Exterior smash Hochschild calculation and equivariant transfer for the named Fermat models

Actual twisted Koszul cochain complex For $A=\mathrm{\Lambda }V$ with odd generators $v_i$, the graded bimodule Koszul resolution has terms $A\otimes {\mathrm{Sym}}^{s}V\otimes A[-s]$. Its exactness is elementary. In $A^e$ change generators to the left-right differences $p_i$ and complementary $q_i$. The resolution is the tensor product of one-variable complexes $\mathrm{\Lambda }(p_i)\otimes C[z_i]$ with $d(z_i^s)=s{p}_i{z}_i^{s-1}$. The contraction $h(p_i{z}_i^s)=z_i^{s+1}\mathrm{/}(s+1),h(z_i^s)=0$ removes every positive resolution degree; only the quotient constant survives. Tensor the finitely many contractions. This proves it is a projective bimodule resolution, not an assumed HKR isomorphism. For a diagonal group element gamma, applying Hom into its twisted bimodule gives $$\begin{gathered} \mathrm{Sym}{V}^{\ast }\otimes \\ LambdaV,d_{\gamma }={\sum }_i({\lambda }_i-1)y_i{v}_i \end{gathered}$$ wedge(-). For ${\lambda }_i$ not 1 the one-variable cohomology is the line $C{v}_i$ in polynomial degree zero: $h(v_i{y}_i^s)=({\lambda }_i-1{)}^{(-1)}{y}_i^{s-1}$ for $s\ge 1$ contracts the rest. For ${\lambda }_i=1$ its differential is zero. Taking the tensor product therefore yields ${\mathrm{Sym}}^s(V^{\gamma }{)}^{\ast }\otimes {\mathrm{\Lambda }}^{r-c}(V^{\gamma })\otimes {\mathrm{\Lambda }}^c(V\mathrm{/}{V}^{\gamma })$, where $c=codim{V}^{\gamma }$ and r is total exterior coefficient degree. The smash-product Hochschild complex is the sum of these twisted complexes followed by tildeGamma invariants. This follows directly by the relative group-bar resolution: insert the separability element $\mathrm{\mid }\mathrm{\Gamma }{\mathrm{\mid }}^{-1}{\sum }_{g}g\otimes g^{-1}$ to contract positive group-bar degrees. Thus averaging is exact and all dimensions below can be read in this explicit small complex. For the source regrading a class at arity s and internal degree t has $r=s+(N\mathrm{/}n)t$; its total Hochschild degree is s+t. This is obtained by adding the exterior degree contributions and cancelling the source/target object shifts around the word. It is essential for the negative-arity slots; ordinary exterior degree must not be substituted silently.

Only the displayed low-degree slots survive In the truncated complex $t\le 0$, total degree one has only the identity-group slot (s,t,r)=(1,0,1). Its invariants are $E_i=y_i{v}_i,i=1,\mathrm{.}\mathrm{.}\mathrm{.},N$. In total degree two the only identity-group slots are (s,t,r)=(2,0,2), (N,2-N,0). The possible extra integral slot for $N=4$ is (3,-1,1), which has nontrivial scalar character and vanishes. The first slot has basis $B_{ij}=y_i{y}_j{v}_i$ wedge $v_j,i<j$. The second has basis w and $y_i^N,i=1,\mathrm{.}\mathrm{.}\mathrm{.},N$. Here is the character check, including the other group sectors. A monomial $y^a\otimes v_J$ is invariant exactly when the exponents $1_J(i)-a_i$ are all congruent modulo N, since the diagonal characters have the sole relation $\prod {\lambda }_i=1$. With $s=2,r=2$ this forces precisely $a_i=1_J(i)$, giving $B_{ij}$. With $s=N,r=0$ it forces either all $a_i=1$, giving w, or one $a_i=N$, giving $y_i^N$. A nonidentity diagonal element has codimension at least two because its determinant is one. The total-one slot has $r=1$ and cannot contain its forced moving-volume factor. In the total-two $s=2$ slot a codimension-two sector would force its moving exponents to be 1 and every fixed exponent congruent to 1; each fixed polynomial exponent would then be at least N-1, exceeding the polynomial degree two. Higher codimension exceeds r. The $r=0$ slot similarly permits only the identity sector. This excludes every nonidentity contribution used here.

