arXiv subjects
Publications and source records attributed to Eike Lau.
We compute the Balmer spectrum of the category of perfect complexes on an algebraic stack admitting a finite locally free cover by an affine scheme and identify it with the homogeneous spectrum of the cohomology ring.
We develop a Dieudonn\'e theory for $p$-divisible groups using sheared Witt vectors.
We show that the mod $p$ fiber of the Shimurian stack $BT_n^{G,\mu}$ constructed by Gardner--Madapusi is a gerbe over the corresponding stack of truncated displays. This confirms a conjecture of Drinfeld.
We consider a Deligne-Mumford stack $X$ which is the quotient of an affine scheme $\operatorname{Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.
We define a category of divided Dieudonn\'e crystals which classifies p-divisible groups over schemes in characteristic p with certain finiteness conditions, including all F-finite noetherian schemes. For formally smooth schemes or locally complete intersections this generalizes and extends known results on the classical crystalline Dieudonn\'e functor.
Deformations of ordinary varieties of K3 type can be described in terms of displays by recent work of Langer-Zink. We extend this to the general (non-ordinary) case using displays with $G$-structure for a reductive group $G$. As a basis we suggest a modified definition of the tensor category of displays and variants which is similar to the Frobenius gauges of Fontaine-Jannsen.
The Dieudonné crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application one obtains a classification of p-divisible groups and commutative finite locally free p-group schemes over perfectoid rings by Breuil-Kisin-Fargues modules if p>2.
Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(μ_{p^r})$. Using crystalline Dieudonné theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(φ,Γ)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonné crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by Kisin--Ren.
We discuss the relation between crystalline Dieudonne theory and Dieudonne displays, with special emphasis on the case p=2. The theory of Dieudonne displays is extended to this case without restriction, which implies that the classification of finite flat group schemes by Breuil-Kisin modules holds for p=2 as well.
We define truncated displays over rings in which a prime p is nilpotent, we associate crystals to truncated displays, and we define functors from truncated displays to truncated Barsotti-Tate groups.
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.
The Galois representation associated to a p-divisible group over a complete noetherian normal local ring with perfect residue field is described in terms of its Dieudonné display. As a corollary we deduce in arbitrary characteristic Kisin's description of the Galois representation associated to a commutative finite flat p-group scheme over a p-adic discrete valuation ring in terms of its Breuil-Kisin module. This was obtained earlier by W. Kim by a different method.
We show that to every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal p-divisible groups and nilpotent displays over p-adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of p-power order and Dieudonne modules over perfect rings.
Let p be an odd prime. We show that the classification of p-divisible groups by Breuil windows and the classification of finite flat group schemes of p-power order by Breuil modules hold over any complete regular local ring with perfect residue field of characteristic p. We use a formalism of frames and windows with an abstract deformation theory that applies to Breuil windows.
It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions.
We propose a notion of frames and windows that allows an alternative proof of the Vasiu-Zink classification of p-divisible groups over ramified complete regular local rings by their Breuil windows.
We show that the Zink equivalence between p-divisible groups and Dieudonne displays over a complete local ring with perfect residue field of characteristic p is compatible with duality. The proof relies on a new explicit formula for the p-divisible group associated to a Dieudonne display.
A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons.