Coherent generators and Fermat deformation for the quartic and quintic mirrors

Use the full Cech nerve: its degree-$p$ term is the product of sections over all ordered $(p+1)$-tuples of cover members, with repetitions allowed. Each degree has finitely many tuples. The differential is the alternating deletion differential, and the Alexander–Whitney cup followed by composition is associative and satisfies the signed Leibniz rule. A permutation of cover members acts on tuples without sorting them; it therefore preserves the cup product as well as the differential. This gives a strict $H$ action. Restriction to increasing tuples is a dg algebra map to the usual ordered Cech algebra. It is a quasi-isomorphism: both augmented Cech sheaf complexes resolve the original sheaf, as is checked at a stalk by inserting any cover member containing the stalk; every term is acyclic for global sections because the finite intersections are affine. Apply the same first-quadrant acyclic-resolution comparison to both augmentations, shifting by the finitely many internal degrees. Thus this full-nerve dg model computes the same derived Hom as the ordered carrier in [F5]. No assertion that sorting indices preserves the ordered cup product is needed.

Denote the two full-nerve complexes over $\mathbb{C}$ by $D_0=C^{\bullet }(P,M_0)$ and $D_1=C^{\bullet }(P,M_1)$. An intersection and its vector bundle are obtained by base extension from $\mathbb{C}$. Sections there extend by tensoring with $R$, since they are modules on an affine scheme and the bundles are extended from the given finite presentation. Consequently the actual algebraic ambient complexes over $R$ are $D_t{\otimes }_{\mathbb{C}}R$, not an assumed completion. Their terms are free $R$-modules: choose a Hamel basis of each actual complex vector space. They need not have finite rank. The totalization has only finitely many internal-degree summands in a fixed total degree, so this assertion does not interchange a tensor product with an infinite product.

For each $D_t$ set $Z_t^j=\ker d$, $B_t^j=\mathrm{im}d$. Choose right inverses of $Z_t^j\to H^j(D_t)$ and $D_t^j\to B_t^{j+1}$ over $\mathbb{C}$, first in each object pair, and average their conjugates over $H$. A finite average of right inverses remains a right inverse, including when $H$ permutes object pairs. It gives the actual decomposition $D_t^j=B_t^j\oplus i_t{H}^j(D_t)\oplus S_t^j\mathrm{.}$ Define $p_t$ to be projection to the middle summand; define $h_t$ on $B_t^{j+1}$ as the inverse of $d:S_t^j\to B_t^{j+1}$ and as zero on the other summands. These are actual linear maps on the specified Cech complex and satisfy $d{h}_t+h_{t}d=1-i_t{p}_t,p_t{i}_t=1,h_t^2=h_t{i}_t=p_t{h}_t=0.$ The identity cochains are closed and nonexact (they already represent nonzero degree-zero maps). Choose the middle summands to include their object ring. Then $h_t$ vanishes on that ring where it occurs. For the optional torus take the complements separately in each weight space; conjugation by $H$ permutes weights, so averaging preserves torus equivariance. Every image of an actual ambient vector is an algebraic section with a finite Hamel expansion, hence finite Laurent-coordinate expressions on the fixed affine intersections. Extend the maps $R$-linearly; this never sums over an infinite basis on one algebraic input. This constructs the splittings; it assumes neither projectivity of hypersurface chart sections nor finite-free relative cohomology.

Form the sheaf Koszul dg algebra $K_f=[{\mathcal{O}}_{P_R}(-d)ϵ\stackrel{dϵ=f}{\to }{\mathcal{O}}_{P_R}],\mathrm{\mid }ϵ\mathrm{\mid }=-1,{ϵ}^2=0,$ with its natural multiplication and signs. Let $A_R$ be the full-nerve total dg algebra of ${\mathrm{Hom}}^{\bullet }({\mathcal{E}}_R,{\mathcal{E}}_R)\otimes K_f$. The augmentation to the restriction on $Y$ is a dg quasi-isomorphism on every intersection: the equation is a non-zero-divisor and the Hom sheaf is locally free. The resulting total comparison is a quasi-isomorphism by filtering the cone by the Cech degree and using the bounded Koszul and internal degrees. Taking normal-subgroup invariants is exact by finite averaging, so the comparison remains a quasi-isomorphism on invariants. This constructs the geometric chain model itself.

Write the underlying complex as $A_R=(D_0\oplus D_1[1])\otimes R,D=d_0+\delta ,$ where $d_0$ is the block differential (minus the $D_1$ differential on its shift) and $\delta$ is multiplication by $f$ from the shifted Koszul summand to the unshifted one. Use the same shift sign on $h_1$. The maps from step 3.1 give a strong deformation retract $(i,p,h)$ for $d_0$ onto $P_R$ with initially zero differential. Each $h$ preserves its Koszul summand, whereas $\delta$ changes it from the shifted to the unshifted summand and vanishes on the latter. Hence $\delta h\delta =0.$ Define, with no infinite series, $i^{\mathrm{\prime }}=i-h\delta i,p^{\mathrm{\prime }}=p-p\delta h,h^{\mathrm{\prime }}=h-h\delta h,d_P=p\delta i\mathrm{.}$ Direct substitution using $d_{0}h+h{d}_0=1-ip$, $d_0\delta +\delta d_0=0$, ${\delta }^2=0$ and $\delta h\delta =0$ gives $D{i}^{\mathrm{\prime }}=i^{\mathrm{\prime }}{d}_P,p^{\mathrm{\prime }}D=d_P{p}^{\mathrm{\prime }},p^{\mathrm{\prime }}{i}^{\mathrm{\prime }}=1,D{h}^{\mathrm{\prime }}+h^{\mathrm{\prime }}D=1-i^{\mathrm{\prime }}{p}^{\mathrm{\prime }},(h^{\mathrm{\prime }}{)}^2=h^{\mathrm{\prime }}{i}^{\mathrm{\prime }}=p^{\mathrm{\prime }}{h}^{\mathrm{\prime }}=0.$ For example $d_{0}h\delta i=\delta i-ip\delta i$, which is exactly the term needed in $D{i}^{\mathrm{\prime }}=i^{\mathrm{\prime }}{d}_P$; every putative second perturbation contains $\delta h\delta$ and is zero. The map $d_P$ is precisely the induced cohomology map of multiplication by $f$, with its cone shift, and $d_P^2=0$ because it too changes the Koszul summand only once. The assumed finite ambient cohomology makes $P_R$ bounded finite-free. If $f_{\ast }=0$, the displayed differential is literally zero. All maps preserve the strict finite-group action and object idempotents.

