Tensor operations on finite flat commutative group schemes over general bases: preservation of flatness and finiteness, and minimal hypotheses for the multilinear theory

For commutative group schemes there is a natural multilinear theory: an inner Hom $\underline{\mathrm{Hom}}(G,H)$ representing homomorphisms after base change, multilinear morphism group schemes, and tensor, symmetric, and alternating constructions characterized by universal properties. The inner Hom of a finite flat source exists over an arbitrary base, and over a field tensor-type constructions exist as profinite group schemes; but the theory has bad behavior at its edges: $\underline{\mathrm{Hom}}(\alpha_p,\alpha_p)\cong\mathbb{G}_a$ is not finite, over $R=\mathbb{F}_p[x]$ the inner Hom of a finite flat Oort–Tate group scheme with itself is not $R$-flat, and tensor products fail to commute with base change. Hedayatzadeh's paper establishing these results states that they 'may hold in a more general context' while explicitly not stating them with minimal hypotheses. The open problem is to build the multilinear theory of finite flat commutative group schemes over general bases: determine the weakest hypotheses on the base and on the group schemes under which representability, flatness and finiteness preservation, base-change commutation, and the exact-sequence decompositions of alternating powers hold, and map the boundary between good and bad behavior with theorems and sharpness counterexamples.

Background

A finite flat commutative group scheme $G$ over a base scheme $S$ is a group object in schemes over $S$ whose structure morphism is finite and flat; equivalently, its coordinate ring is a finite locally free Hopf algebra over the structure sheaf of $S$. Basic examples in characteristic $p$ are the constant group $\mathbb{Z}/p\mathbb{Z}$, the multiplicative $p$-torsion $\mu_p$, and $\alpha_p$, the kernel of the Frobenius endomorphism of the additive group $\mathbb{G}_a$. Oort and Tate classified the group schemes of prime order $p$ over bases in which $p$ is locally nilpotent: the classification is governed by the ring $\Lambda=\mathbb{Z}[\zeta_p]/(1-\zeta_p)^2$, where $\zeta_p$ is a primitive $p$-th root of unity (every $\mathbb{F}_p$-algebra is canonically a $\Lambda$-algebra), and over an $\mathbb{F}_p$-algebra $R$ any pair of elements $a,b$ with $ab=0$ defines a rank-$p$ group scheme $G_{a,b}=\mathrm{Spec}\,R[y]/(y^p-ay)$, with the parameter $b$ entering the comultiplication; these are the Oort–Tate group schemes, with étale and multiplicative degenerations. The Cartier dual $G^{*}=\underline{\mathrm{Hom}}(G,\mathbb{G}_m)$, which represents characters, is the prototype of a general internal Hom.

To develop a group-scheme version of multilinear algebra, one defines the inner Hom $\underline{\mathrm{Hom}}(G,H)$ as the group scheme representing the functor $T\mapsto\mathrm{Hom}_{T}(G_{T},H_{T})$ on $S$-schemes, a multilinear morphism as a morphism of schemes $G_{1}\times\cdots\times G_{r}\to H$ that is a homomorphism in each factor, and the tensor product $G_{1}\otimes\cdots\otimes G_{r}$ as the commutative group scheme equipped with a universal multilinear morphism, i.e. characterized by $\mathrm{Hom}(G_{1}\otimes\cdots\otimes G_{r},H)\cong\mathrm{Mult}(G_{1}\times\cdots\times G_{r},H)$ for all $H$; symmetric and alternating powers $\mathrm{S}^{n}G$ and $\Lambda^{n}G$ are defined analogously via symmetric and alternating multilinear morphisms (those invariant under permutation of factors, resp. vanishing when two factors coincide). This program is due to R. Pink (an unpublished manuscript 'Multilinear theory of commutative group schemes') and was developed by Hedayatzadeh in a 2019 preprint, 'Tensor Operations on Group Schemes'.

What is known over restricted settings: the inner Hom exists whenever $G$ is finite flat over $S$ (and is affine, resp. of finite type, when $H$ is), it commutes with arbitrary base change, and the inner Hom functors are left exact over a field; when $S$ is the spectrum of a field, the tensor product and the symmetric and alternating powers of finite group schemes exist as profinite group schemes, obtained as inverse limits of Cartier duals of finite subgroup schemes of $\mathrm{Mult}(\ldots,\mathbb{G}_m)$. Representability of Hom functors is in fact known very broadly from work on Hom-stacks (Aoki; Olsson; Hall–Rydh: for $Z\to S$ proper flat of finite presentation and $X\to S$ locally of finite presentation with affine stabilizers, the Hom-stack is algebraic), but these results say nothing about the flatness or finiteness of the representing object, nor about tensor-type constructions over general bases.

