Categorical mirror equivalence for elliptic curves

Under Full AC [A1], use the exact datum [F1]. The explicit Fourier-kernel/classification and theta constructions [F2] identify all coherent generators, rational graded lines and source local systems, including every nilpotent Jordan block and torsion jet. The parameter remains $\rho =\tau$; all area sums used by this functor are literally convergent at $q=e^{2\pi i\tau }$.

The all-arity transversal functor of [F2] is extended by the actual fixed-copy and generic-resolution construction [F3]. Its full Hom comparison follows from forward-copy localization and finite triangular filtrations; its cone/evaluation units are proved in the geometric model, including the vertical case by the object-only Fourier rotation. The explicit bar module construction extends it to twisted objects and homotopy-idempotent splittings and gives the specified dg perfect-module enhancement. Thus this is an exact fully faithful functor on the full split-complete categories, not an equivalence transported from a binary category.

The coherent derived/perfect bridge. Here a perfect complex on the curve means a complex locally quasi-isomorphic to a bounded complex of finite-rank locally free sheaves. Let $F$ be any coherent sheaf. Choose an ample line bundle $L$ from a projective embedding of $E_q$. By [F5], $F\otimes L^m$ is globally generated for large $m$. Quasi-compactness selects finitely many generating global sections, giving a surjection $V=(L^{-m}{)}^{\oplus r}\to F$. Its coherent kernel $K$ is locally free. Indeed each local ring of the smooth curve at a closed point is a one-dimensional regular local domain with maximal ideal generated by a local parameter $t$. A nonzero regular germ has finite order in the analytic coordinate at that point, so repeated division by $t$ in the maximal ideal stops at a unit; hence this ring is a discrete valuation ring. Every finitely generated submodule of a finite free module over it is free, as follows by selecting a component of least valuation, using it to eliminate that component, and inducting on the free rank. Applied to $K_x\subset V_x$ this proves freeness at each closed point, and a finite presentation extends a stalk basis to a neighborhood. At the generic point the local ring is a field. Thus $[K\to V]$, in degrees $-1,0$, is a finite vector-bundle resolution of $F$, including torsion sheaves. Every bounded coherent complex is built by its finite canonical truncation triangles from its cohomology sheaves $H^i[-i]$; those triangles follow from the kernels, images and quotients of the complex differential. Therefore every object of $D^b\mathrm{Coh}(E_q)$ is perfect. Conversely a perfect complex on this quasi-compact Noetherian curve has bounded coherent cohomology. The Hom comparison uses these vector-bundle resolutions and the same affine-Cech/Dolbeault computations of [F2], so it is the actual derived Hom, not merely an agreement of object classes. In particular the finite twisted Yoneda construction of [F3], applied to the full coherent sheaf models, realizes the bounded coherent category: the truncation triangles give essential surjectivity, and the matched derived Hom complexes give full faithfulness. Its split completion adds no unbounded objects, since a retract has direct-summand bounded coherent cohomology; the same local finite resolutions and finite truncations place it in the bounded coherent category. This proves the required identification with the coherent perfect-module target of [F3]. The classification in [F2] covers every coherent sheaf, and the source line/local-system and torsion objects cover its represented objects before formal twisted/split closure. Hence the functor's image thickly generates. No hereditary splitting assertion is needed. Use the small dg presentations specified in [F1] and the actual twisted/split extension of [F3]: their object lists, finite morphism matrices and idempotents form sets in the chosen ambient set universe. Transport along the supplied enhancement equivalences, rather than choosing representatives from a proper class. Apply [F4] to these set presentations with [A1], or use the explicitly constructed bar tensor/Hom inverse in [F3], to obtain the equivalence in the unchanged Statement. No general HMS, arbitrary dg-enhancement uniqueness, formal-series numerical specialization or unrelated unobstructed Fukaya construction is assumed.