Transfer the dg multiplication along these actual maps. In suspended notation, put multiplication at binary vertices, $i^{\mathrm{\prime }}$ at leaves, $h^{\mathrm{\prime }}$ on internal edges and $p^{\mathrm{\prime }}$ at the root, with the tensor Koszul signs. For every fixed number of inputs there are finitely many planar binary trees; this remains true although $A_R$ is not bounded or finite rank. The tree sums define the operations on $P_R$ and the comparison maps. Their identities follow by applying $D{h}^{\mathrm{\prime }}+h^{\mathrm{\prime }}D=1-i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}$ at an internal edge: its identity term cancels the contracted-edge term by associativity, and its $-i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}$ term is the required composition of two transferred trees. At a root or leaf the same calculation is the functor equation. Equivalently, extend the contraction to a word of length $r$ by the signed sum placing $h^{\mathrm{\prime }}$ in one position, $i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}$ before it and the identity after it; the multiplication perturbation decreases word length, so its perturbation series has at most $r-1$ factors. This proves the two comparison maps and their two-sided bar homotopy without an infinite summation on a fixed word. The unit cochains are unchanged by $\delta$ and killed by $h^{\mathrm{\prime }}$. The normalized tree calculation therefore gives strict units and no higher operation with a unit input; the remaining unit-adjacent terms cancel in pairs. Thus these are actual strictly unital $A_{\mathrm{\infty }}$ comparisons. Equivariance holds term by term. This verifies transfer for this constructed contraction, rather than invoking a theorem conditional on an unconstructed finite-free input.

Every displayed map is $R$-linear, so preserves $q^s$ times its source. Its quotient modulo $q^s$ satisfies the same finite identities. Extend it to coefficientwise $q$-adic completions by inverse limits; the finite sums, identities and each fixed-arity tree formula continue to hold. The target $P_R$ is already complete because it is finite-free. Thus both $A_R$ and ${\widehat{A}}_R$ retract to this same $P_R$, and $A_R\to {\widehat{A}}_R$ induces the identity under their cohomology identifications with $H(P_R)$. This proves the completed-family comparison without an algebraization or inverse-limit exactness assumption. The same argument works after exact normal-subgroup invariants.

For a flat $R$-algebra $L$, tensor the finite retract identities and the fixed-arity formulas with $L$. The algebraic ambient sections on each intersection base-change to the actual algebraic sections over $L$; only finite products occur in each Cech degree. The Koszul augmentation of step 3.2 therefore identifies this scalar extension with the actual endomorphism complex on $Y_L$. In particular this applies to $\mathbb{C}((q))$ and any extension field. If a completed presentation is used first, use its common finite model $P_R$, tensor that model, and compare with the algebraic model just described. No identification of a completed local chart tensored with $\mathbb{C}((q))$ with its algebraic generic chart is asserted. The unchanged cohomology/binary operations of a minimal model also survive this scalar extension.

The strict full-nerve action, actual ambient splittings, finite Koszul perturbation, finite-arity transfer and common finite model establish all claims. Reduction modulo $q$ retains the optional torus action because the reduced equation and initial splittings are torus equivariant.

The actual two hypersurface families Let $N=4$ or 5, $n=N-2$, $R=\mathbb{C}[\text{\hspace{0.17em}⁣}[q]\text{\hspace{0.17em}⁣}]$, $K=C((q))$, and ${\mathrm{\Lambda }}_Q$ the source algebraic closure of K. Put $Y_q=\{y_1\mathrm{.}\mathrm{.}\mathrm{.}{y}_N+q(y_1^N+\mathrm{.}\mathrm{.}\mathrm{.}+y_N^N)=0\}$ subset $P_R^{N-1}$. The family is R-flat. On each standard affine chart its equation F reduces modulo q to a nonzero polynomial in a domain. If q $f=F$ g, reduction gives $F_0$ $g_0=0$, hence $g=q{g}_1$ and cancellation gives $f=F$ $g_1$. Thus its quotient has no q-torsion and is flat over this DVR. Its generic geometric fibre is smooth. If at least two coordinates vanish, a nonzero coordinate has derivative Nq $y_i^{N-1}$ nonzero. If just one vanishes, its derivative is the product of all the others, nonzero. If none vanishes, setting all partial derivatives zero gives P+Nq $y_i^N=0$ for every i, where $P=\prod$ $y_i$. Multiplying gives $(Nq{)}^N=(-1{)}^N$, impossible for the formal parameter q. Therefore no projective singular point exists. Let tildeGamma consist of the diagonal Nth roots with product one, and $\mathrm{\Gamma }=\tilde{\mathrm{\Gamma }}\mathrm{/}\text{scalar diagonal}$. It has order $N^{N-2}$. Normalize the first diagonal entry to one; this gives an explicit splitting Gamma→tildeGamma because that entry has Nth power one. Thus Gamma acts linearly on the projective coordinates and all the bundles below have actual linearizations. Character values lie in C and every group order is invertible in R and ${\mathrm{\Lambda }}_Q$.