The first differential is finite graph linear algebra The arity-N primary class gives the first higher differential, at order N-1. On these explicit classes its Gerstenhaber action is $d{E}_i=w,d{B}_{ij}=w(E_j-E_i)$. These identities can be checked without a full formality theorem. $E_i$ counts the i-th input degree, so its bracket with the monomial N-product is w. The $B_{ij}$ classes are the alternating cup products $E_i{E}_j$; the bracket is a graded derivation of that cup product, giving the second formula. Consequently the degree-one kernel has dimension N-1. The degree-two arity-two kernel is the cycle space of the COMPLETE GRAPH on N vertices: $ker(C^{\mathrm{binomial}(N,2)}\to C^N,e_{ij}\mapsto e_j-e_i)$, of dimension binomial(N,2)-(N-1). The arity-N part is $(Cw\text{ plus }{⨁}_{i}C{y}_i^N)\mathrm{/}Cw$, of dimension N. Degree positions allow no later differential into or out of these total-two classes for $N=4,5$: substituting (s,t) shifted by a differential of order $r>N-1$ in $r_{exterior}=s+(N\mathrm{/}n)t$ gives a negative or nonintegral exterior degree; incoming slots likewise do not exist. Thus the raw truncated total-two dimensions given by these formulas are 7 for $N=4$ and 11 for $N=5$. This is why the source quintic line “2n+3” cannot be used as a substitute for the displayed computation: its own complete-graph differential has kernel dimension six at $N=5$, not four. Nothing here refutes the full named HMS theorem; its comparison can use the more precise fixed-product part below. A cyclic coordinate permutation's action on this WHOLE group must also be computed through the actual object/character lift. One cannot discard the graph-cycle part merely by saying “the generators sum to zero”: the naive cycle-permutation representation need not have no invariants. We do not use that assertion.

The exact subspace relevant when the binary algebra is fixed For an actual minimal deformation with binary multiplication IDENTICALLY $Q_N$, its leading perturbation has no arity-two term. Its truncated Hochschild class therefore lies in the negative-arity filtration kernel, namely the N-dimensional subquotient direct-sum_i $C[y_i^N]$. The graph-cycle classes above have a nonzero arity-two edge and cannot occur in that leading perturbation. The product monomial w has already been removed by the Euler boundary. This identifies the relevant subspace without a claim about the cyclic action on the unrelated graph cycles. Assume the actual geometric cyclic lift permutes THESE polynomial deformation classes $y_i^N$. This is checked by its action on the actual coordinate-polynomial perturbations; the Fermat sum is then its invariant generator. Hence the invariant FIXED-PRODUCT deformation subspace is one-dimensional, spanned by ${\sum }_i{y}_i^N$. For the fixed-product variant, the q-adic formal-comparison induction stays within fixed products. If an error with zero binary term is c times that generator plus delta h, the arity-two part of delta h is [m2,h1] and must be zero. Thus h1 is a derivation of the binary algebra. Its exponential preserves m2 exactly; higher components of h cannot alter arity two because m0=m1=0. Averaging over the finite cyclic action preserves this property. Therefore each correction in the constructive formal root preserves the binary product, and the induction gives an actual all-higher comparison and parameter automorphism under this PRECISE input. This is not an added restriction on the named HMS target: both of its actual geometric families must prove their cohomology products are the claimed constant $Q_N$ algebra. If they do not, the present fixed-product argument cannot be applied and the missing deformation classification remains.

A stronger actual-symmetry route: even coordinate permutations For the named pencils an additional symmetry gives a valid UNRESTRICTED deformation classification, without imposing the fixed-product condition: use $A_N$, the even coordinate permutations, rather than a naive cyclic-subgroup argument. Both the product monomial and Fermat polynomial are invariant under $A_N$. This group normalizes the determinant-one diagonal group and the full torus. On the geometric collections its action may be lifted through a finite spin/character extension; central object-scalar actions are inner and hence act trivially on Hochschild cohomology. The degree-one classes $E_i-E_N$ are precisely the torus grading derivations. This identifies HH¹ with the standard representation $C^N\mathrm{/}C(1,\mathrm{.}\mathrm{.}\mathrm{.},1)$, naturally for every coordinate permutation normalizing that torus. Its alternating cup products are the graph-cycle classes: $(E_i-E_N)$ cup $(E_j-E_N)=B_{ij}-B_{iN}+B_{jN}$, which form a basis of the complete-graph cycle kernel. Thus this part of HH² is naturally ${\mathrm{\Lambda }}^2(C^N\mathrm{/}C)$, not an arbitrarily chosen list of cyclic triangle generators. It has NO $A_N$-invariants for $N\ge 4$. Indeed represent it in the skew-edge space and choose any two indices i,j and two further indices k,l. The even permutation (ij)(kl) reverses the coefficient of the edge {i,j}; an invariant skew coefficient must equal its negative and is zero. Every edge coefficient therefore vanishes. The negative-arity quotient is the permutation representation on the N pure powers $y_i^N$, whose invariant part is the one-dimensional Fermat sum. Since finite-group invariants are exact, the filtration extension between these two subquotients does not change that result: $H{H}^2(S,S{)}^{\le 0,A_N}=C[{\sum }_i{y}_i^N]$. This proof uses the actual torus-derivation and cup-product action, so it does not assume that a noncanonical smash-product lift acts by naive coordinate permutation on every Hochschild representative. The full q-adic formal comparison root now applies to ANY $A_N$-equivariant minimal deformation with nonzero class, including its binary directions.