The bad behavior at the boundary is documented by explicit examples in the source: over any field of characteristic $p$, $\underline{\mathrm{Hom}}(\alpha_p,\alpha_p)\cong\mathbb{G}_a$ is not finite, so finiteness is not preserved; over the base $R=\mathbb{F}_p[x]$, the Oort–Tate group scheme $G_{0,x}$ (the pair $(a,b)=(0,x)$: underlying algebra $R[y]/(y^p)$, comultiplication twisted by $x$, so this is not $\alpha_p$) is finite and flat, but $\underline{\mathrm{Hom}}(G_{0,x},G_{0,x})\cong\mathrm{Spec}\,R[y]/(xy^p-xy)$ contains the $x$-torsion element $y^p-y$ and is not flat over $R$, so flatness is not preserved either; and the tensor product does not commute with base change in general, already for $\alpha_p\otimes\alpha_p$ over transcendental field extensions, because over such extensions $\mathbb{G}_a$ acquires finite subgroup schemes that do not descend. For modules of finite length over a local ring, alternating powers satisfy $\Lambda^{n+m}(M\oplus N)\cong\Lambda^nM\otimes\Lambda^mN$ and $\Lambda^mM=0$ for $m$ larger than the length; for group schemes the exact-sequence theorem for alternating morphisms is known for short exact sequences of commutative group schemes over a general base, conditional on the vanishing of sufficiently high alternating powers of the subgroup and quotient schemes, while the vanishing inputs themselves are proved only over fields of odd characteristic, where for a local-local group scheme (one with neither étale nor multiplicative part) of order $p^n$ one has $\Lambda^mG=0$ for $m>n$, so the exponent $n$ plays the role of the length.

The good-behavior question is not academic: flatness of Hom-type schemes over integral bases is exactly the crux in the theory of level structures on finite flat group schemes — extra structure, such as a specified homomorphism from a fixed abstract finite group into the group scheme, imposed to rigidify moduli problems (Katz–Mazur full sets of sections, Wake's full homomorphisms, Kottwitz–Wake primitive elements, Guan's full level structures on products of Oort–Tate group schemes) — where naive constructions are known to be non-flat. In neighboring categories the theory does behave well under restrictions: exterior powers of Barsotti–Tate groups ($p$-divisible groups, i.e. inductive systems of finite flat group schemes of $p$-power order, such as the $p$-power torsion of an abelian variety) of dimension at most $1$ exist over arbitrary locally noetherian bases and commute with arbitrary base change, exterior powers of Lubin–Tate groups (the one-dimensional formal groups over a $p$-adic field attached to a choice of uniformizer, used to construct local class field theory) are computable, and the equal-characteristic analogue for $z$-divisible local Anderson modules (the function-field counterpart of $p$-divisible groups) carries tensor constructions. The mixed-characteristic, finite group scheme, general-base case — the setting of the source paper — remains unmapped.

Problem Statement

Let $G_{1},\ldots,G_{r}$ and $H$ be finite flat commutative group schemes over a base scheme $S$. Building on the multilinear theory of commutative group schemes as developed over restricted settings (Hedayatzadeh, 'Tensor Operations on Group Schemes', arXiv:1908.05452, following unpublished work of Pink), determine the weakest hypotheses on the base $S$ and on the group schemes under which the following constructions and properties hold, and map the boundary between the known good behavior and the known counterexamples, for the inner Hom $\underline{\mathrm{Hom}}(G,H)$, the multilinear morphism group schemes $\underline{\mathrm{Mult}}(G_{1}\times\cdots\times G_{r},H)$, and the tensor, symmetric, and alternating constructions $G_{1}\otimes\cdots\otimes G_{r}$, $\mathrm{S}^{n}G$, $\Lambda^{n}G$:

(1) Existence and representability: the constructions exist as commutative group schemes over $S$ under hypotheses as weak as possible — currently, inner Hom is guaranteed only for finite flat source, and tensor-type constructions only over a base field, where they exist merely as profinite group schemes.

(2) Preservation of flatness and finiteness: characterize the conditions on $S$, $G$, and $H$ under which these constructions are again finite and flat over $S$, as well as those under which finite-typeness holds without finiteness, given that finiteness and flatness both fail in general: $\underline{\mathrm{Hom}}(\alpha_p,\alpha_p)\cong\mathbb{G}_a$ is not finite over any field of characteristic $p$, and over $R=\mathbb{F}_p[x]$ the finite flat Oort–Tate group scheme $G_{0,x}$ has $\underline{\mathrm{Hom}}(G_{0,x},G_{0,x})\cong\mathrm{Spec}\,R[y]/(xy^p-xy)$ non-flat over $R$ (the element $y^p-y$ is annihilated by the nonzerodivisor $x$).

(3) Base-change commutation: determine under which hypotheses the tensor construction commutes with base change, given that it fails in general (already $\alpha_p\otimes\alpha_p$ over transcendental field extensions, because finite subgroup schemes of $\mathbb{G}_a$ over such extensions need not descend), while inner Hom does commute with base change.

(4) Exact-sequence decompositions: extend the alternating-power decompositions — the isomorphism $\Lambda^{n+m}(G\oplus H)\cong\Lambda^{n}G\otimes\Lambda^{m}H$ and the vanishing $\Lambda^{m}G=0$ for local-local $G$ of order $p^{n}$, both proved only over fields of odd characteristic, and the exact-sequence theorem for alternating morphisms, known over general bases only under the vanishing of sufficiently high alternating powers of the subgroup and quotient schemes — to minimal hypotheses, clarifying to what extent the exponent of a local-local group scheme plays the role that length plays for finite-length modules.

The four components (1)–(4) are linked: for each of them the target is the weakest hypotheses — necessary and sufficient conditions — together with explicit counterexamples showing that the stated hypotheses cannot be weakened; sufficient conditions without sharpness counterexamples give only one side of the boundary. The source paper explicitly states that its results 'may hold in a more general context' while declining to state them with minimal hypotheses; the minimal-hypothesis boundary is the open target.

References

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