Artin–Mazur Formal Group Functors

• Artin–Mazur formal group functors are contravariant functors that capture deformation data from higher Gₘ cohomology, providing a framework to reconstruct classic formal Picard and Brauer groups.
• They leverage explicit coordinate realizations, Newton polytope computations, and Witt cohomology to establish formal smoothness and p‑adic integrality in deformation theory.
• Modern approaches situate these functors within fppf, crystalline, and unipotent homotopical frameworks, deepening connections to p‑adic Hodge theory and arithmetic geometry.

Artin–Mazur formal group functors are contravariant deformation-theoretic functors extracted from higher $\mathbf G_m$-cohomology of a smooth proper scheme. In their classical form, for a smooth proper variety $X$ over a field or over a suitable base ring, they assign to a local Artinian algebra the kernel of the restriction map in $H^i(-,\mathbf G_m)$; under mild hypotheses they are pro-representable by commutative formal Lie groups, and their tangent spaces are canonically identified with $H^i(X,\mathcal O_X)$ (Vlasenko, 2016). More recent work places these functors simultaneously in fppf, Witt-vector, crystalline, and unipotent-homotopical frameworks, yielding criteria for formal smoothness, explicit coordinate realizations for hypersurfaces, and a reconstruction of the usual Artin–Mazur formal groups without ad hoc cohomology-vanishing assumptions (Grammatica, 3 Oct 2025, Mondal et al., 7 Oct 2025).

1. Definition and cohomological meaning

Let $X$ be a smooth proper scheme over a Noetherian ring $R$ of characteristic $p>0$, or more generally over a complete discrete valuation ring of mixed characteristic $(0,p)$. The $i$-th Artin–Mazur formal group functor is defined on local Artinian $R$-algebras $X$0 by

$X$1

Over an algebraically closed field $X$2 of characteristic $X$3, the fppf formulation used in later work is

$X$4

After fppf-sheafification, these functors are representable by formal groups of finite type over $X$5; when they are representable one recovers the original functor. Via the Kummer sequence $X$6, one also identifies $X$7 with the formal completion of

$X$8

at the identity section (Grammatica, 3 Oct 2025).

The basic linearization is provided by the tangent space. Under mild hypotheses, $X$9 is pro-representable by a commutative formal Lie group whose tangent space is canonically identified with $H^i(-,\mathbf G_m)$0. In hypersurface situations this dimension can be computed explicitly from the interior lattice points of the Newton polytope. This places the functor at the interface of infinitesimal deformation theory, crystalline methods, and $H^i(-,\mathbf G_m)$1-adic invariants. The theory generalizes the classical formal Picard group and the formal Brauer group and is tied, in the formulation of Vlasenko, to $H^i(-,\mathbf G_m)$2-adic variation of zeta- and $H^i(-,\mathbf G_m)$3-functions, period maps, and deformation theory of algebraic cycles (Vlasenko, 2016).

2. Representability, formal smoothness, and cohomological criteria

Formal smoothness is not automatic once representability is known. A precise criterion is given by Grammatica through the auxiliary notion of $H^i(-,\mathbf G_m)$4-smoothness, where $H^i(-,\mathbf G_m)$5 and $H^i(-,\mathbf G_m)$6 is induced by multiplication by $H^i(-,\mathbf G_m)$7 on $H^i(-,\mathbf G_m)$8. For a presheaf of abelian groups $H^i(-,\mathbf G_m)$9, one says that $H^i(X,\mathcal O_X)$0 is $H^i(X,\mathcal O_X)$1-smooth when $H^i(X,\mathcal O_X)$2 is surjective. If $H^i(X,\mathcal O_X)$3 is a formal group of finite type, $H^i(X,\mathcal O_X)$4-smoothness agrees with ordinary formal smoothness. In particular, $H^i(X,\mathcal O_X)$5 is formally smooth if and only if $H^i(X,\mathcal O_X)$6 is $H^i(X,\mathcal O_X)$7-smooth, equivalently if and only if

$H^i(X,\mathcal O_X)$8

is surjective (Grammatica, 3 Oct 2025).

