Hall Integration, Stability, DT/BPS Invariants, and Wall Crossing

For a morphism $f:X\to Y$, its vertexwise kernels form a subrepresentation: if $x\in \ker f_i$, then $f_j{X}_a(x)=Y_a{f}_i(x)=0$ for every arrow $a:i\to j$. The vertexwise cokernels form a quotient representation because $Y_a(\mathrm{im}{f}_i)\subseteq \mathrm{im}{f}_j$. The kernel and cokernel universal properties hold vertex by vertex. The canonical coimage-to-image map is an isomorphism at every vertex, hence an isomorphism of representations. With vertexwise finite direct sums and a zero object, these facts make ${\mathcal{A}}_Q(K)$ abelian.

For $i\in Q_0$, let $P_i(v)$ have basis the directed paths from $i$ to $v$, with arrow maps given by path concatenation. Acyclicity and finiteness of $Q$ make each path set finite. A morphism $P_i\to N$ is uniquely determined by the image of the length-zero path at $i$, so $\mathrm{Hom}(P_i,N)\cong N_i$; evaluation at $i$ preserves epimorphisms, and therefore $P_i$ is projective.

Put $C^0(X,Y)={⨁}_i{\mathrm{Hom}}_K(X_i,Y_i),C^1(X,Y)={⨁}_{a:i\to j}{\mathrm{Hom}}_K(X_i,Y_j),$ and define $\mathrm{\partial }:C^0\to C^1$ by $(\mathrm{\partial }h{)}_a=Y_a{h}_i-h_j{X}_a$. Its kernel is exactly ${\mathrm{Hom}}_{{\mathcal{A}}_Q(K)}(X,Y)$ by the commuting-square definition of a representation morphism.

For a representation $X$, define $ϵ:{⨁}_i{P}_i{\otimes }_K{X}_i\to X$ by $p\otimes x\mapsto X_p(x)$, where $X_p$ is the composite arrow map along the path $p$. Define $d:{⨁}_{a:i\to j}{P}_j{\otimes }_K{X}_i⟶{⨁}_i{P}_i{\otimes }_K{X}_i$ at a path $p:j\to v$ by $d(p\otimes x)=(p\otimes X_a(x){)}_j-((p\circ a)\otimes x{)}_i\mathrm{.}$ Both terms are finite direct sums of the projectives from step 1.2. Then $ϵd=0$ because $X_{p\circ a}=X_p{X}_a$.

Now assume $K={\mathbb{F}}_q$. For each dimension vector $\mathbf{d}$, there are finitely many arrow matrix tuples on the fixed spaces $K^{d_i}$, and the finite group $G_{\mathbf{d}}(K)$ has finitely many orbits on this set. Fix an enumeration of the finite set $K$; the lexicographically least tuple in each orbit gives a representative without making a choice from an arbitrary family. The union over the set ${\mathbb{N}}^{Q_0}$ is a set, so the full subcategory on these representatives is small. It is a skeleton: every finite-dimensional representation becomes one of the matrix tuples after choosing bases at its finitely many vertices. The full skeleton is abelian because each vertexwise kernel and cokernel from step 1.1 is isomorphic to its selected representative, and transporting its kernel or cokernel map across that isomorphism preserves the universal property.

At vertex $v$, $ϵ$ is onto because $x\in X_v$ is the image of ${\mathrm{id}}_v\otimes x$. Modulo $\mathrm{im}d$, the relation in step 2.1 replaces $(p\circ a)\otimes x$ by $p\otimes X_a(x)$. Every positive-length path has a unique first arrow, so repeated replacement sends each path tensor to ${\mathrm{id}}_v\otimes X_p(x)$. Consequently the quotient of the middle term by $\mathrm{im}d$ is isomorphic to $X_v$ through $ϵ$, and $\ker ϵ=\mathrm{im}d$.

The map $d$ is injective. In a nonzero element of its domain, choose the largest length $\mathrm{\ell }$ of a suffix path $p$ with nonzero coefficient. The second term of $d(p\otimes x)$ has length $\mathrm{\ell }+1$. Each such path has a unique first arrow and suffix, so its coefficient cannot cancel with the second term of a different source basis summand. A first term of $d$ has length at most $\mathrm{\ell }$, since no source suffix has length $\mathrm{\ell }+1$. Thus the length-$(\mathrm{\ell }+1)$ part of $d$ is nonzero, proving injectivity. The displayed sequence is a projective resolution of length one; applying $\mathrm{Hom}(-,Y)$ therefore gives ${\mathrm{Ext}}_{{\mathcal{A}}_Q(K)}^r(X,Y)=0$ for $r\ge 2$.