Restricted Euler bundles generate without a Beilinson theorem Let V have dimension N and define $A_p={\mathrm{\Omega }}_P^p(p)$, $0\le p\le N-1$, $F_j=A_{N-j}[N-j]$, $1\le j\le N$. The Euler sequence, checked in a standard coordinate trivialization, is $0\to {\mathrm{\Omega }}^1(1)\to V^{\ast }$ tensor O→O(1)→0. Its exterior-power Koszul sequence is $0\to {\mathrm{\Omega }}^p(p)\to {\mathrm{\Lambda }}^p$ $V^{\ast }$ tensor $O\to {\mathrm{\Lambda }}^{p-1}{V}^{\ast }$ tensor O(1) →...→O(p)→0. It is locally split exact: where one homogeneous coordinate is a unit, wedge/contraction by that coordinate supplies the usual Koszul contraction. Pullback to ANY hypersurface remains exact because this is a locally split sequence of vector bundles. Induction on p shows the two finite collections $\{{\mathrm{\Omega }}^p(p):0\le p\le N-1\}$ and $\{O(p):0\le p\le N-1\}$ have the same thick closure. One direction follows immediately from the displayed sequence. For the other, knowing O(0),...,O(p-1), solve that same finite exact complex for its endpoint O(p). The shifts [p] do not affect the thick closure. Apply the actual ample-power generator root with $r=N-1$ and tensor by O(N-1). The positive consecutive powers therefore generate $\mathrm{Perf}(Y_q)$ and $\mathrm{Perf}(Y_q^{\ast })$. The restrictions $E_j=F_j\mathrm{\mid }{Y}_q$ are thus actual generators of Perf, even though $Y_0$ is singular. On the smooth generic fibre every bounded coherent complex is perfect, by finite regular-local projective dimension and sufficiently negative globally generated equivariant vector-bundle resolutions. Hence these objects generate $\mathrm{DbCoh}(Y_q^{\ast })$. This proves precisely the generation needed; it does not assert the unproved Hom/wedge computation merely by calling the objects a Beilinson basis.

All character twists generate the quotient category Give each $E_j$ its chosen Gamma-linearization and take ALL twists $E_j$ tensor chi, chi in ${\mathrm{\Gamma }}^{\ast }$. There are N $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }=N^{N-1}$ objects: 64 when $N=4$ and 625 when $N=5$. Let G be their direct sum. It is the regular-representation-induced version of the underlying generator. For an equivariant quasi-coherent complex H, derived equivariant Hom from $E_j$ tensor chi is the chi-isotypical part of ordinary derived Hom from $E_j$. This can be checked on the finite affine Čech Hom complex; finite averaging makes invariants exact. If all those groups vanish, ordinary $\mathrm{RHom}(E_j,H)$ vanishes, so $H=0$ by the generator just proved. The same invariant/Čech description shows G is compact, since finite-group invariants and a finite Čech complex commute with arbitrary direct sums. The tensor–Hom adjunction with the ACTUAL dg algebra $S=RHo{m}_{\mathrm{\Gamma }}(G,G)$ therefore gives an equivalence with dg modules: build the tensor functor on semifree modules, verify its unit on the free generator, extend by sums/cones/sequential semifree resolutions, and use the detected-zero cone to prove the counit. Compact modules are finite cell retracts by the explicit finite-generator-support argument in the existing generator root. Consequently $Per{f}_{\mathrm{\Gamma }}(Y_q^{\ast })$ ≅ $Per{f}_{\mathrm{d}\mathrm{g}}(S^{\ast })$. Every equivariant coherent sheaf on this smooth tame quotient is perfect. Indeed equivariant evaluation O(-a) tensor H0(H(a))→H is surjective for sufficiently large a, and its kernel is equivariant coherent. Repeat; regular-local dimension n makes the nth syzygy locally free. Thus $\mathrm{DbCoh}([Y_q^{\ast }\mathrm{/}\mathrm{\Gamma }])$ ≅ $D^{\pi }(S^{\ast })$. This is an actual generation/Morita equivalence, not the final HMS functor. On the quintic the quotient stack here is the source's coherent mirror category itself.