The same paper relates formal smoothness to torsion phenomena in $H^i(X,\mathcal O_X)$9-adic cohomology. One has the split exact sequence

$X$0

so smoothness is controlled by surjectivity and torsion in adjacent degrees. If $X$1 is representable and $X$2 is $X$3-torsion-free in crystalline cohomology, then $X$4 is formally smooth. Conversely, if $X$5 is representable but $X$6 has non-zero torsion, then $X$7 is not formally smooth. For abelian varieties, torsion-freeness of crystalline cohomology yields formal smoothness of all $X$8 (Grammatica, 3 Oct 2025).

A further reformulation uses Serre’s Witt-vector cohomology. If $X$9 is representable, then the following are equivalent: $R$0 is formally smooth; the natural map

$R$1

is surjective; and the Verschiebung $R$2 on $R$3 is injective. This equivalence makes formal smoothness a concrete Witt-cohomological condition rather than merely a property of a functor. It also shows that representability and formal smoothness should be kept distinct: the paper constructs, for every $R$4, smooth proper varieties $R$5 for which $R$6 is formally smooth for $R$7 while $R$8 is not (Grammatica, 3 Oct 2025).

3. Explicit coordinatizations and $R$9-adic integrality

For projective hypersurfaces, Artin–Mazur formal groups admit explicit coordinates. Let

$p>0$0

be a Laurent polynomial over a characteristic-zero ring $p>0$1 with Frobenius lift $p>0$2, and suppose the Newton polytope $p>0$3 has nonempty interior integral set

$p>0$4

For each $p>0$5, define the $p>0$6 matrix $p>0$7 by

$p>0$8

The logarithm series is

$p>0$9

and the associated formal group law over $(0,p)$0 is

$(0,p)$1

Vlasenko proves that in fact

$(0,p)$2

so $(0,p)$3 is an integral $(0,p)$4-dimensional formal group law over $(0,p)$5. If $(0,p)$6 is homogeneous of degree $(0,p)$7 and the corresponding projective hypersurface $(0,p)$8 is smooth over $(0,p)$9, then $i$0 is precisely the coordinate expression of the Artin–Mazur formal group functor $i$1 (Vlasenko, 2016).

This coordinate description interacts with higher Hasse–Witt matrices. Writing $i$2, the reductions $i$3 control $i$4-adic congruences, and when $i$5 is invertible the limits

$i$6

exist $i$7-adically. Their conjectural meaning is that they coincide with the Frobenius and Gauss–Manin connection matrices on the unit-root crystal $i$8, hence encode the crystal dual to the Artin–Mazur formal group (Vlasenko, 2016).

In the one-dimensional case, Vlasenko gives a separate integrality theory for formal group laws in terms of the coefficients of the strict logarithm

$i$9

and the canonical invariant differential

$R$0

The paper gives a criterion of integrality of a one-dimensional formal group law in terms of congruences satisfied by these coefficients, and a $R$1-adic analytic formula for the local characteristic polynomial at $R$2. Applied to Artin–Mazur formal groups of hypersurfaces with exactly one interior lattice point, this yields explicit logarithms

$R$3

with

$R$4

and proves $R$5-integrality under the stated denominator condition on the coefficients of $R$6 (Vlasenko, 2015).

4. Unipotent spectra and reconstruction of the formal groups

A major reformulation comes from the theory of unipotent spectra. Fix a commutative ring $R$7, and let $R$8 be Toën’s $R$9-category of pointed affine stacks over $X$00, equipped with the loop endofunctor $X$01. The $X$02-category of unipotent spectra is defined by

$X$03

the stabilization of pointed affine stacks. There is a left adjoint $X$04 and a right adjoint $X$05; for a stack $X$06, the object $X$07 is its unipotent stable homotopy type. Its homotopy sheaves $X$08 are representable by commutative unipotent affine group schemes, called the unipotent stable homotopy groups of $X$09 (Mondal et al., 7 Oct 2025).

Passing to $X$10-linear objects over a field $X$11, one obtains unipotent homology

$X$12

For a finite-dimensional $X$13-scheme $X$14, this admits an increasing coniveau filtration

$X$15

with associated graded

$X$16

and a convergent homological spectral sequence

$X$17

The local terms are related to flat cohomology by

$X$18

for every commutative unipotent $X$19. This comparison yields exact sequences linking unipotent local homology to Witt-vector cohomology and sets up the reconstruction of the Artin–Mazur functors (Mondal et al., 7 Oct 2025).