A short exact sequence $0\to Y\to E\to X\to 0$ splits as a sequence of vector spaces at each of the finitely many vertices. In the resulting coordinates every arrow of $E$ has block matrix $E_a=\left(\begin{array}{cc}{Y}_a& {\xi }_a\\ 0& X_a\end{array}\right),{\xi }_a\in {\mathrm{Hom}}_K(X_i,Y_j)\mathrm{.}$ Every tuple $\xi \in C^1(X,Y)$ defines such an extension. Changing the vertexwise splittings by $h\in C^0(X,Y)$ changes $\xi$ by $\mathrm{\partial }h$. Conversely, an isomorphism of extensions that is the identity on endpoints has block matrix $\left(\begin{array}{cc}1& h_i\\ 0& 1\end{array}\right)$ at each vertex and hence changes $\xi$ by exactly $\mathrm{\partial }h$. By [F3], the generated endpoint-fixed Yoneda relation consists of these isomorphisms. The Baer sum of the extensions represented by $\xi$ and $\eta$ is represented by $\xi +\eta$: the pullback of their direct sum along $X\to X\oplus X$ followed by pushout along $Y\oplus Y\to Y$ adds the off-diagonal arrow blocks. Pushout along $\lambda {\mathrm{id}}_Y$ sends $\xi$ to $\lambda \xi$. Therefore the Yoneda vector-space structure is the quotient structure ${\mathrm{YExt}}^1(X,Y)\cong C^1(X,Y)\mathrm{/}\mathrm{im}\mathrm{\partial },$ a finite-dimensional $K$-vector space; it is not being identified here with the projective-resolution group of step 4.1.

Both cochain spaces in step 1.3 are finite-dimensional. Rank-nullity for $\mathrm{\partial }$ gives $\dim \ker \mathrm{\partial }-\dim \mathrm{coker}\mathrm{\partial }=\dim C^0-\dim C^1={\sum }_i{d}_i(X)d_i(Y)-{\sum }_{a:i\to j}{d}_i(X)d_j(Y)={\chi }_Q(\mathbf{d}(X),\mathbf{d}(Y))\mathrm{.}$ The kernel and cokernel identifications already proved give the Hom–Yoneda-Ext${}^1$ formula and finite-dimensionality over any $K$. When $K={\mathbb{F}}_q$, the underlying sets are finite. The full skeleton constructed in step 2.2 has the same Hom sets on its objects; transporting extension middle terms to their representatives identifies its Yoneda classes with those in ${\mathcal{A}}_Q(K)$. Thus this small abelian skeleton satisfies the finitary hypotheses of [F2], while ${\mathcal{A}}_Q(K)$ is essentially small.

Suppose $\arg z>\arg w$. Then $\mathrm{Im}(\bar{w}z)=\mathrm{\mid }w\mathrm{\mid }\mathrm{\mid }z\mathrm{\mid }\sin (\arg z-\arg w)>0$. Consequently $\mathrm{Im}(\bar{w}(z+w))>0$, which, by [F1], gives $\arg (z+w)>\arg w$. Also $\mathrm{Im}(\overline{(z+w)}z)=\mathrm{Im}(\bar{w}z)>0$, so [F1] gives $\arg z>\arg (z+w)$.

If $\arg w>\arg z$, interchange $z,w$ in step 1.1. If the arguments are equal, $z,w$ are positive real multiples of the same ray, so their sum has that same argument. These cases prove the stated between-phase rule.

