2 repaired, I hope.

# DoIsthetangentspacefunctorfrom PD envelopesforgetp-infinitesimalgroupstructureformalgroupstoLiealgebrasanequivalence?

The following is my understanding, which may be previous version of this question was rather badly broken:

If X , and I hope this version makes some sense.

There have been a few questions on MathOverflow about how much representation-theoretic information is lost when passing from a smooth varietyLie group to its Lie algebra, then e.g., away from the semisimple case, Lie algebras have many more representations. In the algebraic setting, there is an intermediate construction between an algebraic group and its Lie algebra, given by the formal completion of X at a closed point x (isomorphic group. One completes the algebraic group along the identity to Spf of a Cohen ring) has get a canonical formal scheme equipped with a group structurelaw, with multiplication arising and one can pass from there to the completion tangent space to get the Lie algebra. In characteristic zero, the tangent space functor is an equivalence of $X \times X$ at (x,x). These categories from formal groups encode some kind to Lie algebras, but in positive characteristic, formal groups form an honest intermediate category since the tangent space can lose a lot of informationabout infinitesimal translations near x. If I'm not mistakenFor example, this formal group there is always a product only one isomorphism class of one-dimensional Lie algebra, but one-dimensional formal additive groups in characteristic zero, because once we choose have a generating set of the augmentation idealrich arithmetic theory, we can recursively construct with a formal logarithm homomorphismmoduli space stratified by positive integer heights. In characteristic p, The completions at the standard counterexamples are identity of the multiplicative formal additive group and the multiplicative group have very distinct formal group of an elliptic curve. Herestructures, and one way to explain the logarithm would require denominators with factors lack of p. I think there isomorphism is a fancier explanation involving p-divisible groups.

Instead by the presence of a formal group at a closed pointdenominators in the usual logarithm and exponential power series.

It seems to me that in positive characteristic, we can consider a more global objectthere could be an intermediate construction between formal groups and Lie algebras, given by completing $X \times X$ along passing to PD rings and replacing the diagonal embedding coordinate ring of X. This has a natural the formal groupoid structure, and taking a quotient group with the divided power envelope of X by its action the identity section. If I'm not mistaken, this construction yields an a group object called the de Rham stack of Xin PD formal schemes.As

Here is a functor its R-points bit of explanation for the uninitiated (see Berthelot-Ogus for more): PD rings are X(R/I), triples $(A,I,\gamma)$, where I $A$ is the nilradical of R. There a commutative ring, $I$ is an equivalence of categories between quasicoherent modules over the sheaf of ordinary differential operators on X ideal, and quasicoherent sheaves on the de Rham stack (which seems $\gamma = \{ \gamma_n: I \to mean a quasicoherent sheaf on X that A \}_{n \geq 0}$ is equivariant under the groupoid action)a system of divided power operations.

When trying I think they arose when Grothendieck tried to make de get De Rham cohomology to give the expected answers for proper varieties in characteristic p, one typically adds divided power (PD) structures (which are canonical in characteristic zero) to $p$, since the defining ideals. In particular, one considers formal groups and groupoids given by PD completions instead of plain formal completionsnaïve definition tended to yield infinite dimensional spaces. There is a canonical map forgetful functor $(A,I,\gamma) \mapsto (A,I)$ from the usual formal group/groupoid PD rings to ring-ideal pairs, and it has a left adjoint, called the formal PD group/groupoid that is an isomorphism in divided power envelope. In characteristic zero. Sheaves that are equivariant under the PD groupoid action are called crystals (I'm not sure about this - the usual definition involves the crystalline site, but Berthelot's thesis seems to say the groupoid $\gamma$ is universal)canonically given as $\gamma_n(x) = x^n/n!$, so both functors are equivalences in that case. There is a theorem giving an equivalence The notion of categories between crystals on X PD ring can be sheafified and quasicoherent sheaves on X with a flat connectionlocalizations have canonical PD structures, so one has notions of PD scheme and PD formal scheme.

My question is

Question: Are these Do PD formal groups always isomorphic to a power of contain any more information than the formal (PD) additive groupunderlying Lie algebra?

I have a suspicion that the answer is "no" and the answer to the title question is "yes". Vague word-association suggests that the divided power structure is exactly what one needs to get a formal logarithm, but maybe there is a more fundamental geometric obstruction.

(More motivation: I'm looking for a crystalline

I was originally motivated by the question of how Gelfand-Kazhdan equivalence, like formal geometry would differ in charateristic $p$ if I switched between ordinary and PD structures (cf. David Jordan's question). IdeallyUnfortunately, this would already exist in the literature. Is it I was laboring under some misconceptions about formal completions, and I'm still a stupid idea?bit confused about the precise structure of the automorphism group of the completion (PD or ordinary) of a smooth variety at a point in characteristic $p$.

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# Do PD envelopes forget p-infinitesimal group structure?

The following is my understanding, which may be broken:

If X is a smooth variety, then the formal completion of X at a closed point x (isomorphic to Spf of a Cohen ring) has a canonical formal group structure, with multiplication arising from the completion of $X \times X$ at (x,x). These formal groups encode some kind of information about infinitesimal translations near x. If I'm not mistaken, this formal group is always a product of formal additive groups in characteristic zero, because once we choose a generating set of the augmentation ideal, we can recursively construct a formal logarithm homomorphism. In characteristic p, the standard counterexamples are the multiplicative formal group and the formal group of an elliptic curve. Here, the logarithm would require denominators with factors of p. I think there is a fancier explanation involving p-divisible groups.

Instead of a formal group at a closed point, we can consider a more global object, given by completing $X \times X$ along the diagonal embedding of X. This has a natural formal groupoid structure, and taking a quotient of X by its action yields an object called the de Rham stack of X. As a functor its R-points are X(R/I), where I is the nilradical of R. There is an equivalence of categories between quasicoherent modules over the sheaf of ordinary differential operators on X and quasicoherent sheaves on the de Rham stack (which seems to mean a quasicoherent sheaf on X that is equivariant under the groupoid action).

When trying to make de Rham cohomology give the expected answers for proper varieties in characteristic p, one typically adds divided power (PD) structures (which are canonical in characteristic zero) to the defining ideals. In particular, one considers formal groups and groupoids given by PD completions instead of plain formal completions. There is a canonical map from the usual formal group/groupoid to the formal PD group/groupoid that is an isomorphism in characteristic zero. Sheaves that are equivariant under the PD groupoid action are called crystals (I'm not sure about this - the usual definition involves the crystalline site, but Berthelot's thesis seems to say the groupoid is universal). There is a theorem giving an equivalence of categories between crystals on X and quasicoherent sheaves on X with a flat connection.

My question is: Are these PD formal groups always isomorphic to a power of the formal (PD) additive group?

Vague word-association suggests that the divided power structure is exactly what one needs to get a logarithm, but maybe there is a more fundamental geometric obstruction.

(More motivation: I'm looking for a crystalline Gelfand-Kazhdan equivalence, like in David Jordan's question. Ideally, this would already exist in the literature. Is it a stupid idea?)