The dg model and scalar extension are explicit Choose the projective coordinate cover, which is stable under the diagonal group and permuted by coordinate permutations. The full Čech nerve with all ordered tuples gives the strict symmetry-compatible dg cup-composition model; restriction to increasing tuples is a dg quasi-isomorphism to the ordered Čech carrier, as proved in [step 8.1]. Both compute the same derived endomorphisms. The finite-free ambient resolution and its actual comparison to this geometric carrier are constructed in steps 15.2, 16.2, 16.3 and 10.1. Over R its terms are R-flat: $Y_q$ is flat and End(G) is locally free. They are sections over affine flat R-schemes, hence flat R-modules. The ordered presentation has bounded Čech direction and R-flat terms, and is therefore an underlying K-flat model. The full-nerve symmetry model is compared to it by [step 8.1]; for the field extension here, ordinary tensor is exact. Tensoring either algebraic Čech presentation with ${\mathrm{\Lambda }}_Q$ computes the generic-fibre endomorphisms, with the SAME restriction and cup-composition maps. Invariants commute with this tensor by the averaging idempotent. Therefore $S_q$ $tenso{r}_R^L$ ${\mathrm{\Lambda }}_Q$ $=$ $S_q$ $tenso{r}_R$ ${\mathrm{\Lambda }}_Q$ ≃ $S^{\ast }$. This is the actual all-higher/dg scalar-extension comparison. It uses no automatic uniqueness-of-enhancement assertion. The cyclic permutation of coordinates acts on the GEOMETRIC collection and permutes its character linearizations through the chosen splitting. Its lift must be computed on those objects, not replaced by a naive permutation of vector-space coordinates after a noncanonical smash-product identification. The invariant Hochschild action is supplied by the declared exterior/smash Hochschild lemma.

Quartic resolution and the exact McKay hypotheses For $N=4$ the target is the minimal resolution of the coarse quotient. Apply the declared algebraic surface derived-McKay supplier after verifying its hypotheses below; the required equivalence is $\mathrm{DbCoh}(Z_q^{\ast })$ ≅ $\mathrm{DbCoh}([Y_q^{\ast }\mathrm{/}\mathrm{\Gamma }])$. The family-specific hypotheses can all be checked. The geometric generic quartic is a smooth projective surface by step 9.1. The equation sequence $O(-4)\to O\to O_Y$ and the projective-space cohomology of [F4] give $H^0(Y,O_Y)={\mathrm{\Lambda }}_Q$: H0(O(-4)) and H1(O(-4)) vanish. Hence Y is connected. Over the algebraically closed field ${\mathrm{\Lambda }}_Q$ its smooth irreducible components are disjoint, so connectedness makes it integral. In particular its nonempty open set with all coordinates nonzero is dense. A diagonal element fixing a point of that open set is scalar and thus trivial in Gamma; this proves faithfulness of the action. Residue adjunction gives ${\omega }_Y=O$ and the determinant-one action preserves its residue generator. A point with all coordinates nonzero has trivial projective stabilizer; so does a point with just one coordinate zero, by the determinant condition on the remaining normalized diagonal entries. A point with three zero coordinates is not on $Y_q^{\ast }$. With exactly two nonzero coordinates its equation is $y_i^4=-y_j^4$, giving four points for each of six coordinate pairs, hence 24 points. Their stabilizer is ${\mu }_4$ acting on the two missing tangent coordinates by $(zeta,zet{a}^{-1})$. Each orbit has four points, so the coarse quotient has exactly six singular points. At each, average parameter lifts to obtain equivariant formal coordinates; its completed local invariant ring is $K[\text{\hspace{0.17em}⁣}[x,y,z]\text{\hspace{0.17em}⁣}]\mathrm{/}(xy-z^4)$, $K={\mathrm{\Lambda }}_Q$, with $x=u^4,y=v^4,z=uv$. These are all invariant monomials, since an invariant exponent difference is divisible by four. Blow up each ACTUAL singular closed point. In the x and y charts the strict transform is smooth; in the z chart its equation is $XY=z^2$. Blow up its remaining singular point to obtain a smooth surface with a three-P1 exceptional chain. Blowups commute with the flat completion, so these chart computations prove regularity of the actual algebraic surface, not just of a separate formal resolution. The resulting morphism is projective. Crepancy is equally explicit. The $A_{r-1}$ resolution charts have coordinates $s_i,t_i$, $i=0,\mathrm{.}\mathrm{.}\mathrm{.},r-1$, with $x=s_i^{r-i}{t}_i^{r-i-1}$, $y=s_i^i$ $t_i^{i+1}$, $z=s_i$ $t_i$. Adjacent charts are glued by $s_{i+1}=s_i^2{t}_i$, $t_{i+1}=s_i^{-1}$. The quotient two-form dx wedge dz/x pulls back to $d{s}_i$ wedge $d{t}_i$ on EVERY chart. Thus it has neither zero nor pole and the resolution is crepant. Adjunction then makes each exceptional P1 self-intersection -2, so this is the minimal resolution in the statement. The coarse quotient is projective: the invariant homogeneous coordinate ring is a finite module over the ring generated by the coordinate Nth powers, since reducing exponents modulo N leaves only finitely many invariant monomials, and its projective spectrum is the finite diagonal quotient. Thus the displayed projective resolution is itself projective over ${\mathrm{\Lambda }}_Q$. Together with connectedness, faithfulness, and the invariant nonzero form proved above, this verifies every family-specific hypothesis of [F10] over the exact algebraically closed characteristic-zero field ${\mathrm{\Lambda }}_Q$.