Work over $\bar{k}$. By [F2] and [F7], the Plucker equations define a closed projective subscheme whose $\bar{k}$-points are the $r$-planes. On a chart $p_I\mathrm{\ne }0$, normalize $p_I=1$ and use the signed one-outside coordinates as the entries of a matrix $A$. The Plucker relation with tuples $J\setminus \{j\}$ and $I\cup \{j\}$ expresses each $p_J$, for $j\in J\setminus I$, using coordinates with fewer indices outside $I$ and one-outside coordinates. Induction identifies every $p_J$ in the quotient chart ring with the corresponding maximal minor of $(I_r\mid A)$; conversely those minors satisfy every Plucker relation. Hence the chart ring is $\bar{k}[A]$. The row spans of $(I_r\mid A)$ give a rank-$r$ subbundle of the trivial bundle, and its quotient is locally free; the pivot changes on overlaps are invertible and glue these bundles and charts. Thus the equations construct the Grassmannian scheme with its closed Plucker embedding. The endpoint cases $r=0,n$ are points. For the Segre equations in [F3], on $z_{i_0{j}_0}\mathrm{\ne }0$ normalize $z_{i_0{j}_0}=1$ and put $u_i=z_{i{j}_0}$ and $v_j=z_{i_{0}j}$, with $u_{i_0}=v_{j_0}=1$. The minor relations give $z_{ij}=u_i{v}_j$ in the quotient ring, and substitution and coordinate extraction are inverse maps with $\bar{k}[u_i(i\mathrm{\ne }{i}_0),v_j(j\mathrm{\ne }{j}_0)]$. These charts construct the closed Segre embedding. Iterating it embeds the finite product $P_{\mathbf{e}}={\prod }_i\mathrm{Gr}(e_i,k^{d_i}{)}_{\bar{k}}$ as a closed subscheme of one projective space; if $Q_0$ is empty, $P_{\mathbf{e}}$ is a point. The chart ring surjections remain surjective after adjoining the arrow-matrix coordinates, so ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}\times P_{\mathbf{e}}$ is closed in a relative projective space over ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}$. The chart calculations show these projective subschemes are reduced with the point loci in [F2] and [F3]; [A2] records their classical projective coordinate interpretation over $\bar{k}$.

For each subvector $\mathbf{e}$ with $0<\mathbf{e}\le \mathbf{d}$ and ${\varphi }_Z(\mathbf{e})>{\varphi }_Z(\mathbf{d})$, impose in ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}\times P_{\mathbf{e}}$, for each arrow $a:i\to j$, that the composition ${\mathcal{S}}_i⟶k^{d_i}\stackrel{A_a}{\to }{k}^{d_j}⟶{\mathcal{Q}}_j$ vanish. Here ${\mathcal{S}}_i$ and ${\mathcal{Q}}_j$ are the tautological subbundle and quotient bundle constructed in step 1.1; their fibers at $U_i,U_j$ are $U_i$ and $k^{d_j}\mathrm{/}{U}_j$. On the charts from step 1.1 these are polynomial equations, so their common zero locus $I_{\mathbf{e}}$ is closed. Its geometric points are exactly pairs consisting of an arrow-matrix tuple and a subrepresentation of dimension $\mathbf{e}$. The closed inclusion $I_{\mathbf{e}}\hookrightarrow {\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}\times P_{\mathbf{e}}$ composes with the relative Plucker–Segre closed immersion of step 1.1, so the projection to ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}$ is projective by [F4].

By [A1] and [F5], the projection in step 2.1 is proper; [F6] then makes its image closed in ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}$. Its $\bar{k}$-points are exactly the representations with a destabilizing subrepresentation of dimension $\mathbf{e}$.

There are finitely many subvectors $\mathbf{e}$ with $0<\mathbf{e}\le \mathbf{d}$. The unstable locus is the finite union of the closed images from step 3.1 over the subvectors satisfying the strict phase inequality. Its complement is therefore open and, by [F1], is precisely ${\mathrm{Rep}}_{\mathbf{d}}^{ss}(Q,Z)$. If $\mathbf{d}=0$ the union is empty and the whole one-point representation space is open.

The two declared axiom branches are [A2], used in step 1.1 through [F7] for the scheme-theoretic projective constructions, and [A1], used in step 3.1 through [F5] for properness. The semistability and HN counting arguments do not otherwise use AC.

Let $M\mathrm{\ne }0$ have dimension vector $\mathbf{d}$. The nonzero dimension vectors of its subrepresentations form a nonempty subset of the finite set $\{\mathbf{e}:0\le e_i\le d_i\}$. Choose the largest attained phase and, among subobjects attaining it, the largest attained total dimension. Such a phase and dimension are maxima of finite sets of attained values, so they are attained without an infinite choice. Let $S\subseteq M$ attain both maxima. Every nonzero subobject of $S$ is a subobject of $M$, so its phase is at most ${\varphi }_Z(S)$; hence $S$ is semistable.

If $A,B$ are semistable and ${\varphi }_Z(A)>{\varphi }_Z(B)$, then $\mathrm{Hom}(A,B)=0$. Indeed, a nonzero map has a nonzero image $I$. From $0\to \ker f\to A\to I\to 0$, semistability of $A$ and [F3] force ${\varphi }_Z(I)\ge {\varphi }_Z(A)$: if the kernel is zero the phases are equal; if the kernel is nonzero and ${\varphi }_Z(I)<{\varphi }_Z(A)$, the phase of the sum lies strictly below ${\varphi }_Z(A)$, contradicting $Z(A)=Z(\ker f)+Z(I)$. But $I\subseteq B$ and semistability of $B$ give ${\varphi }_Z(I)\le {\varphi }_Z(B)$, a contradiction.

