Cohen–Macaulay isospectral schemes give all-n surface Hilbert kernel equivalence

Define $X_n(S)=(Y{\times }_{Q}M{)}_{\mathrm{r}\mathrm{e}\mathrm{d}}$, the reduced isospectral fiber product, and write $p:X_n\to Y$, $r:X_n\to M$. The remaining input is:

$(H_n)$ $X_n(S)$ is Cohen–Macaulay, for every n.

The projection p is finite, since M→Q is finite, and generically has degree n!. Its reduced fiber product equals the reduced closure of the ordered distinct-point locus: that closure is $S_n$-stable and its image under the finite p is closed and contains the dense distinct-point locus, hence all of Y. Over any closed y its nonempty fiber therefore contains an ordering of the support cycle. $S_n$ acts transitively on all such orderings, so this fiber contains every closed point of the reduced fiber product. The two reduced closed schemes consequently coincide. The ordered distinct-point locus is irreducible, so $X_n$ is integral of dimension 2n. In the local finite extension over the regular local ring of Y, the parameters of Y are a system of parameters on $X_n$ and are regular by $(H_n)$. Auslander–Buchsbaum makes $p_{\ast }{O}_X$ locally free, of rank n!. This is actual flatness, not a generic-degree argument. The remaining Haiman hypothesis is precisely Cohen–Macaulayness, not an additional assumed component equality.

The $S_n$-action splits this bundle by the constant character projectors. Its isotypic ranks are constant on connected Y and equal the regular-representation multiplicities on the distinct-point locus. Thus every fiber is a regular $S_n$-module. The closed embedding $X_n\subset Y\times M$ is an actual family of $S_n$-clusters and defines a morphism $\psi :Y\to S_n-Hilb(M)$, into the reduced free-orbit component.

The reverse morphism is constructed on families. For any $S_n$-cluster algebra P over T, $P^{S_{n-1}}$ is a direct summand of rank n. It is generated, as an $O_T$-algebra, by functions on the remaining single S factor, because symmetric functions in the other n−1 factors are expressible using functions in that factor and fully symmetric functions, and the latter act as scalars on $P^{S_n}=O_T$. Here is the invariant generation used in this assertion: orbit sums of tensors $f_1\otimes \cdots \otimes f_k$ with different positions are obtained by inclusion–exclusion over set partitions from products of the power sums ${\mathrm{\Sigma }}_j$ $f(P_j)$; divide by the finite factorial multiplicities in characteristic zero. For $S_{n-1}$, use ${\mathrm{\Sigma }}_{j<n}$ $f(P_j)={\mathrm{\Sigma }}_{j\le n}$ $f(P_j)-f(P_n)$. Exact finite-group averaging lifts invariants through a quotient. This proves the assertion for every affine coordinate algebra, not only for polynomial functions on A².

Choose an affine open in S containing the finitely many support points (an ample hyperplane avoiding them has affine complement). The preceding generation gives a closed length-n flat family in S, hence a morphism $\phi :S_n-Hilb(M)\to Y$. These morphisms glue canonically. Both φψ and ψφ are identity on the distinct free-orbit locus; since the relevant components are reduced integral and their targets are separated, they are identities everywhere. Therefore Y is exactly the free-orbit $S_n$-Hilbert component and $X_n$ its universal cluster. In particular its Kodaira–Spencer map into equivariant module deformations is injective: an equivariantly trivial deformation of a cyclic cluster quotient is an ideal-trivial deformation, since its invariant unit generator is unique up to a scalar unit.

This paragraph supplies the full scheme-level moduli interface from $(H_n)$. It does not manufacture $(H_n)$. Haiman's proof of that input uses the all-n polygraph theorem, not the small cases or the formal-local transfer alone.

Put $d=2n$ and $K=O_{X_n}$ on Y×M with its natural $S_n$-linearization; let $S_n$ act trivially on Y. Define

$\mathrm{\Phi }(A)=R{\pi }_{M,\ast }(K{\otimes }^L{\pi }_Y\ast A)$, $\mathrm{\Psi }(B)=[R{\pi }_{Y,\ast }(K^{\vee }\otimes {\pi }_M\ast {\omega }_M[d]{\otimes }^L{\pi }_M\ast B){]}^{S_n}$.

Both ambient varieties are smooth projective, so their kernels are perfect and these are bounded coherent functors. Finite-group invariants are exact by averaging. Smooth projective trace and tensor evaluation give Ψ⊣Φ. These operations can be computed by finite locally free resolutions and finite Čech models, so convolution commutes with derived closed-point restriction. If Q is the perfect kernel for ΨΦ on Y×Y, then

$Ho{m}^i(Q,k_{(y,z)})=Ex{t}_{S_n,M}^i(O_{X_y},O_{X_z})$.

For different clusters, equivariant Hom is zero. A map takes the invariant unit 1 to a scalar multiple of the invariant unit of the other cyclic cluster; a nonzero scalar makes a surjection, and equal lengths force equality of the defining ideals. Top Ext in degree d also vanishes, by Serre duality and the canonical comparison in the geometric Hilbert supplier: on each support orbit ${\omega }_M$ is a constant equivariant line pulled from Q, so twisting does not change that Hom argument. All remaining Ext degrees lie between 1 and d−1. If the quotient supports differ, all Ext vanish by disjoint support.

