Derived Hall Algebras from dg-Enhanced Categories

Fix a component $x\in {\pi }_0(X)$ and put $y=f(x)$. The homotopy fiber of $u^{\mathrm{\prime }}$ over $x$ is the pullback of the homotopy fiber of $u$ over $y$, hence is equivalent to it by the homotopy-cartesian condition. Since $u$ is proper, this fiber has finitely many components; therefore $u^{\mathrm{\prime }}$ is proper on components.

Fix $y^{\mathrm{\prime }}\in {\pi }_0(Y^{\mathrm{\prime }})$ and put $y=u(y^{\mathrm{\prime }})$. The homotopy fiber of $f^{\mathrm{\prime }}$ over $y^{\mathrm{\prime }}$ is equivalent to the homotopy fiber of $f$ over $y$, with its map to $X$ equal to the composite of the fiber map to $X^{\mathrm{\prime }}$ and $u^{\mathrm{\prime }}$. Thus the summands, their higher homotopy groups, and the values of $\alpha$ pulled back along $u^{\mathrm{\prime }}$ agree term by term. Both sides evaluated at $y^{\mathrm{\prime }}$ are the same finite sum from [F1].

Replace $f$ by the mapping-path fibration $p_f$ in [F2]. For a component $x$ mapping to $y$, let $P_x\subseteq {\pi }_0(F_y)$ be the components mapping to $x$. A point $z\in P_x$ bases the fiber over $y$; $p_f$ is a Hurewicz, hence Serre, fibration, and $j_f$ identifies its total-space homotopy groups with those of $X$, so [F3] applies. Its pointed-set tail identifies $P_x$ as one orbit of ${\pi }_1(Y,y)$, with stabilizer the image of ${\pi }_1(X,x)\to {\pi }_1(Y,y)$; hence $\mathrm{\mid }{P}_x\mathrm{\mid }=\mathrm{\mid }{\pi }_1(Y,y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{im}({\pi }_1(X,x)\to {\pi }_1(Y,y))\mathrm{\mid }$. For $z\in P_x$, the group tail includes ${\pi }_2(Y,y)\to {\pi }_1(F_y,z)\to {\pi }_1(X,x)\to {\pi }_1(Y,y)$, so ${\pi }_1(F_y,z)$ need not be the kernel of the last map. All groups are finite, and exactness makes the fiber groups vanish in high degrees. Expanding each finite group order as the product of its kernel and image orders through this entire tail, and using the displayed orbit count at its pointed-set end, gives $\mathrm{\mid }{P}_x\mathrm{\mid }=r_f(x,y){\prod }_{j>0}\mathrm{\mid }{\pi }_j(F_y,z){\mathrm{\mid }}^{(-1{)}^{j+1}}\mathrm{.}$ The action transports components within $P_x$, so their higher homotopy groups have the same orders.

The total weight of the fiber components mapping to $x$ is $\mathrm{\mid }{P}_x\mathrm{\mid }$ times the weight of one of them, since their homotopy-group orders agree by step 1.1. Multiplying its formula by ${\prod }_{j>0}\mathrm{\mid }{\pi }_j(F_y,z){\mathrm{\mid }}^{(-1{)}^j}$ cancels the fiber factors and gives $r_f(x,y)$.

Substitute the componentwise identity of step 2.1 into the defining fiber sum for $f_{!}$ and group terms by their image component $x$. Each $P_x$ is finite by step 1.1, and only finitely many $x$ have $\alpha (x)\mathrm{\ne }0$, so this gives the stated ratio formula and the finiteness of $f_{!}(\alpha )(y)$.

If $x\in {\pi }_0(X)$ maps to $y\in {\pi }_0(Y)$ and $y$ maps to $z\in {\pi }_0(Z)$, then [F1] gives $r_f(x,y)r_g(y,z)=r_{g\circ f}(x,z)$, because the exponents of every $\mathrm{\mid }{\pi }_i(Y,y)\mathrm{\mid }$ sum to zero.

