Let H be a bialgebra and let A be an associative algebra. The algebra A is said to be an H-module-algebra if there is an H-module structure on A such that the multiplication on A becomes an H-module morphism. For example, if S denotes the Landweber-Novikov algebra [15, 21], then the complex cobordism MU(X) of a topological space X is an S module-algebra. Likewise, the singular mod p cohomology ...

0.1. The kinds of homotopy theories under consideration in this paper are Waldhausen ∞-categories [2, Df. 2.7]. (We employ the quasicategory model of∞-categories for technical convenience.) These are ∞-categories with a zero object and a distinguished class of morphisms (called cofibrations or ingressive morphisms) that satisfies the following conditions. (0.1.1) Any equivalence is ingressive. ...

In their book Rapoport and Zink constructed rigid analytic period spaces F for Fontaine’s filtered isocrystals and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism F → F of rigid analytic spaces and of interesting local systems of Qp-vector spaces on F. For those period spaces possessing perio...

It is considered a smooth projective morphism p : T ! S to a smooth variety S. It is proved, in particular, the following result. The total direct image Rp (Z=nZ) of the constant etale sheaf Z=nZis locally for Zarisky topology quasi-isomorphic to a bounded complex L on S consisting of locally constant constructible etale Z=nZ-module sheaves.

In this paper, projective modules over a quantale are characterized by distributivity, continuity, and adjointness conditions. It is then show that a morphism Q // A of commutative quantales is coexponentiable if and only if the corresponding Q-module is projective, and hence, satisfies these equivalent conditions.

Let p denote an odd prime. We show that the spectrum [ E(n), the In-adic completion of Johnson and Wilson’s E(n), admits a unique topological A∞ structure compatible with its canonical ring spectrum structure. Furthermore, the canonical morphism of ring spectra [ E(n) −→ K(n) admits an A∞ structure whichever of the uncountably many A∞ structures of A. Robinson is imposed upon K(n), the n th Mor...

We study the rational Cherednik algebra attached to complex reflection group G(r, 1, 2). Each irreducible representation Sλ of 2) corresponds a standard module Δ(λ) for algebra. give necessary and sufficient conditions existence morphism between two these modules explicit formulas them when they exist.

Let A be a fixed ring, and consider the projective space PA = ProjA[x0, . . . , xn] over A. On PA we have the invertible sheaf O(1), and the homogenous coordinates x0, . . . , xn give rise to global sections ẋ0, . . . , ẋn ∈ Γ(PA,O(1)). Throughout this section we drop the dot in this notation and just write xi for the global section of O(1). One sees easily that the sheaf O(1) is generated by t...

Let H be a bialgebra. An H-module-algebra is an associative algebra A that is also an H-module such that the multiplication map on A becomes an H-module morphism. This algebraic structure arises often in algebraic topology, quantum groups [11, Chapter V.6], Lie and Hopf algebras theory [3, 16, 19], and group representations [1, Chapter 3]. For example, in algebraic topology, the complex cobordi...

We show that on an Abelian variety over algebraically closed field of positive characteristic, the obstruction to lifting automorphism a characteristic zero as morphism vanishes if and only it for derived autoequivalence. also compare deformation space these two types deformations.