Finite étale stability lifting and Hilbert norm-line intertwining

1. A complete finite abelian cover lifting theorem. Let $\pi :T\to Y$ be a finite étale torsor under a finite abelian group H, between smooth projective complex varieties. Its cover algebra is A=pi_*O_T=direct_sum_(χ∈H^vee) L_χ, with invertible character eigensheaves and their actual associative multiplication maps. The primitive character projectors come from averaging. The multiplication $L_{\chi }$ tensor $L_{\psi }\to L_{\chi \psi }$ is an isomorphism: étale locally the torsor is H times the base and this is the Fourier decomposition of its regular function algebra. In particular every $L_{\chi }$ has finite order. No arbitrary choice of multiplication identifications is made. Suppose ${\sigma }_Y=(P,Z)$ is a locally finite Bridgeland stability condition with support property for $v:K(Y)\to \mathrm{\Lambda }$, and EVERY tensor functor by $L_{\chi }$ preserves its slicing and charge. Then there is a stability condition on T with P_T(phi)={E : pi_*E belongs to P(phi)}, Z_T(E)=Z(pi_*E). Here are the actual heart, HN, support and local-finiteness arguments. Finite étaleness gives an exact equivalence Coh(T)≃Coh(A-modules). On derived categories restriction of scalars is conservative and faithful: ${\pi }^{\ast }{\pi }_{\ast }E\simeq {⨁}_{h\in H}{h}^{\ast }E$, naturally, so ${\pi }_{\ast }f=0$ implies $f=0$. This formula is proved étale locally by the torsor identity $T{\times }_{Y}T=\coprod H$ copies of T and descends. It also proves boundedness detection. Let $B=P((0,1])$ be the sigma heart. Each $L_{\chi }$ tensor(-) is exact on B. Define $B_A$ to be A-module objects in B, with the cover algebra multiplication just specified. It is abelian: kernels and cokernels in B inherit the action because A tensor(-) is a finite sum of these exact tensor functors. It is the heart of a bounded t-structure on Db(T). One can construct truncations directly: apply the B truncation to the underlying ${\pi }_{\ast }$ complex. The action maps $L_{\chi }$ tensor ${\pi }_{\ast }E\to {\pi }_{\ast }E$ induce actions on its truncations, since tensor $L_{\chi }$ is t-exact; the unit/associativity equalities persist by naturality of truncation. Thus they define actual cover objects, give the truncation triangle, and remain bounded by conservativity. For an explicit construction avoiding a module-enhancement assumption, the étale algebra has its diagonal separability idempotent $e=\sum a_i$ tensor $b_i$, with sum $a_i{b}_i=1$ and $ae=ea$. It makes every derived module E a RETRACT of the free module ${\pi }^{\ast }{\pi }_{\ast }E$. Morphisms between free modules ${\pi }^{\ast }F,{\pi }^{\ast }G$ are ${\mathrm{Hom}}_Y(F,A\otimes G)=⨁{\mathrm{Hom}}_Y(F,L_{\chi }\otimes G)$. Truncating each matrix entry is functorial because all $L_{\chi }$ tensor functors are t-exact, and the given algebra multiplication makes these truncations preserve composition. Truncate a retract by truncating its idempotent; idempotents split in Db(T). This constructs the asserted bounded t-structure and its heart $B_A$ directly on the actual derived category. Conversely an A-module object in B is a retract of its free module by e, so gives precisely such a heart object. The familiar bar contraction $h(c)(a1,\mathrm{.}\mathrm{.}\mathrm{.})=\sum a_{i}c(b_i,a1,\mathrm{.}\mathrm{.}\mathrm{.})$ has $dh+hd=1$ in positive degrees and verifies that no higher action obstruction is being suppressed. For E in $B_A$ take the UNIQUE HN filtration of its underlying object U in B. Every multiplication map $L_{\chi }$ tensor $U\to U$ preserves this filtration: tensor $L_{\chi }$ preserves all phases and the HN filtration, and a morphism preserves the HN cutoff subobject by the usual no-Hom inequality between its two phase ranges. The induced HN subobjects therefore carry the actual A-actions and the same associative relations. Their quotients are objects of $B_A$ with semistable underlying objects. This supplies a finite HN filtration of E. Conversely a nonzero object of $B_A$ with semistable underlying U has nonzero charge on the indicated ray, so these definitions give the claimed slicing; shifts and vanishing follow from the faithful pushforward and the original slicing. Local finiteness is inherited without a hidden boundedness estimate. Choose an interval of phases on which $P_Y$ has finite length. The corresponding quasi-abelian category on T has every strict subobject chain mapping to a strict chain in the base category; conservativity makes a nontrivial quotient stay nontrivial. Thus it is of finite length. The central charge factors through the coimage lattice Λ_T=im(K(T) --pi_*--> K(Y) --v--> Λ)/torsion. This has finite rank. It embeds in Λ modulo its torsion, so use the restricted norm. A ${\sigma }_T$-semistable E has ${\sigma }_Y$-semistable ${\pi }_{\ast }E$ and therefore ||v_T(E)|| <= C |Z(pi_*E)| = C |Z_T(E)|. All support and local-finiteness conclusions are actual, not just a pre-stability claim. No condition about geometric stability is needed for this lifting theorem.