Exact finite-group transfer once a strict chain model is constructed Here is the precise algebraic interface, without replacing its geometric input by a coherent action. Let $R=C\text{［［}q\text{］］}$ (or the stated characteristic-zero coefficient field). Let C be a bounded finite-free cochain complex over R, with finite-free cohomology H, and a strict R-linear action of a finite group G by cochain maps. Suppose all uncurved A-infinity operations on C, including the differential, are strictly G-equivariant, filtered and continuous. The group acts trivially on q. Then C has a G-equivariant strong deformation retract onto H, and its transferred minimal operations and comparison functors are strictly G-equivariant. The same assertion holds over the semisimple ring of object idempotents if G permutes these idempotents: all splittings below are taken idempotentwise before averaging their conjugates. Proof of the contraction, including its integral/formal scope. Over the PID R the boundaries $B^j$ and cycles $Z^j$ are finite-free. The sequence $0\to Z^j\to C^j\to B^{(j+1)}\to 0$ splits because its last term is free. The sequence $0\to B^j\to Z^j\to H^j\to 0$ splits because $H^j$ is free. Average each chosen right inverse over G. Averaging preserves its right-inverse equation, so it gives a G-stable direct decomposition $C^j=B^j\oplus i(H^j)\oplus S^j$. The differential maps $S^j$ isomorphically onto $B^{(j+1)}$. Let p be projection to H, and let h on $B^{(j+1)}$ be this inverse and zero on i(H) and S. These definitions give $dh+hd=1-ip,pi=1,h^2=hi=ph=0$, all exactly and G-equivariantly. Finite matrices are q-adically continuous, so this is a contraction over R, not a noncontinuous vector-space splitting after forgetting R. If the relative chain model is not bounded finite-free, its author must instead supply a continuous equivariant contraction or a finite-free chain-equivalent replacement; the preceding proof does not silently cover that case. For completeness the transfer is by actual rooted planar trees. Put the old operations at vertices, i at leaves, h on internal edges, and p at the root, with the suspended Koszul signs. For each fixed arity there are finitely many trees after the differential has been absorbed into the contraction. Their sum defines $m_H$. In the expansion of the bar equation, apply $dh+hd=1-ip$ to a distinguished internal edge. Its identity term cancels the contracted-edge term using the old bar equation; its −ip term is the corresponding two-tree composite. Root and leaf versions give the two comparison functor equations. The empty edge gives first component i or p, respectively. These cancellations are algebraic identities over R. Each summand commutes with G because every assigned vertex, edge and endpoint map does. Thus the transferred action on H is the actual linear representation, and $m_H$ is fixed by it in every arity. The q-adic filtration is preserved term by term. For a finite invariant brane collection, take its total endomorphism complex with its object idempotents. The precise sufficient geometric input is therefore a strict finite-group natural chain-valued correspondence model on a finite-free replacement, not an invariant transverse Morse function on every Lagrangian. Orbit-indexed auxiliary choices can supply such a model only after the actual perturbation construction proves both its boundary-product compatibility and its strict group law. Neither rational averaging of finished non-equivariant operations nor an unspecified coherent group action implies that input: averaging products separately would destroy their nonlinear bar relations. A central finite lift acting by object-scalar conjugations acts inner on the resulting endomorphism algebra; the standard insertion homotopy identifies its induced Hochschild action with the identity. Consequently the invariant computation of §5 applies to the descended even-coordinate action once that exact chain-level supplier is attached.

All-arity identification of the central model Let two normalized minimal uncurved structures have the same binary source algebra $Q_N$ and the same nonzero arity-N class w. The binary small Koszul calculation above has total-degree-two slots only in arities 2 and N; the possible arity-three slot at $N=4$ vanishes by its character. First kill every operation of arity 3 through N−1 in increasing arity using the corresponding zero binary Hochschild group. Normalize their arity-N representatives: their difference, after matching the scalar w, is a binary Hochschild boundary, so an arity-(N−1) normalized primitive removes it. Suppose the resulting structures first differ in arity $d>N$ by $e_d$. Subtract their Stasheff equations in arity d+1. Every term involving only smaller operations cancels; minimality removes unary terms, leaving $[\mu 2,e_d]=0$. The binary Hochschild group in arity d and internal degree 2−d is zero: in the Koszul model its exterior degree is 2(N−d)/(N−2)<0. Thus $e_d=[\mu 2,h_{(d-1)}]$ for an object-ring linear normalized primitive. No averaging is required for this central comparison: equality of the primary class means equality of the entire arity-$N$ binary Hochschild class, including its pure-power components. Choose the primitive without any symmetry condition. Exponentiating that coderivation changes no smaller arity and removes $e_d$; brackets with μN start in arity $d+N-2>d$. Repeat. The minimum arity of every successive gauge tends to infinity, so each fixed bar word is changed finitely many times. The limit and its inverse are actual strictly unital all-arity isomorphisms. This proves that equality of the binary and primary classes here identifies the TWO full central A∞ models; it is not an assertion that equality of those data always suffices. The vanishing slots proved above are essential.

The twisted Koszul computation, even-coordinate invariant calculation, finite-group contraction and central arity induction prove every assertion.