Evaluate $g_{!}{f}_{!}\alpha$ at $z$. Its finite sum is over pairs $(x,y)$ with $f(x)=y$ and $g(y)=z$; each $x$ determines the unique component $y=f(x)$. By step 1.1 its weight is $r_{g\circ f}(x,z)$, exactly the component formula for $(g\circ f{)}_{!}\alpha (z)$ from [F1]. Hence the two pushforwards agree at every component.

For representables $h_a=\mathcal{T}(-,a)$ and $h_b=\mathcal{T}(-,b)$, evaluation at $1_a$ identifies the dg Hom complex $\underset{}{\mathrm{Hom}}(h_a,h_b)$ with $\mathcal{T}(a,b)$. Indeed, a module map $\varphi :h_a\to h_b$ is determined by ${\varphi }_a(1_a)$, and for any $\gamma :c\to a$ naturality forces ${\varphi }_c(\gamma )={\varphi }_a(1_a)\circ \gamma$; conversely composition by any homogeneous element of $\mathcal{T}(a,b)$ defines such a map and respects the differential. The standard projective presentation in [F1] computes derived Hom by this dg Hom complex, so ${⨁}_i{\mathrm{Ext}}^i(h_a,h_b)$ is finite-dimensional.

Fix $x=h_a$. Shifts and finite sums of the second argument preserve finite total Ext. If $y_1\to y_2\to y_3\to y_1[1]$ is a triangle, applying ${\mathrm{Hom}}_{\mathcal{D}(\mathcal{T})}(h_a,-[i])$ gives an exact sequence; each ${\mathrm{Ext}}^i(h_a,y_2)$ is a subquotient of the neighboring finite groups, and outside their finite degree range it is zero. Direct summands give subgroups. Induction through the finite construction of a perfect $y$ proves the claim for $(h_a,y)$.

For fixed perfect $y$, the contravariant long exact Hom sequence gives the same closure under shifts, finite sums, cones, and retracts in the first argument. Induction through the finite construction of $x$ extends step 2.1 to every pair of perfect modules. The finite support in degrees is preserved at each finite construction step, proving the statement without a choice of resolutions or simultaneous representatives.

Fix a perfect module $a$. The based loop space of $X^{(0)}$ at $a$ is the equivalence component of $\mathrm{Map}(a,a)$. Its components are $\mathrm{Aut}(a)$, a subset of the finite vector space ${\mathrm{Ext}}^0(a,a)$, and for $i>1$, [F2] gives ${\pi }_i(X^{(0)},a)={\mathrm{Ext}}^{1-i}(a,a)$. These groups are finite by [F1] and vanish in high degrees, so $X^{(0)}$ is locally finite.

Fix perfect modules $a,b$. A point of the homotopy fiber of $s\times t$ over $(a,b)$ is an arrow with source identified with $a$ and homotopy cofiber identified with $b$. Its connected components are distinguished triangles $a\to z\to b\to a[1]$ with both endpoints identified; the connecting map classifies the component in ${\mathrm{Hom}}_{\mathcal{D}(\mathcal{T})}(b,a[1])={\mathrm{Ext}}^1(b,a)$. The latter is finite by [F1], so this fiber has finitely many components and $s\times t$ is proper on components.

For arrows and composable pairs, the derived mapping complexes are finite homotopy limits of finite sums of the complexes $\mathrm{R}\mathrm{H}\mathrm{o}\mathrm{m}(a,b)$ between their finitely many entries. By [F1], each such complex has finite-dimensional cohomology in a finite degree range. A finite direct sum or homotopy fiber of these complexes again has finite-dimensional cohomology in a finite degree range. The same loop-space calculation as in step 1.1 therefore shows that $X^{(1)}$ and $X^{(2)}$ are locally finite.