If $T\subseteq M\mathrm{/}S$ is nonzero, let $P\subseteq M$ be its inverse image. Then $0\to S\to P\to T\to 0$ is exact. If ${\varphi }_Z(T)>{\varphi }_Z(S)$, [F3] gives ${\varphi }_Z(P)>{\varphi }_Z(S)$, contrary to maximality of the phase of $S$. If ${\varphi }_Z(T)={\varphi }_Z(S)$, additivity and [F3] give ${\varphi }_Z(P)={\varphi }_Z(S)$, while $\mathrm{\mid }\mathbf{d}(P)\mathrm{\mid }>\mathrm{\mid }\mathbf{d}(S)\mathrm{\mid }$, contrary to the second maximum. Thus every nonzero subobject of $M\mathrm{/}S$ has phase strictly less than ${\varphi }_Z(S)$.

Consider two filtrations with first subobjects $S,S^{\mathrm{\prime }}$ of phases $p,p^{\mathrm{\prime }}$. If $p>p^{\mathrm{\prime }}$, every factor in the quotient filtration of $M\mathrm{/}{S}^{\mathrm{\prime }}$ has phase below $p^{\mathrm{\prime }}$. Step 1.2 therefore makes every map from $S$ to each such factor zero. Induction along the finite quotient filtration shows $\mathrm{Hom}(S,M\mathrm{/}{S}^{\mathrm{\prime }})=0$: a map whose projection to the last factor is zero factors through the preceding subobject. Thus the inclusion of $S$ in $M$ factors through $S^{\mathrm{\prime }}$, giving a nonzero map $S\to S^{\mathrm{\prime }}$, contrary to $\mathrm{Hom}(S,S^{\mathrm{\prime }})=0$ from step 1.2. Interchanging $S,S^{\mathrm{\prime }}$ similarly rules out $p^{\mathrm{\prime }}>p$; hence $p=p^{\mathrm{\prime }}$. Applying step 1.2 to all factors of both quotients now gives $S\subseteq S^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}\subseteq S$, hence $S=S^{\mathrm{\prime }}$. Induction on the quotient proves equality of the full filtrations.

Induct on $\mathrm{\mid }\mathbf{d}(M)\mathrm{\mid }$. The zero representation has the empty filtration. For $M\mathrm{\ne }0$, the quotient $M\mathrm{/}S$ has smaller total dimension, so by induction it has a filtration with semistable factors in strictly decreasing phase order. Step 2.1 places every one of those phases below ${\varphi }_Z(S)$. Pulling that filtration back along $M\to M\mathrm{/}S$ and placing $S$ first gives the required filtration of $M$.

Let $M$ be defined over $k$, set $M_{\bar{k}}=M{\otimes }_k\bar{k}$, and let $\sigma (x)=x^q$ be Frobenius. Applying $\sigma$ to a filtration preserves the arrow maps of $M_{\bar{k}}$, every dimension vector, and every phase. It therefore gives another HN filtration; uniqueness proved above fixes each term. At each vertex, a Frobenius-invariant subspace has a unique reduced row-echelon basis matrix. Applying $\sigma$ gives another such matrix for the same subspace, so uniqueness makes the matrix fixed entrywise. Its entries lie in ${\mathbb{F}}_q$, and the arrow maps of $M$ preserve these descended subspaces. Thus the filtration descends to $k$.

If an ${\mathbb{F}}_q$-object is geometrically unstable, it has a geometric subrepresentation of phase greater than its total phase. Step 1.1 chooses a subobject of maximal phase, so the first HN term has phase greater than the total phase. Step 4.1 descends that term and its phase inequality to ${\mathbb{F}}_q$, contradicting ${\mathbb{F}}_q$-semistability. Conversely, an ${\mathbb{F}}_q$-subrepresentation extends to $\bar{k}$ with the same dimension vector and phase. This proves the stated equivalence and the result.

