Let k be a perfect field of characteristic p > 2 and K a totally ramified extension of K₀ = Frac W(k) with uniformizer π. Let F ⊆ K be a subfield with ϖ, ring of integers O, and residue field k(F) ⊆ k with |k(F)| = q. Let W(F) = O⊗(W(k(F))) W(k) and consider the ring 𝔖 = W(F)⟦u⟧ with an endomorphism φ that lifts the q-power Frobenius of k on W(F) and satisfies φ(u) ≡ u^q mod ϖ and φ(u) ≡ 0 mod u. In this dissertation, we use O-divided powers to define the analogue of Breuil-Kisin modules over the rings 𝔖 and S, where S is an O-divided power envelope of the surjection 𝔖 ↠ O(K) sending u to π. We prove that these two module categories are equivalent, generalizing the case when F = Q(p) and ϖ - p. As an application of our theory, we generalize the results of Kisin [17] and Cais-Lau [8] to relate the Faltings Dieudonné crystal of a ϖ-divisible O-module, which gives a Breuil module over S in our sense, to the modules of Kisin-Ren, providing a geometric interpretation to the latter.

The objects studied in this thesis are families of cycles on schemes. A space — the Chow variety — parameterizing effective equidimensional cycles was constructed by Chow and van der Waerden in the first half of the twentieth century. Even though cycles are simple objects, the Chow variety is a rather intractable object. In particular, a good functorial description of this space is missing. Consequently, descriptions of the corresponding families and the infinitesimal structure are incomplete. Moreover, the Chow variety is not intrinsic but has the unpleasant property that it depends on a given projective embedding. A main objective of this thesis is to construct a closely related space which has a good functorial description. This is partly accomplished in the last paper. The first three papers are concerned with families of zero-cycles. In the first paper, a functor parameterizing zero-cycles is defined and it is shown that this functor is represented by a scheme — the scheme of divided powers. This scheme is closely related to the symmetric product. In fact, the scheme of divided powers and the symmetric product coincide in many situations. In the second paper, several aspects of the scheme of divided powers are discussed. In particular, a universal family is constructed. A different description of the families as multi-morphisms is also given. Finally, the set of k-points of the scheme of divided powers is described. Somewhat surprisingly, cycles with certain rational coefficients are included in this description in positive characteristic. The third paper explains the relation between the Hilbert scheme, the Chow scheme, the symmetric product and the scheme of divided powers. It is shown that the last three schemes coincide as topological spaces and that all four schemes are isomorphic outside the degeneracy locus. The last paper gives a definition of families of cycles of arbitrary dimension and a corresponding Chow functor. In characteristic zero, this functor agrees with the functors of Barlet, Guerra, Kollár and Suslin-Voevodsky when these are defined. There is also a monomorphism from Angéniol's functor to the Chow functor which is an isomorphism in many instances. It is also confirmed that the morphism from the Hilbert functor to the Chow functor is an isomorphism over the locus parameterizing normal subschemes and a local immersion over the locus parameterizing reduced subschemes — at least in characteristic zero. / QC 20100908