By [F4], pullback along $s\times t$ preserves finite support, and pushforward along $c$ sends a finite-support function to a finite-support function. The homotopy groups of every fiber used in the pushforward are finite and eventually zero by steps 1.1–1.2, so each weight is a finite rational number. Thus the convolution in Toën derived Hall algebra by homotopy-cofiber convolution is a well-defined finite-support function.

The loop space at $f$ of the restricted arrow core over the fixed target $z$ is the homotopy fiber over $f$ of postcomposition $f\circ -:\mathrm{Map}(x,x{)}^{\simeq }\to \mathrm{Map}(x,z)$, based at ${\mathrm{i}\mathrm{d}}_x$. Here $\mathrm{Map}(x,x{)}^{\simeq }$ is the union of components represented by isomorphisms in $\mathcal{D}(\mathcal{T})$; using the full mapping space would also include noninvertible endomorphisms and would not compute loops in the core. Since this is a union of components, its positive homotopy groups at ${\mathrm{i}\mathrm{d}}_x$ are ${\mathrm{Ext}}^{-i}(x,x)$, while [F2] gives ${\pi }_i\mathrm{Map}(x,z)={\mathrm{Ext}}^{-i}(x,z)$. The long exact sequence in [F3] therefore gives, for each $i\ge 1$, ${\mathrm{Ext}}^{-i-1}(x,x)\to {\mathrm{Ext}}^{-i-1}(x,z)\to {\pi }_{i+1}(C_f)\to {\mathrm{Ext}}^{-i}(x,x)\to {\mathrm{Ext}}^{-i}(x,z)$. Its low-degree tail is ${\mathrm{Ext}}^{-1}(x,x)\to {\mathrm{Ext}}^{-1}(x,z)\to {\pi }_1(C_f)\to \mathrm{Aut}(x)\to {\mathrm{Hom}}_{\mathcal{D}(\mathcal{T})}(x,z)$; the last map has kernel $\mathrm{Aut}(f\mathrm{/}z)$, so the preceding image in $\mathrm{Aut}(x)$ is exactly that stabilizer and the induced map onto it is surjective.

All Ext groups in these sequences are finite and vanish outside a finite range by [F2]. The group $\mathrm{Aut}(f\mathrm{/}z)$ is a subgroup of the finite group $\mathrm{Aut}(x)\subseteq {\mathrm{Ext}}^0(x,x)$. Thus all groups in the displayed exact sequence are finite and the sequence terminates. For a finite exact sequence of finite groups, the product of group orders with alternating exponents is $1$: at each term, its order is the product of the kernel order and image order, and adjacent image factors cancel. Applying this cancellation to step 1.1 yields $\mathrm{\mid }{\pi }_1(C_f){\mathrm{\mid }}^{-1}{\prod }_{j\ge 2}\mathrm{\mid }{\pi }_j(C_f){\mathrm{\mid }}^{(-1{)}^j}=\mathrm{\mid }\mathrm{Aut}(f\mathrm{/}z){\mathrm{\mid }}^{-1}{\prod }_{i>0}\mathrm{\mid }{\mathrm{Ext}}^{-i}(x,z){\mathrm{\mid }}^{(-1{)}^i}{\prod }_{i>0}\mathrm{\mid }{\mathrm{Ext}}^{-i}(x,x){\mathrm{\mid }}^{(-1{)}^{i+1}}$, which is the claimed homotopy weight.

For chain maps $x\stackrel{f}{\to }y\stackrel{g}{\to }z$, put $h=gf$. With the mapping-cone convention $\mathrm{Cone}(f{)}^n=y^n\oplus x^{n+1}$ and $d(y,x)=(d_{y}y+fx,-d_{x}x)$, the induced map is $k(y,x)=(gy,x)$. A point of the first homotopy pullback consists of arrows $f$ and $g$ with the target of $f$ identified with the source of $g$, together with an object identified with the cofiber of $g$. To compare this cofiber coordinate with the image of a composable pair, write a degree-$n$ element of $\mathrm{Cone}(k)$ as $(z,u,y^{\mathrm{\prime }},x^{\mathrm{\prime }})\in z^n\oplus x^{n+1}\oplus y^{n+1}\oplus x^{n+2}$. Its differential is $(d_{z}z+gfu+g{y}^{\mathrm{\prime }},-d_{x}u+x^{\mathrm{\prime }},-d_y{y}^{\mathrm{\prime }}-f{x}^{\mathrm{\prime }},d_x{x}^{\mathrm{\prime }})$. Under the invertible coordinate change $\tilde{y}=y^{\mathrm{\prime }}+fu$, $\tilde{x}=x^{\mathrm{\prime }}-d_{x}u$, this becomes $(d_{z}z+g\tilde{y},\tilde{x},-d_y\tilde{y},0)$, the direct sum of $\mathrm{Cone}(g)$ and a contractible identity pair. Thus $\mathrm{Cone}(k)\simeq \mathrm{Cone}(g)$. The homotopy pullback of the source and target maps on arrow cores is the core of composable pairs: its data are two arrows and an equivalence between their middle objects; transporting one arrow along that equivalence gives a composable pair, and the same transport identifies equivalences of such data with objectwise equivalences of pairs. The additional cofiber coordinate is an object equipped with an equivalence to $\mathrm{Cone}(g)$, whose space of choices is contractible over each $g$. The coordinate change supplies the equivalence from $\mathrm{Cone}(k)$, so this coordinate adds no further homotopy data. Hence the first square is homotopy cartesian.

A point of the homotopy pullback in the second square consists of an object $x$, an arrow $h:x\to z$, an arrow $k:a\to b$, and an equivalence $b\simeq \mathrm{Cone}(h)$. Form $y:=z{\times }_{\mathrm{Cone}(h)}^{h}a$, the homotopy pullback of the cofiber map $z\to \mathrm{Cone}(h)$ along $k$. Pulling the fiber sequence $x\to z\to \mathrm{Cone}(h)$ back along $k$ gives a fiber sequence $x\to y\to a$, so $\mathrm{Cone}(x\to y)\simeq a$. The pullback square also identifies $\mathrm{Cone}(y\to z)$ with $\mathrm{Cone}(k)$. The composite $x\to y\to z$ is $h$, and the induced cofiber map is $k$ by the defining pullback square. The object $y$ is perfect because the perfect modules form a stable subcategory closed under finite homotopy limits. These constructions are inverse, functorially on equivalences, to the map from composable pairs to $(h,k)$; hence the second square is homotopy cartesian.

The constructions in steps 1.1 and 1.2 are functorial in maps and equivalences of the input diagrams, and their displayed inverse maps recover every object, arrow, and comparison path. They therefore give equivalences of the full homotopy pullbacks with $X^{(2)}$, proving both squares homotopy cartesian. The argument uses the enhancement's functorial cones; an unenhanced triangulated category alone would not supply these maps.

Fix a representative $x$. The fiber in [F1] fixes the target $z$ and restricts the source and cofiber to the isomorphism classes of $x$ and $y$; it retains no chosen equivalence of the source to $x$ or of the cofiber to $y$. For an arrow $x^{\mathrm{\prime }}\to z$ in that restriction, choose a source equivalence $x\simeq x^{\mathrm{\prime }}$ to express its derived morphism as $f:x\to z$ with cofiber isomorphic to $y$. A different source identification precomposes $f$ by an element of $\mathrm{Aut}(x)$. Two such representatives lie in the same component precisely when there is a source equivalence over $z$ up to homotopy: after the identifications this says $f^{\mathrm{\prime }}=f\circ a$ in the derived category for some $a\in \mathrm{Aut}(x)$ (using its inverse if the equivalence is oriented oppositely). Conversely that equality gives a homotopy between the maps and hence an equivalence of arrows in the target homotopy fiber. The cofiber condition restricts components and adds no framing. Thus the component set is exactly $[x,z{]}_y\mathrm{/}\mathrm{Aut}(x)$. This identifies only components; their higher homotopy and stabilizer weights are retained in the following weight calculation.

