Exact classical principal reduction on equivariant arc modules

Fix a closed point $y$ of the constraint fibre at stage $m$. Put $A=\widehat{(R_m{)}_y}$, let $a_1,\dots ,a_s$ be the independent moment constraints, put $B=A\mathrm{/}(a_1,\dots ,a_s)$, and let $M=\widehat{(U_m{)}_y}$. F1's formal transverse chart is an isomorphism $\phi :A\to C:=B[\text{\hspace{0.17em}⁣}[u_1,\dots ,u_s]\text{\hspace{0.17em}⁣}]$. Choose the transverse basis $q_i$ used in that chart and write ${\delta }_i$ for its infinitesimal action on functions. With F1's pullback convention, $\phi (a)$ is the image in $B[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]$ of $\exp (-{\sum }_i{u}_i{\delta }_i)a$; the minus sign compensates for the action on functions being pullback by the inverse adjoint action. Every ${\delta }_i$ extends to $A$, lowering maximal-ideal order by at most one. The Leibniz identity likewise extends $D_{q_i}$ to $M$: it sends the $k$th maximal-ideal filtration into the $(k-1)$st. Thus the following exponentials are legitimate $u$-adic formal series with continuous coefficients.

Choose a finite presentation $M=F\mathrm{/}K$ with $F=A^r$ and $K$ finitely generated. For each $i$, lift $D_{q_i}$ on the images of the standard basis to a matrix ${\mathrm{\Gamma }}_i\in {\mathrm{Mat}}_r(A)$ and define ${\mathrm{\nabla }}_i(v)={\delta }_i(v)+{\mathrm{\Gamma }}_{i}v$ on $F$. This operator descends to $D_{q_i}$, so ${\mathrm{\nabla }}_i(K)\subseteq K$. Put $E_A=\exp (-{\sum }_i{u}_i{\delta }_i)$ and $E_F=\exp (-{\sum }_i{u}_i{\mathrm{\nabla }}_i)$ on $A[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]$ and $F[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]$. The iterated Leibniz rule gives $E_F(av)=E_A(a)E_F(v)$. Both exponentials have inverses obtained by changing the sign, and both preserve their relevant relation modules; in particular $E_F(K[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}])=K[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]$. No commutativity of the chosen transverse operators is needed: each exponential uses the single operator $-{\sum }_i{u}_i{\mathrm{\nabla }}_i$.

Let $P$ be the matrix whose columns are $E_F(e_j)$, so $E_F(v)=P{E}_A(v)$ coefficientwise and $P\equiv I(\mathrm{m}\mathrm{o}\mathrm{d}(u))$. Hence $P$ is invertible over $A[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]$. Apply the quotient $A[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]\to B[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]=C$ to the equality in step 2.1. If $K_{\phi }\subseteq C^r$ is generated by the images under $\phi$ of a finite set of generators of $K$, and $K_0\subseteq B^r$ is generated by their images modulo the constraints, the resulting equality is $P_B{K}_{\phi }=K_{0}C$, where $P_B$ is the image of $P$. Indeed $E_A$ transforms every coefficient of a relation by the scalar automorphism $E_A$, and the inverse transform gives the reverse inclusion; specialization then preserves this equality of generated submodules. Therefore multiplication by $P_B$ gives an isomorphism $M{\otimes }_{A,\phi }C=C^r\mathrm{/}{K}_{\phi }⟶C^r\mathrm{/}(K_{0}C)=(M\mathrm{/}(a_1,\dots ,a_s)M)[\text{\hspace{0.17em}⁣}[u_1,\dots ,u_s]\text{\hspace{0.17em}⁣}]\mathrm{.}$ Its inverse is multiplication by $P_B^{-1}$. For the last identity, lift each series coefficient in $M\mathrm{/}(a)M$ to $B^r$. A series maps to zero exactly when each coefficient is in $K_0$; express those coefficients in its fixed finite generating set and collect the resulting coefficient series. A1 permits these choices. Thus the kernel is exactly $K_{0}C$, without assuming flatness of $M$. This supplies the missing actual module transport, rather than inferring flatness from finite presentation.