Consequently Q off the diagonal is supported on (Y×_QY)\Δ, of codimension at least d in the smooth ambient Y×Y, by the geometric Hilbert supplier. Its minimal free models have terms in degrees [−d+1,−1], of length d−2. the regular-base acyclicity supplier makes it zero there. Thus $E_y=\mathrm{\Psi }\mathrm{\Phi }(k_y)$ is a perfect complex supported only at y, with a model in [−d,0]. The adjunction counit $E_y\to k_y$ is an isomorphism on scalar Hom in degree zero. The induced degree-one map

$Ex{t}_Y^1(k_y,k_y)\to Ex{t}_{S_n,M}^1(O_{X_y},O_{X_y})$

is the injective universal-cluster Kodaira–Spencer map from the argument beginning at step 1.1. For the triangle $C\to E_y\to k_y$, the long exact Hom sequence then gives $Ho{m}^i(C,k_y)=0$ for i≤0. A minimal free model of C has no terms in degrees≥0, so $H^j(C)=0$ for j≥0. Thus $H^0(E_y)=k_y$. The regular d-dimensional diagonal criterion of the regular-base acyclicity supplier gives $E_y=k_y$ and identifies the counit as an isomorphism. The cone of the kernel counit $Q\to O_{\mathrm{\Delta }}$ has zero derived restriction at every closed point, so minimal-model Nakayama makes it zero. This proves $\mathrm{\Psi }\mathrm{\Phi }=id$ and full faithfulness of Φ for EVERY n.

Essential surjectivity does not require a CY hypothesis. Since q $r=\tau$ p on $X_n$, the geometric Hilbert supplier identifies ${\pi }_Y\ast {\omega }_Y$ and ${\pi }_M\ast {\omega }_M$ on the kernel. Therefore Φ commutes with the respective Serre functors, including their common shift [d]. Its fully faithful image is admissible (both adjoints exist for a perfect kernel between smooth projective varieties), and this Serre compatibility makes its left and right orthogonal complements equal. Thus it gives an orthogonal decomposition of $D_{S_n}^b(M)$.

That equivariant category is indecomposable. Choose an $S_n$-linearized ample line bundle A on M (tensor the translates of any ample line). Each $A^{-m}\otimes \rho$, for an irreducible representation ρ, is indecomposable, with endomorphisms C. Every such object has a nonzero map to a fixed free-orbit sheaf $O_{S_{n}x}$, which is equivariantly simple; all these objects therefore lie in the same member of any orthogonal decomposition. They form a spanning family: for a nonzero bounded equivariant complex B, choose m large enough that positive cohomology of all its cohomology sheaves twisted by $A^m$ vanishes; some global cohomology module is then nonzero and has an irreducible ρ constituent, yielding a nonzero Hom from $A^{-m}\otimes \rho$ to a shift of B. The complementary component must be zero. The image of Φ is nonzero, hence Φ is an equivalence

$D^b(\mathrm{Coh}{S}^{[n]})\simeq D_{S_n}^b(\mathrm{Coh}{S}^n)$.

This is a genuine all-n kernel proof from $(H_n)$, not an application of the three-dimensional McKay lemma to $S^n$.

Every automorphism of S acts on $X_n$ through its canonical reduced-closure construction and commutes with $S_n$. The kernel linearization is the actual action on its structure sheaf, so the transform intertwines these automorphisms with their canonical cocycles, without projective scalar choices.

For a line bundle A on S, the permutation-linearized $A^{⊠n}$ has trivial stabilizer actions and descends to the norm line $A_{(n)}$ on Q by the same descent proof as the geometric Hilbert supplier. Put $A_n=\tau \ast A_{(n)}$. This is a LINE BUNDLE, not the rank-n tautological bundle usually denoted $A^{[}n]$. On $X_n$ the two pullbacks of this line agree canonically. Projection formula gives

$\mathrm{\Phi }(F\otimes A_n)=\mathrm{\Phi }(F)\otimes A^{⊠n}$.

The isomorphisms are coherent under tensor products and powers. Thus if $A^r=O$ with a specified trivialization, the same specified r-fold action is intertwined; torsion-cover lifting uses this actual equivariant identity. In rational cohomology $ch(A)=1$ for torsion A, so the corresponding tensor actions are the identity on Chern characters. No integral-cohomology triviality is inferred.

If a Fourier–Mukai equivalence between two surfaces has a perfect kernel P and inverse kernel R, its n-fold exterior product has a canonical $S_n$-linearization, with the usual Koszul signs for permutation of complex factors. Convolution and the canonical permutation maps commute, giving $(P^{⊠n})\ast (R^{⊠n})=(P\ast R{)}^{⊠n}=O_{{\mathrm{\Delta }}^n}$ equivariantly. This proves the induced equivalence on $S_n$-equivariant product categories. If the kernels also intertwine finite group actions, combine their n coordinate linearizations and the permutation linearization; the semidirect-product identity is precisely the permutation of these n linearizations. Thus the same construction is compatible with the wreath group $G^n⋊S_n$. Composing with Φ or its inverse gives the required Hilbert-scheme lift whenever those kernel linearizations are part of the actual product equivalence data; their existence is not assumed for an arbitrary unlinearized functor.