By [F4], the weight of the component represented by $f$ is its stabilizer reciprocal times the higher-Ext factor displayed in the first formula. This uses the homotopy exact sequence in [F4], including its low-degree stabilizer tail; in particular the stabilizer is not the component set of the mapping-space fiber.

Summing the component weights from step 1.2 over the component set in step 1.1 gives the first displayed formula by [F1].

For a finite group $G$ acting on a finite set $S$, each orbit of $s$ has size $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{Stab}(s)\mathrm{\mid }$, so $\mathrm{\mid }S\mathrm{\mid }\mathrm{/}\mathrm{\mid }G\mathrm{\mid }={\sum }_{[s]\in S\mathrm{/}G}1\mathrm{/}\mathrm{\mid }\mathrm{Stab}(s)\mathrm{\mid }$. Apply this with $G=\mathrm{Aut}(x)$ and $S=[x,z{]}_y$, then collect the common Ext factor in step 2.1. This gives the second formula.

Fix $\alpha ,\beta ,\gamma \in {\mathbb{Q}}_c(X^{(0)})$ and write $p=s\times t:X^{(1)}\to X^{(0)}\times X^{(0)}$. For any map $a$ and finite-support functions $\mu ,\nu$, one has $(a_{!}\mu )\otimes \nu =(a\times \mathrm{i}\mathrm{d}{)}_{!}(\mu \otimes \nu )$: the fiber of $a\times \mathrm{i}\mathrm{d}$ at $(y,z)$ is $F_y\times \{z\}$, with the same homotopy groups and weight as $F_y$, and its component sum is the defining one for $a_{!}$. The first square in [F4], transposed so that its lower map is $p$, is homotopy cartesian and $p$ is proper by [F2]. Base change therefore gives $(\alpha \ast \beta )\ast \gamma =c_{!}{p}^{\ast }(c\times \mathrm{i}\mathrm{d}{)}_{!}((p\times \mathrm{i}\mathrm{d}{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))=(c\circ g{)}_{!}((s\circ f,t\circ f,t\circ k{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))$. Here $f,g,h,k$ are the maps on $X^{(2)}$ from Toën derived Hall algebra by homotopy-cofiber convolution, and the three entries are source of $f$, cofiber of $f$, and cofiber of $k$; [F4] identifies the last entry with the cofiber of $g$.

Let ${\delta }_0={\chi }_0$. For ${\delta }_0\ast {\chi }_x$, arrows with source zero have target equivalent to their cofiber; the target map on this restricted homotopy type has contractible fibers over the component of $x$. Its pushforward is therefore ${\chi }_x$, and is zero on other components.

The second square in [F4], again transposed with lower map $p$, is homotopy cartesian and gives $\alpha \ast (\beta \ast \gamma )=c_{!}{p}^{\ast }(\mathrm{i}\mathrm{d}\times c{)}_{!}((\mathrm{i}\mathrm{d}\times p{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))=(c\circ h{)}_{!}((s\circ f,s\circ k,t\circ k{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))$. By definition of the induced cofiber map, $s\circ k=t\circ f$; also $c\circ h=c\circ g$ because $h$ is the composite of $f$ and $g$. Thus the last expression is the same as in step 1.1. This proves associativity on finite-support functions.

For ${\chi }_x\ast {\delta }_0$, the selected arrows have source in the component of $x$ and zero cofiber, hence are quasi-isomorphisms. Their core is the path homotopy type of the component of $x$; the target map is an equivalence onto that component with contractible homotopy fibers. The pushforward is again ${\chi }_x$ and vanishes elsewhere. Thus ${\delta }_0$ is a two-sided unit.

Steps 1.1–1.2 prove associativity on arbitrary finite-support functions, and steps 1.2–2.2 prove both unit identities. Finiteness of each convolution is supplied by [F2].