2. Free quotient descent and product Fourier–Mukai compatibility. For a free finite G-action q:A0->S, equivariant coherent sheaves descend by the actual finite faithfully flat equalizer, and the averaging inverse to q^* is q_*(-)^G. These are exact on complexes; their descent unit/counit are checked after faithfully flat q, where the torsor is the disjoint union of group copies. Thus Db(S) ≃ Db_G(A0). Tensor by a line bundle M on S corresponds EXACTLY to tensor by q^*M with its natural descent G-linearization. Exterior products preserve these identities by tensoring their actual kernels and unit/counit; no arbitrary product stability assertion is involved. For Enriques S use the source Kummer surface resolution followed by its free residual involution. Its composed surface Fourier–Mukai equivalence is equipped with the actual finite-group linearization. For every smooth connected projective surface S, [F9] proves the Cohen–Macaulay premise of [F10]. Applying [F10] therefore gives the actual universal-isospectral-kernel equivalence $D^b(S^{[n]})\simeq D_{S_n}^b(S^n)$ for every n, with its natural automorphism intertwiners and their cocycles. Its exterior-product construction carries specified equivariant surface-kernel linearizations to the actual permutation and wreath linearizations; no linearization is inferred for an arbitrary unlinearized equivalence. Compose this Hilbert equivalence with the exterior product of the free-quotient descent equivalence just constructed. For the bielliptic quotient this gives the actual equivalence to the $G^n⋊S_n$-equivariant elliptic-product category used below. The norm-line tensor compatibility of [F10] is also reconstructed directly in the subsequent norm-line construction, retaining all specified torsion trivializations.

3. Norm line bundles and the actual Hilbert kernel identity. Let S be a smooth projective surface and $Y=S^{[n]}$. For a line bundle M on S define its norm line on the Hilbert scheme by N_n(M)=det(p_*(O_U tensor q^*M)) tensor det(p_*O_U)^-1, where $U\subset Y\times S$ is the universal length-n family, finite flat over Y. The two determinants are therefore of vector bundles of rank n. The norm is multiplicative, including on families: locally trivialize M and use that its transition unit h acts on the finite free cluster algebra by multiplication, so its transition determinant is ${\mathrm{Norm}}_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}(h)$. Determinants of compositions give the cocycle. Let $M^n$ be the line bundle on ${\mathrm{Sym}}^n(S)$ descended from $M^{⊠n}$. It exists without freeness of the Sn-action: a permutation stabilizing a tuple permutes identical one-dimensional tensor factors and acts trivially, and transition functions are the symmetric products of the transitions of M. Equivalently construct these product transition functions on neighborhoods of each unordered support. If $\rho :Y\to {\mathrm{Sym}}^n(S)$ is Hilbert–Chow, then N_n(M)=rho^*M^(n). Indeed the determinant norm of a unit on a finite length algebra is the product of its values at the associated ordered cycle with multiplicity. This equality follows for families from the characteristic polynomial of the multiplication matrix, not merely from equality on reduced clusters. Thus it is the actual line bundle equality with its multiplicative identifications. Let Z be the isospectral Hilbert correspondence used in the separately constructed Hilbert–Sn equivalence, with $p:Z\to Y$ and $q:Z\to S^n$. Its Hilbert–Chow square immediately gives p^*N_n(M) = q^*(M^boxtimes n) as Sn-linearized line bundles. Pulling through its Fourier–Mukai kernel proves Phi(F tensor N_n(M)) = Phi(F) tensor M^boxtimes n naturally, with all iterated tensor cocycles. This uses no assumption that the correspondence is normal: both line bundles are pulled back from the same line bundle on the symmetric product.

