\in k^{\times }$. The zero scalar is one orbit, and all nonzero scalars form one orbit: for $\lambda ,\mu \mathrm{\ne }0$, $g_1=1$ and $g_2=\mu {\lambda }^{-1}$ send $\lambda$ to $\mu$. Hence the only middle classes are $S_1\oplus S_2$ and $P_1$.

A short exact sequence with fixed endpoints has middle dimension vector equal to the sum of their dimension vectors. There are finitely many arrow matrices for that dimension vector and finitely many injection and quotient matrices, so there are finitely many short exact sequences; their Yoneda classes are a quotient of this finite set. Thus $\mathcal{A}$ is small finitary as required by [F1].

In either middle representation the only subrepresentation isomorphic to $S_2$ has vertex spaces $(0,k)$, and its quotient is $S_1$. The short exact sequence fixes dimension vector $(1,1)$, so no other middle classes contribute. Thus $g_{S_1,S_2}^{S_1\oplus S_2}=1$ and $g_{S_1,S_2}^{P_1}=1$.

By steps 1.1–1.2 and [A1], $\mathcal{A}$ satisfies the finitary, hereditary, boundedness and exact-heart hypotheses of [F4]. Applying [F4] to the two coefficients from step 2.2 gives the displayed derived Hall product; the order is the opposite-heart order, as stated.