Grading restriction of finitely generated C2-cofinite weak joint modules

First take a module cyclic under all components, with generator $w$. For $d\in {\mathbb{Z}}^k$, define $H_d$ as the span of all words of total component multidegree $d$ applied to $w$. Component commutativity reorders every word into $k$ consecutive component blocks. For each component choose its generator cutoff $N_i\ge 1$ on the finite representative set $A$. Each block may be reduced by [F1] on the cyclic module of that component generated by $w$: commute all other blocks to the left, apply the finite linear relation on $w$, then multiply those other blocks back. Cross-component commutation makes this operation preserve all other multidegrees. Applying this successively to the $k$ blocks gives strict-index block strings with the same total multidegree. The coefficients and truncations may depend on the particular block word; only the final representative cutoff $N_i$ is uniform, precisely as in [F1].

For each component the bound in [F1] bounds its normal-string length in terms of the actual integer $d_i$, independently of initial filtration. For each bounded length there are finitely many state choices and, with largest index less than $N_i$ and fixed index sum, finitely many index choices. Products of these finite choices across the finitely many components give a finite spanning set for $H_d$. There are integer lower bounds $D_i$ such that $H_d=0$ if any $d_i<D_i$. Furthermore $W={\sum }_d{H}_d$, initially an algebraic sum rather than a direct grading. Each $L_i(0)$ preserves $H_d$: its mode has multidegree zero, and left multiplication by it is a permissible additional mode in the definition of the same degree span. Thus every $H_d$ is a finite-dimensional invariant space for all the conformal operators.

In particular $T=H_0$ is finite dimensional, contains $w$, and is invariant under all $L_i(0)$. Commuting endomorphisms of a finite-dimensional complex vector space have a finite joint generalized-eigenspace decomposition: decompose for the first operator using its split minimal polynomial and pairwise coprime primary factors; every other commuting operator preserves each primary space, so repeat on each space for the remaining operators. Let $R\subset {\mathbb{C}}^k$ be the resulting finite joint spectral set, and write $w={\sum }_{\rho \in R}{w}_{\rho }$ accordingly. Each projection is a polynomial in the commuting operators on $T$, so $w_{\rho }\in H_0$. The commutator [F2] implies that a component word of multidegree $d$ sends $w_{\rho }$ into joint generalized weight $\rho +d$. This also puts it in $H_d$, because $w_{\rho }$ is a finite linear combination of multidegree-zero words on $w$.

Consequently $H_d$ is spanned by its joint generalized-weight vectors at the finite weights in $R+d$, and $W={\sum }_d{H}_d$ is spanned by joint generalized-weight vectors. Distinct joint weights are linearly independent: any proposed linear relation uses finitely many weights and finitely many vectors, and polynomial primary projections in the commuting operators isolate each weight on the finite-dimensional space spanned by their finite nilpotent orbits. This proves the asserted algebraic direct sum. At a fixed weight $\lambda$, only the finitely many multidegrees $d=\lambda -\rho \in {\mathbb{Z}}^k$, $\rho \in R$, can contribute. Hence $W_{[\lambda ]}$ lies in the finite sum of those finite-dimensional $H_d$ and is finite dimensional. Its support lies in $R+D+{\mathbb{N}}^k$ by step 2.1. These conclusions were derived from finite degree spans; they were not assumed in defining them.

For finitely many generators, form the finite direct sum of their cyclic joint submodules and map it onto $W$ by addition. The preceding arguments give a finite joint spectral set, coordinate lower bounds, and finite joint spaces for that finite direct sum. Its kernel is stable under the conformal operators and decomposes into its joint components: for any vector of the kernel, finitely many polynomial primary projections isolate its finitely many joint components and remain in the kernel. The quotient therefore inherits the joint decomposition, finite joint-space dimensions, and support within the same finite union of shifted nonnegative lattices. Taking coordinatewise minima of the finitely many integer bounds gives one $D$ and a finite set $R$ as in the Statement.

Within each of the finitely many distinct joint lattice cosets choose a representative $\alpha$, and for each coordinate take the minimum integer shift $m_i$ among occurring weights $\alpha +n$. These minima exist by the proved coordinatewise lower bounds. Put $\beta =\alpha +m$; it need not itself occur. The sum of $W_{[\beta +n]}$ over those distinct cosets defines finite-dimensional pieces indexed by $n\in {\mathbb{N}}^k$. Their direct sum is $W$, and [F2] gives exactly the component mode-degree rule. These auxiliary semisimple degree operators do not replace the original conformal operators. Finally at fixed total weight, or below a fixed upper bound on its real part, the finite-coset support and the coordinatewise lower bounds leave finitely many joint weights; all corresponding joint spaces are finite dimensional. The total conformal generalized-weight space is exactly their direct sum, since on each joint space the sum of the commuting nilpotent parts is nilpotent.

If a compatible grading is already specified, enlarge the finite generating set by all its finitely many graded components, obtaining finitely many homogeneous generators. For any such generator at multidegree $m$, the proof of steps 1.1–2.1 gives finite-dimensional operator-multidegree spans $H_d$ with $d_i\ge D_i$. A fixed specified multigrading piece of degree $n$ is spanned by terms with $d=n-m$, so is finite dimensional. For a single specified total grading, its degree-$n$ piece is spanned by terms with ${\sum }_i{d}_i=n-m$; the coordinate lower bounds leave finitely many such integer tuples, and each corresponding $H_d$ is finite dimensional. Summing over the finite homogeneous generating set proves the assertion for the original pieces. This proves all remaining assertions without replacing a specified grading by an assumed conformal grading.