4. Why every finite cover character on a Hilbert scheme is such a norm. For $n\ge 2,\pi 1(S^{[n]})$ is canonically H1(S,Z). Here is the required topology, rather than a citation. Remove the diagonals from $S^n$. Their real codimension is four, so transversality of paths and path homotopies identifies the pure configuration fundamental group with $\pi 1(S{)}^n$. Passing to unordered configurations adds the symmetric-group permutation generators, whose elementary exchanges have square null-homotopic in a small real four-ball. The needed double-collision model can be constructed directly. In a holomorphic coordinate ball of $S$ with coordinates $(x,y)$, let $(c_x,c_y)$ be half the traces of multiplication by $x,y$ in the length-two algebra, and put $u=x-c_x$, $v=y-c_y$. The trace-zero part of that algebra has dimension one. On the open set where $1,u$ is a basis its ideals therefore have the unique form $(v-tu,u^2-a)$ with coordinates $(c_x,c_y,t,a)$: trace zero removes the linear coefficient of $u^2$. On the other open set they are $(u-sv,v^2-b)$; the overlap is $s=t^{-1}$, $b=t^{2}a$. These two charts cover because $u,v$ cannot both vanish in a length-two algebra. With barycenter fixed they are precisely the total space of ${\mathcal{O}}_{{\mathbb{P}}^1}(-2)$. The Hilbert–Chow coordinates are $(u^2,uv,v^2)=(a,ta,t^{2}a)$ on the first chart and satisfy $AC=B^2$; the zero section is contracted to the double point. A small disk-bundle neighborhood retracts to its exceptional ${\mathbb{P}}^1$ and is simply connected. Its punctured part has fundamental group $\mathbb{Z}\mathrm{/}2$: it is the quotient of a punctured complex two-ball by $(u,v)\mapsto (-u,-v)$, whose cover retracts to the simply connected three-sphere. Its nontrivial loop interchanges the two ordered points. Disjoint small support balls split a nearby length-$n$ family into this length-two family and $n-2$ singletons, so the remaining factors are disks. Van Kampen thus kills exactly the simple transposition when this boundary neighborhood is attached. The deeper collision boundary has complex codimension at least two in the smooth Hilbert scheme and does not change pi1. Van Kampen therefore gives the quotient of $\pi 1(S{)}^n$ semidirect Sn by the normal closure of Sn. It identifies all factors and forces them to commute, giving exactly the abelianization H1(S,Z). The map is the product of the individual point loops. A finite-order line bundle is a finite unitary flat character: use a chosen trivialization of its r-th power and local r-th roots of its transition functions. Conversely these characters give its finite étale Kummer local system and line bundle. Construct M ALGEBRAICALLY as follows. Fix n−1 distinct points and pull the given finite-order line L back under the embedding of S minus those points into Y obtained by adjoining them. Since S is smooth, its Cartier-divisor class group is unchanged on removing finitely many points, so this line extends uniquely to an algebraic line M on S; the r-th-power trivialization extends by normal Hartogs, so M has finite order. The pi1 calculation says that $N_n(M)$ and L have the same flat character: on configurations it is the product of the point characters, exactly the abelianization map. Their difference is therefore analytically trivial. This implies algebraic triviality here without assuming a GAGA statement: its finite étale Kummer torsor is analytically a disjoint union of copies of Y; each algebraic connected smooth projective component is analytically connected (connect points by algebraic curve sections and normalize those curves). Thus it has the same number of algebraic components, each degree one over normal Y, hence each is isomorphic to Y. Its algebraic section trivializes the difference line. No Hartogs extension across the Hilbert boundary divisor, nor an unconstructed Riemann-existence theorem, is used. For disconnected S, its finitely many connected components $S_a$ are open and closed. A length-n family splits uniquely by these components, and the locally constant lengths $n_a$ give the displayed disjoint-union/product decomposition of the Hilbert scheme. Each nonempty factor $S_a^{[n_a]}$ is connected; for $n_a\ge 2$ the preceding argument applies and for $n_a=1$ it is $S_a$ itself with $N_1$ the identity. Restrict a finite-order line on a product component to the individual factors by fixing points in the others. Its finite flat character is the product of these restrictions, since the fundamental group of a product is the product of the factor groups. The difference line has trivial flat character, and the same finite Kummer-torsor component argument makes it algebraically trivial. Apply the connected norm assertion to each factor. The construction retains the finite order of the restricted lines; any simultaneous order bound for the finitely many factors may be their least common multiple. Multiplicativity of the finite-algebra determinant then proves that $N_n(M)$ restricts to the external product of $N_{n_a}(M\mathrm{\mid }{S}_a)$, giving precisely the stated compatibility criterion. This criterion is necessary, not automatic: take $S=S_1$ disjoint-union $S_2$ with each $S_a=E$ times P1 and $n=2$. Choose a nontrivial finite-order line T pulled back from E. Put the trivial line on both pure Hilbert components $S_a^{[}2]$, and $p{r}_1^{\ast }T$ on the mixed component $S_1$ times $S_2$. If this were $N_2(M)$, restricting the two trivial pure norms to the moving-one-point copies of $S_a$ minus a fixed point would force $M\mathrm{\mid }{S}_a$ trivial, since finite-point removal preserves the divisor class group of a smooth surface. Its mixed restriction would then be trivial, a contradiction. Thus no unqualified single-global-norm assertion is made for disconnected surfaces. The all-n Hilbert geometry supplier is used for smoothness, the local disjoint-support decomposition and the deeper-collision codimension bound. The transverse simple-collision resolution and the meridian killed by it were constructed in the two charts above, rather than attributed to that supplier.