Under the inherited assumption [A1], the semistable locus is the open subscheme from [F1]. An object of its quotient groupoid over ${\mathbb{F}}_q$ is a tuple of ${\mathrm{GL}}_{d_i}$-torsors together with an equivariant semistable representation. A ${\mathrm{GL}}_n$-torsor is the frame torsor of its associated $n$-dimensional vector space. Every finite-dimensional vector space over ${\mathbb{F}}_q$ has a basis, so that associated space is isomorphic to ${\mathbb{F}}_q^n$ and its frame torsor is trivial. Applying this at the finitely many vertices trivializes the product $G_{\mathbf{d}}$-torsor. Thus every quotient-stack object is isomorphic to a representation on the standard spaces $k^{d_i}$.

Under a chosen trivialization, an equivariant map to ${\mathrm{Rep}}_{\mathbf{d}}^{ss}$ is an arrow-matrix tuple. Two trivialized objects are isomorphic exactly when their tuples differ by change of bases in $G_{\mathbf{d}}({\mathbb{F}}_q)$. If $\mathbf{d}\mathrm{\ne }0$, [F2] identifies the rational tuples in the geometric semistable locus with the finite-field semistable representations. If $\mathbf{d}=0$, the geometric condition is vacuous, the representation space is a point, and $G_{\mathbf{0}}$ is trivial. Its quotient groupoid therefore has one object with trivial automorphism group, matching the zero representation included by the statement's convention. This proves the claimed equivalence in every dimension.

There are finitely many arrow-matrix tuples on the fixed finite vector spaces, and each automorphism group is a subgroup of the finite group $G_{\mathbf{d}}({\mathbb{F}}_q)$. Hence the groupoid is finite. Under the equivalence of step 2.1, the stabilizer of a tuple is exactly ${\mathrm{Aut}}_{{\mathcal{A}}_Q}(M)$. Substituting these stabilizers into the groupoid-cardinality definition in Finite-field stability data for an acyclic quiver gives the displayed sum.

Fix the middle object to be $M$. The coefficient of ${\delta }_M$ in ${\delta }_U\star {\delta }_V$ counts subobjects $S\subseteq M$ with $S\cong U$ and $M\mathrm{/}S\cong V$, by [F1] and [F2]. For each such $S$, the monomorphisms $U\to M$ with image $S$ form a set of size $\mathrm{\mid }\mathrm{Aut}(U)\mathrm{\mid }$; the epimorphisms $M\to V$ with kernel $S$ form a set of size $\mathrm{\mid }\mathrm{Aut}(V)\mathrm{\mid }$. These choices are independent. Hence the number of endpoint-fixed short exact sequences with middle object literally $M$ is $g_{V,U}^M\mathrm{\mid }\mathrm{Aut}(U)\mathrm{\mid }\mathrm{\mid }\mathrm{Aut}(V)\mathrm{\mid },$ where $g_{V,U}^M$ is the Hall coefficient of [F1].

The finite group $\mathrm{Aut}(M)$ acts on those sequences by $(i,p)\mapsto (gi,p{g}^{-1})$. Its orbits are exactly the endpoint-fixed Yoneda classes with middle term isomorphic to $M$: an isomorphism from any middle object $E\cong M$ transports its sequence to one on $M$, and changing that isomorphism changes the result by this action. The stabilizer of a sequence $(i,p)$ consists of $g$ with $gi=i$ and $pg=p$. Put $h=g-1$. Since $hi=0$ and $p$ is an epimorphism, $h$ factors uniquely as $h=ifp$ for a unique $f:V\to U$. Conversely, $1+ifp$ is invertible because $(ifp{)}^2=0$, with inverse $1-ifp$. Thus every stabilizer has exactly $\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }$ elements.

For a fixed sequence, the map from $\mathrm{Aut}(M)$ to its orbit has fibers exactly the cosets of its stabilizer, so that orbit has size $\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }$. Summing over the $\mathrm{\mid }{\mathrm{YExt}}^1(V,U{)}_M\mathrm{\mid }$ orbits gives $g_{V,U}^M\mathrm{\mid }\mathrm{Aut}(U)\mathrm{\mid }\mathrm{\mid }\mathrm{Aut}(V)\mathrm{\mid }=\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }\frac{\mathrm{\mid }{\mathrm{YExt}}^1(V,U{)}_M\mathrm{\mid }}{\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }}\mathrm{.}$ Multiplying the coefficient of ${\delta }_M$ in ${\kappa }_U\star {\kappa }_V$ by the definition of ${\kappa }_M$ yields precisely the claimed coefficient. Summing over the finitely many possible middle terms is valid by [F3] and [F4].