The Hom computation is an explicit homogeneous Koszul cancellation The Euler resolution of $A_q(-j)$, $0\le j\le N-1$, has terms ${\mathrm{\Lambda }}^{q-b}{V}^{\ast }$ tensor O(b-j), $b=0,\mathrm{.}\mathrm{.}\mathrm{.},q$, in cochain degree b. All their higher cohomology vanishes: $-N<b-j\le N-1$, so the actual projective-space line-bundle calculation has neither top cohomology nor intermediate cohomology. Its degree-zero section complex consists of the terms $b\ge j$ and is exactly the degree-(q-j) homogeneous part of the polynomial Koszul complex ${\mathrm{\Lambda }}^s$ $V^{\ast }$ tensor $Sy{m}^{q-j-s}{V}^{\ast }$. Write $s=q-b$ for the exterior degree and $e=b-j$ for the polynomial degree. The differential is $su{m}_i$ $y_i$ $contractio{n}_{v_i}$, and take $h=su{m}_i$ $(v_i^{\ast }wedge-)$ $partia{l}_{y_i}$. Their anticommutator is $(s+e)Id=(q-j)Id$. Thus division by q-j contracts every positive homogeneous degree, with no appeal to a Bott theorem; the letter q in this Euler calculation is the integer bundle index, not the family parameter. If $q=j$ the sole remaining constant is in cochain degree q. If $q<j$ there are no terms. Hence $H^i(P,A_q(-j))=C$ if $i=q=j$, and zero otherwise. (E) This proves every vanishing used here, rather than saying “Bott computation”. The tautological line injection O(-1)→V tensor O gives a locally split Koszul resolution of $A_p^{\ast }={\mathrm{\Lambda }}^p$ Q: terms ${\mathrm{\Lambda }}^{p-j}V$ tensor O(-j) in degrees -j, $j=0,\mathrm{.}\mathrm{.}\mathrm{.},p$. Tensor with $A_q$ and apply (E). The associated finite double complex has just one nonzero cohomology term, when $j=q\le p$. Therefore ${\mathrm{Ext}}_P^i(A_p,A_q)={\mathrm{\Lambda }}^{p-q}V$ if $p\ge q$ and $i=0$, 0 otherwise. Contraction by constant vectors v in V gives $A_p\to A_{p-1}$. Its iterates realize this isomorphism: on the one surviving Koszul term the induced map is exactly the corresponding exterior vector. Anticommutation of contractions makes compositions the wedge product, with its graded order convention fixed once. With the cohomological shift convention, ${\mathrm{Hom}}^r(A_p[p],A_q[q])={\mathrm{Ext}}^{r+q-p}(A_p,A_q)$, so it is ${\mathrm{\Lambda }}^{p-q}V$ precisely when $p\ge q$ and $r=p-q$. For the consistently reversed source labels $F_j=A_{N-j}[N-j]$, this becomes ${\mathrm{Hom}}^r(F_j,F_k)={\mathrm{\Lambda }}^{k-j}V$ precisely when $j\le k$ and $r=k-j$, and zero otherwise. This is GL(V)-equivariant by the natural contractions.

Restriction to a degree-N hypersurface gives a CONSTANT product algebra Let $i:Y\to P_R$ be any of the flat Cartier degree-N hypersurfaces used above, over $R=C$ or C［［q］］. The length-one equation resolution gives the triangle $RHo{m}_P(F_j,F_k(-N))\to RHo{m}_P(F_j,F_k)$ $\to RHo{m}_Y(i^{\ast }{F}_j,i^{\ast }{F}_k)$. Projective-space Serre duality identifies the first term with the reverse Hom dual shifted by -(N-1), with the determinant character retained. The groups in §1 are free over R, in their single displayed degrees. In one directed order the reverse term is zero; in the other the forward term is zero; on the diagonal their degrees differ. Thus the triangle's map is zero in the derived category of R-modules and its cohomology is the unique directed-plus-dual splitting ${\mathrm{Hom}}_Y^{\ast }(E_j,E_k)={\mathrm{Hom}}_P^{\ast }(F_j,F_k)$ direct-sum ${\mathrm{Hom}}_P^{\ast }(F_k,F_j{)}^{\ast }[-n]$ tensor det V. For SL(V), det V is trivial. The second summand is the dual bimodule of the first. Its product with itself is zero: before shifts these are ${\mathrm{Ext}}^n$ between vector bundles, while the displayed computation has no ${\mathrm{Ext}}^{2n}$. Cross products are uniquely the dual directed products by the perfect Serre trace. Normalize the trace using the equation resolution: its connecting map $H^n(Y,O_Y)\to H^{N-1}(P,O(-N))$ is an isomorphism, and the latter has the coefficient trace taking the Cech class $(y_1\mathrm{.}\mathrm{.}\mathrm{.}{y}_N{)}^{-1}$ to 1 in the chosen volume convention. For a locally free Hom sheaf use evaluation followed by this same connecting map. The boundary of a Cech product satisfies the signed Leibniz identity, so this trace is compatible with left and right composition; its duality pairing is exactly the Cartier residue pairing, including its determinant character. This specifies one normalization independent of the hypersurface equation, rather than choosing unrelated rank-one traces on different fibres. It is perfect over R, either from the actual regular-base projective embedding duality or by reducing its free-module determinant modulo q. Hence all products are exactly the trivial-extension product, INDEPENDENT OF THE HYPERSURFACE EQUATION. This proves the coherent family's fixed binary algebra, not merely its constant graded ranks. Its GL(V) action includes the determinant action on the dual summand. Cyclic symmetry must retain that action and the actual character linearizations; choosing a volume form and then forgetting its character would give a wrong lifted action.