5. Bielliptic finite cover lifting without uniqueness. Take the source free quotient $S=(E1\times E2)\mathrm{/}G$. If M is finite-order flat on $S,q^{\ast }M$ is finite-order flat on E1×E2. Since $\pi 1(E1\times E2)=\pi 1(E1)\oplus \pi 1(E2)$, its character splits and q^*M=pr1^*M1 tensor pr2^*M2 for finite-order flat lines Mi on Ei. The two factor lines can be obtained by restricting the algebraic line $q^{\ast }M$ to the two coordinate elliptic curves, so they are algebraic and finite order without an existence assertion for arbitrary characters. The difference between their external product and $q^{\ast }M$ has trivial flat character by the direct-sum pi1 calculation; the finite étale Kummer-torsor component argument of the Why every finite cover character on a Hilbert scheme is such a norm construction makes that difference algebraically trivial. No assertion that an arbitrary product line bundle splits is made. Assume the separately authored product stability condition is invariant under tensor by these factorwise finite-order flat lines, and under the actual wreath group $G^n$ semidirect Sn. Induce it equivariantly using the installed finite-group induction construction. The induced slicing and charge are defined by forgetting the linearization. Consequently tensoring by the actual naturally linearized $(q^{\ast }M{)}^{⊠n}$ preserves it; its underlying tensor functor preserves the product slicing and charge, and any character discrepancy in a chosen linearization is invisible under forgetting. The free-quotient product equivalence and the Norm line bundles and the actual Hilbert kernel identity construction exact Hilbert kernel identity therefore show that the resulting condition on $S^{[n]}$ is invariant under EVERY finite-order flat norm line. Any source finite abelian étale cover of $S^{[n]}$ has precisely such character lines. the A complete finite abelian cover lifting theorem construction now constructs an actual stability condition upstairs. This route never invokes Cheng4.5 or an unproved assertion that identical central charges determine the slicing. It also does not treat a merely cohomological equality $ch(L)=1$ as equality of tensor functors.

6. Enriques canonical cover lifting. For the source Enriques surface, the universal degree2 cover of $Y=S^{[n]}$ is its canonical cover. The underlying geometric identification can be constructed separately by the fundamental-group and canonical-line calculation; no stability argument is used to assert it here. Under the composed product/finite-group/Hilbert Fourier–Mukai equivalence, tensor by ${\omega }_Y$ is the canonical-character tensor functor on the equivariant product. This follows from the actual Serre functors: on smooth projective Y it is $(-\otimes {\omega }_Y)[2n]$; on the equivariant product it is $(-\otimes {\omega }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}})[2n]$, with its natural canonical group linearization. A Fourier–Mukai equivalence with the projective trace intertwines these functors, including the natural transformations. The underlying canonical line of the elliptic product is trivial, and the permutation of two-dimensional surface blocks has determinant+1, so its remaining action is exactly the product of the surface canonical characters. The installed equivariant-induction stability condition is invariant under EVERY character twist, since its forgetful object and charge are unchanged. It is therefore invariant under ${\omega }_Y$ and the A complete finite abelian cover lifting theorem construction lifts it to the canonical double cover. This is stronger than the source's central-charge calculation and supplies its missing slicing invariance directly.