The main reconstruction theorem states that over any perfect field $X$20 of characteristic $X$21, for every smooth proper $X$22 and each $X$23, the Cartier dual of the fppf-sheafified Artin–Mazur functor is canonically isomorphic to the term $X$24 on the second page of the coniveau spectral sequence: $X$25 Accordingly,

$X$26

is the formal group functor arising from unipotent stable homotopy. The conceptual point is that the usual Artin–Mazur formal groups are recovered without any vanishing assumptions on the intermediate $X$27-cohomology (Mondal et al., 7 Oct 2025).

5. Heights, Witt cohomology, and quasi-$X$28-split geometry

When the Artin–Mazur functor is one-dimensional, its height provides a refined measure of its $X$29-typical structure. Let $X$30 be a perfect field of characteristic $X$31, and let $X$32 be a proper log-smooth scheme of Cartier type over $X$33. For each $X$34, the Artin–Mazur functor

$X$35

has height

$X$36

with the convention $X$37 if all such Frobenius maps vanish (Nakkajima, 2019).

Nakkajima relates this invariant to Yobuko’s height

$X$38

where $X$39 is the truncation map. The condition $X$40 is called quasi-$X$41-splitness. Under the stated hypotheses—one-dimensional $X$42, vanishing $X$43, vanishing Bockstein operators, and pro-representability of $X$44—one has the fundamental inequality

$X$45

This exhibits Artin–Mazur height as bounded above by a splitting invariant defined directly on Witt sheaves (Nakkajima, 2019).

The same framework yields finiteness statements for Witt cohomology. If $X$46 is quasi-$X$47-split, then for every $X$48,

$X$49

is a finitely generated $X$50-module; in particular $X$51 is finitely generated for every $X$52. For a proper smooth threefold $X$53, if $X$54 is finitely generated over $X$55, then the $X$56-primary torsion subgroup $X$57 is of finite cotype. Combining the results, a quasi-$X$58-split proper smooth threefold has $X$59 of finite cotype. In the Calabi–Yau threefold case, finite third Artin–Mazur height implies the same conclusion through the equality $X$60 noted in the paper (Nakkajima, 2019).

6. Comparisons, standard examples, and duality refinements

Several standard examples clarify the range of the theory. In degree $X$61, the unipotent-spectral reconstruction gives

$X$62

This is the Artin–Mazur formal Picard group. In degree $X$63 for surfaces, the same formalism recovers the Brauer-type example. Under the classical connectivity assumptions

$X$64

one has

$X$65

recovering the Mondal–Reinecke result. A recurrent oversimplification is that such vanishing conditions are intrinsic to the Artin–Mazur construction itself; the unipotent-spectral reconstruction shows instead that they are only needed for the older identification with unipotent $X$66, not for recovering the formal group functor (Mondal et al., 7 Oct 2025).

The same paper extends the formalism to syntomic cohomology. For each weight $X$67 and proper $X$68, the functor

$X$69

is represented by a quasi-finite type perfect unipotent spectrum $X$70. If $X$71 is smooth and proper of dimension $X$72, then

$X$73

in perfect unipotent $X$74-modules, refining Milne’s Poincaré duality. In the $X$75-complete limit this becomes an autoduality on the $X$76-module spectrum $X$77, recovering and extending Milne’s duality to all $X$78-complete coefficients (Mondal et al., 7 Oct 2025).

Taken together, these results place Artin–Mazur formal group functors at a nexus of deformation theory, $X$79-adic Hodge theory, crystalline and Witt-vector cohomology, and unstable-to-stable unipotent homotopy theory. Their classical role as formal Picard or formal Brauer groups persists, but the modern picture is broader: explicit coordinatizations are available for hypersurfaces, formal smoothness can be read off from crystalline or Witt-cohomological conditions, and the functors themselves arise naturally from unipotent stable homotopy without auxiliary cohomology-vanishing hypotheses (Vlasenko, 2016, Grammatica, 3 Oct 2025).