The actual ambient complex, not flat hypersurface terms Keep precisely the shifted Euler bundles $F_j$ and every character twist from steps 10.1–11.1, and denote their direct sum on $P_R$ by tildeG. For the projective coordinate cover use the full Čech nerve of the dg sheaf End(tildeG) tensor [O(-N) epsilon → O], with $\mathrm{\mid }ϵ\mathrm{\mid }=-1$, ${ϵ}^2=0$ and d $ϵ=F_q=w+q$ sum $y_i^N$. Its augmentation is an actual dg quasi-isomorphism to the Čech End(G) model on $Y_q$: on each affine intersection the Cartier equation resolution is exact, and tensoring with the locally free Hom sheaf preserves that exactness. Take Gamma invariants by its averaging idempotent. Before the hypersurface differential is inserted, the two ambient complexes are obtained by extending actual C-vector-space Čech complexes for $\mathrm{End}(tilde{G}_C)$ and $\mathrm{End}(tilde{G}_C)(-N)$. They therefore have R-free terms; no freeness or splitting of the hypersurface chart modules is asserted. In [steps 3.1–5.1], choose objectwise complex cycle/boundary complements in these ambient complexes, extend them R-linearly, and let i,p,h be the resulting actual maps. The hypersurface perturbation delta changes the epsilon summand only once, so delta h $\delta =0$. Thus the new maps are the finite formulas $i^{\mathrm{\prime }}=i-h$ delta i, $p^{\mathrm{\prime }}=p-p$ delta h and $h^{\mathrm{\prime }}=h-h$ delta h, with finite-model differential $d_H=p$ delta i. They satisfy the displayed strong-deformation-retract equations in [step 8.1], not merely a cohomology dimension comparison.

The coordinate-point Koszul complex detects nonformality For v nonzero in V, contraction by v gives the locally free resolution of the point [v] in P: $[A_{N-1}\to A_{N-2}\to \mathrm{.}\mathrm{.}\mathrm{.}\to A_0]\simeq O_{[}v]$. This is the Koszul complex of the section of the tautological quotient Q induced by v; its zero scheme is the single point [v], cut regularly in its standard affine chart. Restrict it DERIVEDLY to $Y_0=\{\prod y_i=0\}$. It is zero when [v] is outside $Y_0$. Apply RHom from $E_1=A_{N-1}[N-1]$ to this restricted complex. The directed-plus-dual calculation of §2 gives the finite spectral sequence with bottom row $C\to V\to {\mathrm{\Lambda }}^{2}V\to \mathrm{.}\mathrm{.}\mathrm{.}\to {\mathrm{\Lambda }}^{N-1}V$ and one additional C in position (column0,row n). Bottom differential is wedge by v. It leaves precisely one C at the last bottom column and the additional top C. If [v] is outside $Y_0$ the total complex is acyclic, so the only possible remaining differential, of order N-1, MUST be nonzero. If the central dg algebra were formal, model the point restriction as a twisted complex over its binary trivial extension. Its strictly triangular twisting matrix can have only adjacent v entries: the cohomological degree of a directed jump of length r is r, and a twisting entry must have degree one. Thus all longer entries are zero. The Hom differential in that binary formal model has only its first wedge differential, so the displayed spectral sequence could not kill its two surviving classes. This contradiction proves nonformality. For the central algebra with its diagonal torus action, the negative Hochschild obstruction space is one-dimensional at arity N and zero at other arities $>2$. This follows directly from the small Koszul cochain calculation in the previous packet, now taking invariants of the FULL determinant-one torus: its character equality is equality of integer exponents, not only congruence mod N, and it forces $y_1\mathrm{.}\mathrm{.}\mathrm{.}{y}_N$ at arity N. Weight projection is exact for torus representations. Thus all operations below N can be killed equivariantly, and nonformality forces the arity-N class to be a nonzero multiple of $w=y_1\mathrm{.}\mathrm{.}\mathrm{.}{y}_N$. Rescaling the arrow generators normalizes that multiple; a nonzero complex Nth root exists. This supplies the central-class input of the Hochschild/formal-comparison roots.

The finite model is actually minimal and has the stated product Here is the zero-differential input of [step 8.1] for this collection. On a (j,k) summand, step 14.1 puts $H^r(P,F_j^{\ast }\otimes F_k)$ in degree $r=k-j$ when $j\le k$, and makes it zero otherwise. Projective-space Serre duality puts $H^r(P,F_j^{\ast }\otimes F_k(-N))$ in degree $r=N-1-(j-k)$ when $j\ge k$, with the actual determinant character, and makes it zero otherwise. The $\text{multiplication-by-}{F}_q$ map has the same ambient cohomological degree on source and target. If $j<k$ its source is zero; if $j>k$ its target is zero; if $j=k$ its source has degree N-1 and its target has degree zero. Consequently the map is identically zero for $N=4,5$. This remains true after all character twists and Gamma invariants. The constructed finite model therefore has literally $d_H=0$, with underlying bounded finite-free module $H^{\ast }(En{d}_P(tildeG))$ plus $H^{\ast }(En{d}_P(tildeG)(-N))[1]$, followed by invariants. Its cohomology identification is the actual directed-plus-dual one from step 7.1. Use the ambient Serre trace on the epsilon component to identify the dual summand. The correction -h delta i has no epsilon component, so it does not change that trace. The directed classes are the actual constant contraction maps of step 14.1; their compositions and their dual trace products are exactly those in step 15.1, while dual times dual is zero in the unshifted ${\mathrm{Ext}}^{2n}$ group. Thus the transferred binary product, in this specified identification, is the source exterior/trivial-extension product over R, not a deformation of it or a rank argument.

