Cartier duality via Mittag-Leffler modules

We construct the Cartier duality equivalence for affine commutative group schemes $G$ whose coordinate ring is a flat Mittag-Leffler module over an arbitrary base ring $R$. The dual $G^\vee$ of $G$ turns out to be an ind-finite ind-scheme over $R$. When $R$ is Noetherian and admits a dualizing complex, we construct a Fourier-Mukai transform between quasicoherent derived categories of $G$ and of $BG^\vee$ and also between those of $G^\vee$ and $BG$.

💡 Research Summary

The paper “Cartier duality via Mittag‑Leffler modules” develops a comprehensive extension of Cartier duality from the classical setting of finite group schemes over a field to a much broader context: affine commutative group schemes whose coordinate rings are flat Mittag‑Leffler modules over an arbitrary base ring R. The authors begin by recalling the classical Pontryagin and Cartier dualities, emphasizing that while Pontryagin duality is an antiequivalence on locally compact abelian groups, the algebraic analogue (Cartier duality) traditionally connects finite group schemes with formal groups (ind‑finite ind‑schemes) only over a field. Their goal is to lift this restriction to arbitrary bases.

The technical heart of the work lies in the linear duality theory for modules. Section 2 introduces flat Mittag‑Leffler modules (following Raynaud–Gruson and Drinfeld) and the notion of pro‑R‑modules, including pro‑finite, pro‑projective, and Mittag‑Leffler pro‑modules. The key result (Theorem 2.3.1) establishes a perfect antiequivalence between the category of flat Mittag‑Leffler modules and the category of pro‑projective Mittag‑Leffler pro‑modules via the functors

• (t(M)=M\otimes_R-) (viewed as an endofunctor of (\mathrm{Mod}_R)),
• (h(N)=\mathrm{Hom}_R(N,-)) (the left‑exact functor associated to a pro‑module (N)). The theorem shows that a flat module (M) admits a representing pro‑module (N) with (h(N)\simeq t(M)) precisely when (M) is flat, and conversely a pro‑module (N) admits such an (M) precisely when (N) is pro‑projective and Mittag‑Leffler. Moreover, the duality preserves the Mittag‑Leffler condition on both sides.

Using this linear duality, the authors define the Cartier dual of an affine commutative group scheme (G=\operatorname{Spec}A) (with (A) a flat Mittag‑Leffler algebra) as (G^\vee=\operatorname{Spec}A^\vee), where (A^\vee) is the linear dual Hopf algebra. Since (A^\vee) is a pro‑projective Mittag‑Leffler pro‑module, its spectrum is an ind‑finite ind‑scheme, which the authors call a “co‑flat ind‑scheme”. Theorem A (Corollary 3.1.6) asserts that the assignment (G\mapsto G^\vee) yields an antiequivalence between the category of flat Mittag‑Leffler affine commutative group schemes and the category of co‑flat commutative group ind‑schemes.

The paper then moves to the “1‑duality” level. Two notions are distinguished:

1. Geometric 1‑duality, defined via a pairing (P:X\times Y\to B\mathbf{G}_m) that is perfect (induces equivalences (Y\simeq X^D) and (X\simeq Y^D)).
2. Categorical 1‑duality, defined via a Fourier–Mukai transform with kernel (P) that yields an equivalence of derived categories of quasi‑coherent sheaves.

For a flat Mittag‑Leffler group (G), the natural pairing (G\times B G^\vee\to B\mathbf{G}_m) is generally not perfect (e.g. (G=\mathbf{G}_a) in characteristic 0). The authors remedy this by redefining the classifying stack (B G^\vee) using the h‑topology, showing that under suitable hypotheses the geometric 1‑dual of (G) is precisely this h‑classifying stack.

The categorical side is more robust. Theorem B (Theorem 5.2.12) states that for a Noetherian base (R) admitting a dualizing complex, and for flat Mittag‑Leffler affine commutative groups (G) and (H) equipped with a bilinear pairing (G^\vee\times H^\vee\to \mathbf{G}_m), there is an equivalence of (\infty)‑categories \