F1's same formal inverse replaces $u$ by the original constraints $a$, with the transverse coordinates fixed: their derivative with respect to $u$ is the invertible triangular matrix in that supplier. Applying this substitution coefficientwise to step 3.1 identifies $M$ with $(M\mathrm{/}(a)M)[\text{\hspace{0.17em}⁣}[a_1,\dots ,a_s]\text{\hspace{0.17em}⁣}]$, with the original constraint operators acting by the displayed variables. Successive multiplication by $a_i$ is therefore injective after quotienting by its predecessors, by coefficient comparison, even when $M\mathrm{/}(a)M$ has torsion or nilpotent scalar action. The finite Koszul mapping-cone argument and faithful completed-local detection in F1 now apply to the finite module $U_m$, yielding zero higher Koszul homology and degree-zero quotient $U_m\mathrm{/}{I}_m{U}_m$. Outside the constraint fibre a unit constraint contracts the complex. Thus no freeness of $U_m$ has been assumed.

On the torsor $N_m\times S_m$, the constrained module $L=U_m\mathrm{/}{I}_m{U}_m$ is an equivariant module over H tensor B, where $H=O(N_m),B=O(S_m)$. Here is the explicit descent, including its inverse. Write its rational coaction as $\lambda (l)=\sum l_{-1}\otimes l_0$, with compatibility $\lambda (al)=\mathrm{\Delta }(a)\lambda (l)$ for a in H and with B coinvariant. If $S_H$ is the Hopf antipode, put $\pi (l)=\sum S_H(l_{-1})l_0$. The Hopf identities and coassociativity imply $\lambda (\pi (l))=1\otimes \pi (l)$. The multiplication map $H\otimes L^N$ to L has inverse l maps to sum $l_{-1}\otimes \pi (l_0)$; composing in either order uses sum $a_1{S}_H(a_2)=ϵ(a)1$ and gives the identity. All sums are finite because the coaction is rational. This proves actual equivariant-module descent, without a statement about closed points or a choice of local trivializations.

The finite classical BRST complex is the same constraint-Koszul/Chevalley bicomplex as in F1, now with coefficients $U_m$. Its finite filtration first computes the Koszul cohomology in step 4.1. Its remaining Chevalley complex, by step 5.1, is $L^N$ tensor the algebraic de Rham complex of the affine space $N_m$. F1's Euler homotopy tensor the identity of $L^N$ contracts it to $L^N$. Thus its only total cohomology is $(U_m\mathrm{/}{I}_m{U}_m{)}^{n_m}$. This is valid without any dimension or smoothness hypothesis on the coefficient module.

For a short exact sequence of finite-stage equivariant modules, the Koszul complexes form a short exact sequence and their higher homology is zero. The long exact sequence therefore makes constrained modules exact, including injectivity at the left. Their constrained modules have the Hopf descent of step 5.1, and the Chevalley higher cohomology is zero by step 6.1. The cohomology long exact sequence now makes degree-zero invariants exact, including surjectivity at the right. For possibly different finite stages enlarge the common stage; every finite collection of generators, relations and witnesses is contained in one such stage.

Pass to the stated restricted unions. A polynomial cochain and the finite equality saying it is a cycle occur at one finite stage. The finite-stage primitives in nonzero degree remain polynomial primitives in the union. Degree-zero cohomology is the filtered colimit; its transition maps need not be injective. Exactness of that colimit follows directly on representatives: a kernel element or a relation between two representatives is witnessed at some common finite stage, and the exact finite-stage construction then supplies the required preimage or primitive. Invariants here are the degree-zero cohomology of the restricted complex, equivalently the vectors killed by all n[t] modes. This proves concentration and exactness on the entire stated module class.