The finite symmetry is the actual object/character action No arbitrary permutation of a chosen smash-product basis is used. Write s:Gamma→tildeGamma for the section in step 1.1. For an even coordinate permutation sigma let $ta{u}_{\sigma }$ be its induced conjugation on Gamma and set $c_{\sigma }(\gamma )=s(ta{u}_{\sigma }(\gamma ){)}^{-1}$ sigma s(gamma) ${\sigma }^{-1}$ in ${\mu }_N$. The natural GL(V)-linearization of $F_j$ has its actual scalar character $a_j=-(N-j)$. Under sigma define the new twist $ch{i}_{\sigma }$ by $ch{i}_{\sigma }(ta{u}_{\sigma }(\gamma ))=c_{\sigma }(\gamma {)}^{a_j}$ chi(gamma). This is an existing twist, because $c_{\sigma }$ is a character. The identity $c_{\sigma eta}(\gamma )=c_{\sigma }(ta{u}_{e}ta(\gamma ))c_{e}ta(\gamma )$, together with the natural bundle maps $F_j\to {\sigma }_{\ast }{F}_j$, proves the group law on the actual direct sum. The chosen bases of the one-dimensional character lines identify these maps without additional scalar choices. Thus Gamma semidirect $A_N$ has a strict finite action on the full-nerve ambient dg model and preserves $F_q$ and the object idempotents. Choose the initial complements by finite averaging as in [step 8.1]. On the central fibre also use the determinant-one torus weight spaces before averaging: the Čech terms are rational weight modules, and even permutations normalize that torus. Since w is torus invariant, the reduced contraction and transfer retain the full central torus action used in step 8.1. All arities of the family remain equivariant for the actual finite even-coordinate action.

The moving point calculates the first-order polynomial class The same coordinate-point construction over $R=\mathbb{C}[\text{\hspace{0.17em}⁣}[q]\text{\hspace{0.17em}⁣}]$ computes the derived restriction of a formal point $[v_q]$ to $Y_q$. Locally its hypersurface tensor resolution is the two-term complex $[R--F_q(v_q)-\to R]$. Its relevant Hom spectral sequence has the same two surviving free rank-one terms as in step 16.1. Locally, trivializing the tautological line at the point and the rank-one bundle $A_{N-1}$, its abutment is the two-term complex $[R--F_q(v_q)-\to R]$, up to an overall shift and invertible changes of its two generators. This identifies the vanishing of its last differential with the equation, without yet assuming a point-independent normalization. To pin down that normalization globally rather than allow an arbitrary point-dependent unit, kill the family's operations of arity 3,...,N-1 equivariantly. Their binary Hochschild obstruction groups vanish by the same grading/character calculation, so coefficientwise primitives over R give a convergent formal gauge; higher components of that gauge preserve the fixed binary product. Here is why the last differential is the polynomial evaluation of that operation. Order the N summands of the point complex as $E_1,\mathrm{.}\mathrm{.}\mathrm{.},E_N$. A directed morphism crossing r positions has degree r. A component of a twisting cochain must have degree one, so its only entries are the adjacent copies of v. Every higher comparison component on a string of r adjacent arrows has degree $r+(1-r)=1$ and crosses r positions; it therefore vanishes for $r>1$. The all-arity comparison consequently carries the actual geometric point complex to this exact adjacent-arrow twisted complex, without hidden longer entries. Its Maurer–Cartan equation holds: for $r=2$ the wedge v wedge v is zero, and for $r>2$ the operation has output degree 2 in a forward Hom space concentrated in degree r, hence vanishes. In $\mathrm{Hom}(E_1,-)$, after the first wedge differential, the only survivors are the dual top class at the first position and the last bottom class. Their column distance is N-1. All operations of arities 3,...,N-1 have been removed, and a strictly triangular path contains at most N-1 arrows. Thus the unique remaining differential is exactly ${\mu }^N$ applied to the top input and N-1 adjacent copies of v. Its value in the rank-one last bottom quotient is a homogeneous degree-N polynomial $p_q(v)$: the first N-1 factors come from the arrows, and identifying ${\mathrm{\Lambda }}^{N-1}V\mathrm{/}(vwedge{\mathrm{\Lambda }}^{N-2}V)$ with the chosen line uses the map a↦v wedge a, giving the last factor. On any projective coordinate chart this identification is an isomorphism for v nonzero, and it uses the same global volume convention as the residue trace. Binary Hochschild boundaries do not change this polynomial: after passing to the two surviving terms their contributions are wedge-v boundaries. Equivalently, in the exterior Koszul cochain resolution [F1], the scalar-output polynomial summand is obtained by symmetrizing the N inputs; diagonal evaluation is its polynomial symbol. Explicitly, the polarized cochain of a symbol P evaluated on the top exterior input and N-1 copies of v gives, up to the common nonzero symmetrization factor, $su{m}_i$ $(partia{l}_{i}P)(v)$ $contractio{n}_{v_i^{\ast }}(vol)$. Wedging with v gives $(su{m}_i{v}_i(partia{l}_{i}P)(v))vol=N$ P(v)vol. This is the homogeneous Euler identity and proves that the evaluation is a nonzero constant multiple of the polynomial symbol on the whole scalar-output summand; all symmetrization factors are invertible in characteristic zero. Fix this single constant in the trace convention. Polarization recovers the symbol from all v, so the evaluation is injective on that summand. The same argument applies to Gamma-invariant cochains and after extension of the coefficient field. On the generic fibre the two surviving Koszul terms are killed precisely when $F_q(v)$ is nonzero; if $F_q(v)=0$, derived restriction is the nonzero point plus its shift. Thus $p_q$ and $F_q$ have the same projective zero hypersurface. For completeness, the hypersurface is geometrically connected because the equation sequence and [F4] give $H^0(O_Y)={\mathrm{\Lambda }}_Q$; smoothness then makes it geometrically irreducible. Consequently its reduced prime equation $F_q$ divides $p_q$ over the algebraically closed generic field. Both have degree N and $p_q$ is nonzero, so $p_q=c(q)F_q$. Compare the coefficient of w, whose coefficient in $F_q$ is 1: this gives c(q) equal to the w-coefficient of $p_q$, in R itself. The torus-equivariant central calculation of step 16.1 gives $p_0=c(0)w$ with c(0) nonzero. Hence c(q) is an R-unit. This coefficient argument also descends the identity from the algebraic closure to R, without needing an extension of every generic point to a chosen formal point. Hence the normalization's derivative changes the first-order polynomial only by a multiple of w. After dividing once by the nonzero central constant c(0), coefficient comparison gives $p_0=w$ and $p_1=su{m}_i$ $y_i^N+(c^{\mathrm{\prime }}(0)\mathrm{/}c(0))w$. By the injective polynomial-symbol calculation just given this is also the arity-N scalar-output Hochschild class, not merely a pointwise nonvanishing test. The central arity-N class is therefore w, and the first-order polynomial is the Fermat sum modulo the displayed scalar multiple of w. That multiple is the Euler boundary already identified in the Hochschild calculation. Consequently the class in the FIXED-BINARY-product subquotient is nonzero and is the Fermat class. Summing the point over its finite Gamma orbit makes the test equivariant without changing this conclusion. The coordinate cyclic permutation permutes these actual polynomial deformations; hence its invariant negative-arity class is exactly their sum. This verifies the coherent side of the precise one-dimensional formal-comparison input. The normalization ambiguity does not turn this into a claim that arbitrary unrestricted deformations have one-dimensional HH². The graph-cycle binary deformations are excluded here because §2 proves the binary algebra is literally fixed.