Apply [F4] to the small finitary abelian full skeleton supplied by [F3]. Its inclusion preserves isomorphism classes and subobject classes: transporting a monomorphism's domain to its skeleton representative gives a bijection on monomorphisms modulo domain isomorphism, and transports the quotient to the same quotient class. Thus its Hall coefficients are exactly those in [F1]. Write ${\ast }_q$ for this quotient-first product. By [F1], $f\star g=g{\ast }_{q}f$, so $(f\star g)\star h=h{\ast }_q(g{\ast }_{q}f)=(h{\ast }_{q}g){\ast }_{q}f=f\star (g\star h)$ by [F4]. The zero-object class is the unit for ${\ast }_q$, hence also for $\star$.

The degree of an extension is the sum of the endpoint dimension vectors. For a fixed output degree $\mathbf{n}$, a product coefficient uses only the finitely many pairs $\mathbf{d}+\mathbf{e}=\mathbf{n}$; each $H_{Q,\mathbf{d}}$ is finite-dimensional by the representation-matrix definition. Thus [F5] extends the product degree by degree to ${\widehat{H}}_Q$, and truncating above total dimension $D$ changes no coefficient of dimension at most $D$. This is continuity for the total-dimension filtration.

On a normalized basis vector, $I({\kappa }_M)=X^{\mathbf{d}(M)}$ because ${\kappa }_M=\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }{\delta }_M$. For $\mathbf{d}=\mathbf{d}(U)$ and $\mathbf{e}=\mathbf{d}(V)$, [F2] gives $I({\kappa }_U\star {\kappa }_V)=\frac{\mathrm{\mid }{\mathrm{YExt}}^1(V,U)\mathrm{\mid }}{\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }}{X}^{\mathbf{d}+\mathbf{e}}\mathrm{.}$ By [F3], both sets are finite vector spaces and the ratio of their cardinalities is $q^{\dim {\mathrm{YExt}}^1(V,U)-\dim \mathrm{Hom}(V,U)}=q^{-{\chi }_Q(\mathbf{e},\mathbf{d})}\mathrm{.}$ This is exactly $I({\kappa }_U)I({\kappa }_V)$ in the Euler target.

On the same basis, $J({\kappa }_M)=t^{{\chi }_Q(\mathbf{d},\mathbf{d})}{x}^{\mathbf{d}}$. Applying [F2] and [F3] to ${\kappa }_U\star {\kappa }_V$ gives its $J$-image coefficient $t^{-2{\chi }_Q(\mathbf{e},\mathbf{d})+{\chi }_Q(\mathbf{d}+\mathbf{e},\mathbf{d}+\mathbf{e})}\mathrm{.}$ By bilinearity, ${\chi }_Q(\mathbf{d}+\mathbf{e},\mathbf{d}+\mathbf{e})={\chi }_Q(\mathbf{d},\mathbf{d})+{\chi }_Q(\mathbf{e},\mathbf{e})+{\chi }_Q(\mathbf{d},\mathbf{e})+{\chi }_Q(\mathbf{e},\mathbf{d})\mathrm{.}$ Since $⟨\mathbf{e},\mathbf{d}⟩={\chi }_Q(\mathbf{e},\mathbf{d})-{\chi }_Q(\mathbf{d},\mathbf{e})$ and $q=t^2$, the coefficient becomes $t^{{\chi }_Q(\mathbf{d},\mathbf{d})+{\chi }_Q(\mathbf{e},\mathbf{e})-⟨\mathbf{e},\mathbf{d}⟩},$ which is exactly the coefficient of $J({\kappa }_U)\ast J({\kappa }_V)$ in the skew target.

Bilinearity of ${\chi }_Q$ makes the Euler twist associative: both parenthesizations of $X^{\mathbf{d}}{X}^{\mathbf{e}}{X}^{\mathbf{f}}$ have exponent $-{\chi }_Q(\mathbf{e},\mathbf{d})-{\chi }_Q(\mathbf{f},\mathbf{d})-{\chi }_Q(\mathbf{f},\mathbf{e})$. Bilinearity of $⟨-,-⟩$ gives the same cocycle identity for the skew product. Both products have the degree-zero basis vector as unit. Step 1.2 gives finite-dimensional homogeneous components, so the formulas for $I$ and $J$ define degree-preserving linear maps and are continuous on the completion. Step 1.1 establishes the source unit and associativity, and steps 1.3–1.4 establish multiplicativity on normalized basis vectors. The cocycle calculation above establishes the target unit and associativity, so linearity and continuity extend $I$ and $J$ to unital algebra homomorphisms on the completions.