Finite generation of the multipoint Jacobi quotient

Every displayed sum is finite on $\mathbf{w}$. For ordinary $u_k$ this is module truncation; for $u_{-1-k}^{\ast }$ it is upper truncation of the opposite field. The operator $L(1)$ is locally nilpotent on each module because it lowers real weight by one and the weights are bounded below. Hence both exponentials in $K_i$ are finite on the vector where they act, and the $k$ sum is finite after the right exponential is expanded. Expanding $K_i$ gives Laurent-polynomial scalar coefficients: a term $L(1{)}^a{u}_{k}L(1{)}^b$ has weight shift $d-k-1-a-b$. Thus the three families are actual elements of $T$.

Choose finite homogeneous representatives $P_i\subset W_i$ for $W_i\mathrm{/}{C}_1(W_i)$. They exist because $C_1(W_i)$ is graded and the quotient is finite dimensional. Choose $M$ at least the largest sum of weights in $P_0\otimes \cdots \otimes P_{n+1}$. Decompose each factor as a representative plus a sum of $u_{-1}w$ terms. Consequently every homogeneous tensor of weight $s>M$ is a sum of tensors with at least one such factor; any term with all representative factors has weight at most $M$ and hence cannot occur in its weight-$s$ component.

Reduce a factor $u_{-1}{w}_i$ with $1\le i\le n$ by $A_i$, the last factor by $C$, or factor zero by $D$. The designated term has total weight $\sigma +d$, where $\sigma$ is the weight before insertion. Every other term of $A_i$ or $C$ has weight either $\sigma -d-k$ or $\sigma +d-k-1$, hence strictly less. In $D$, step 1.1 shows that every finite-factor term has weight $\sigma +d-k-1-a-b<\sigma +d$ and its last term has weight $\sigma -d$. All changes are integral, and all tensor weights have a common real lower bound. Thus each sequence of reductions terminates after finitely many strict integer weight decreases. Applying step 1.2 at each remaining weight above $M$ proves $T=J+R_n\otimes ({⨂}_i{W}_i{)}_{\le M}$. This argument needs no induction over a nondiscrete ordering of real numbers.

For completeness the annihilation assertion uses actual Jacobi residues rather than an arbitrary quotient. Insert the current $Y(u,t)$ into the formal product of intertwiners. Apply intertwiner Jacobi successively across each insertion; it equates the expansion at infinity paired with $w_0$ with the sum of residues at $t=z_i$ and $t=0$. Take the residue with multiplier $(t-z_i{)}^{-1}$. At infinity its expansion is ${\sum }_{k\ge 0}{z}_i^k{t}^{-1-k}$; at $z_j$, $j\mathrm{\ne }i$, its expansion is ${\sum }_{k\ge 0}(-1{)}^k(z_j-z_i{)}^{-1-k}(t-z_j{)}^k$; at zero it is ${\sum }_{k\ge 0}(-1{)}^k(-z_i{)}^{-1-k}{t}^k$. At $z_i$ the residue is $u_{-1}$. Pairing the infinity expansion transposes ordinary modes to $u_{-1-k}^{\ast }$ on $W_0$. These four contributions are precisely $A_i$, since $(-1{)}^k(-z_i{)}^k=z_i^k$. Taking multiplier $t^{-1}$ instead gives $C$, by the expansion $t^{-1}={\sum }_{k\ge 0}(-1{)}^k{z}_i^{-1-k}(t-z_i{)}^k$ at $z_i$. All equalities are formal residues of Jacobi with the displayed binomial expansions, so no analytic contour deformation is assumed.

If $B_i$ is a lower weight bound for $W_i$, each factor of a tensor of total weight at most $M$ has weight at most $M-{\sum }_{j\mathrm{\ne }i}{B}_j$. Its span is therefore contained in the tensor product of finitely many finite cutoff spaces, finite dimensional by F1 (use a strict cutoff larger by one). Choose a finite basis of that subspace. Its image generates $T\mathrm{/}J$ over $R_n$, proving finite generation.

Apply the same $t^{-1}$ residue after changing the global coordinate to $t^{-1}$, interchanging zero and infinity. The finite insertion coordinates become $z_i^{-1}$; their state transformation is $A(z_i)$ and its inverse. F2 gives the exact conjugation of the local $u_k$ action as $K_i(u_k)$; the coefficient $(-1{)}^k(z_i^{-1}{)}^{-1-k}$ is $(-1{)}^k{z}_i^{1+k}$. The infinity and zero terms are respectively $u_{-1}$ on $W_0$ and $u_{-1}^{\ast }$ on $W_{n+1}$. Thus this residue is exactly $D$. Steps 2.2 and 3.2 prove annihilation of every generator, and $R_n$-linearity in its specified radial expansion proves annihilation of $J$.

The field $\mathbb{C}$ is Noetherian, since its only ideals are zero and itself. Iterating the polynomial result and localizing at the multiplicative system of coordinates and their differences makes $R_n$ Noetherian. For fixed $l$ set $b_j=[w_0\otimes \cdots \otimes L(-1{)}^j{w}_l\otimes \cdots \otimes w_{n+1}]\in T\mathrm{/}J$. Its $R_n$-span is a submodule of the finite module in step 3.1, hence is finitely generated. Express finitely many generators as finite combinations of the $b_j$, and let $m_l-1$ be at least every index that occurs. Thus $b_0,\dots ,b_{m_l-1}$ generate that span and $b_{m_l}+{\sum }_{k=1}^{m_l}{a}_{k,l}{b}_{m_l-k}=0$ for some $a_{k,l}\in R_n$. Applying the coefficient map from steps 2.2–2.2 and using the derivative identity of each intertwiner turns $b_j$ into ${\mathrm{\partial }}_{z_l}^j\phi$. This proves the claimed scalar differential equations as formal identities. It gives no pole-order estimate for their coefficients at a boundary divisor.