Completed family and actual generic scalar extension The preceding computations use the convenient Euler labels $E_j=A_{N-j}[N-j]{\mathrm{\mid }}_{Y_q}$. For the grading in [F1], replace each character-twisted object by $E_j^{\mathrm{♯}}=E_j[s_j]$, where $s_j=\lfloor 2j\mathrm{/}N\rfloor$. Index the twists by the characters $m$ of $\tilde{\mathrm{\Gamma }}$ whose scalar restriction is $j\equiv {\sum }_i{m}_i(\mathrm{m}\mathrm{o}\mathrm{d}N)$, using $j\in \{1,\dots ,N\}$. A singleton arrow has target label $k=j+1$ if $j<N$ and $k=1$ if $j=N$. Its old degree is $d_{jk}=1$ in the first case and $d_{N1}=n-(N-1)=-1$ in the second. Equivalently $d_{jk}=n\mathrm{/}N+2(k-j)\mathrm{/}N$. Since ${\mathrm{Hom}}^r(E_j[s_j],E_k[s_k])={\mathrm{Hom}}^{r+s_k-s_j}(E_j,E_k)$, its new degree is $d_{jk}+s_j-s_k=n\mathrm{/}N+{\theta }_{m+e_i}-{\theta }_m$, where ${\theta }_m=2j\mathrm{/}N-s_j$. This is exactly [F1], including the wrap from $N$ to $1$. For example, at $N=4$ the $1\to 2$ arrow now has degree $1+0-1=0$. Apply the usual suspension signs to compositions and to every higher operation; explicitly the shifted Hom complex is the old Hom complex shifted by $s_k-s_j$, with the tensor Koszul signs. This is an invertible object shift of the entire dg category, so it carries the contraction, transfer, units and group action to the shifted category. Even coordinate permutations preserve $j$ and $s_j$. Around a closed Hochschild word the added degrees telescope to zero, so the polynomial-symbol, primary-class and first-order computations of steps 16.1–17.1 are unchanged. Individual shifts preserve the thick closure and the perfect-module equivalence of step 11.1. Thus the Statement uses these explicitly shifted objects, while the point-Koszul calculation can and does remain in the unshifted $E_j$ labels. Transfer multiplication along i',p',h'. The binary-tree sums and the bar comparison/homotopy construction in [step 5.1] are finite on each fixed input word, so they give actual strictly unital all-arity operations on this bounded finite-free module and comparison quasi-isomorphisms to the ambient dg model. They require no pre-existing finite-free model. Every map is R-linear and hence q-adically continuous. The same identities hold modulo $q^s$ and in the coefficientwise completion; the finite target is already complete. Both the algebraic and completed ambient presentations therefore have this same finite model. For generic extension tensor the algebraic ambient presentation and its Koszul augmentation with $K=C((q))$ and then ${\mathrm{\Lambda }}_Q$. They become the actual affine-cover endomorphism model of the restricted Euler collection on $Y_q^{\ast }$, including Gamma invariants, because these field extensions are flat. The finite model and each comparison tensor to the same model there. Completion is never interchanged with localization of an individual chart: its comparison passes through the common finite model. Combining with the actual generator/Morita construction of step 11.1 gives the claimed generic perfect quotient-stack category and, on the smooth generic fibre, its bounded coherent category. The transferred maps induce precisely the Euler/residue cohomology identifications; on finite upper-triangular twisted complexes they are finite bar-word maps. Hence the actual point Koszul calculations in steps 16.1 and 17.1 compute the primary and first-order classes of this model. Its construction in steps 15.2, 16.2 and 16.3 uses only steps 9.1–15.1, independently of those point calculations, so there is no circular finite-model assumption.

Generation, the complete Hom and residue calculation, the moving-point class, finite replacement, and the checked McKay hypotheses give the asserted coherent model over the precise coefficient field.