In this note, we report on a work jointly done with C. Simpson on a generalization of Reznikov’s theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees > 1) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi–projective variety with an irreducible smooth divisor at infinity. We defi...

In this paper, we prove a generalization of Reznikov's theorem on the torsion-property of the Chern-Simons classes and in particular the torsion–property of the Deligne Chern classes (in degrees > 1) of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi–projective variety with an ir-reducible smooth divisor at infinity. We define the Chern-Simons...

Our work is a foundational study of the notion of approximation inQ-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (...

In this paper, we study the left orthogonal class of max-flat modules which are homological objects related to s-pure exact sequences and module homomorphisms. Namely, a right $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any $R$-module $B$, strongly ${Ext}^{i}_{R}(A,B)=0$ all $R$-modules $B$ $i\geq 1$. Firstly, give some properties $MF$-projective SMF-projective modules. Then intro...

‎we say that a module $m$ is a emph{cms-module} if‎, ‎for every cofinite submodule $n$ of $m$‎, ‎there exist submodules $k$ and $k^{'}$ of $m$ such that $k$ is a supplement of $n$‎, ‎and $k$‎, ‎$k^{'}$ are mutual supplements in $m$‎. ‎in this article‎, ‎the various properties of cms-modules are given as a generalization of $oplus$-cofinitely supplemented modules‎. ‎in particular‎, ‎we prove tha...

A module $M$ is called $emph{H}$-cofinitely supplemented if for every cofinite submodule $E$ (i.e. $M/E$ is finitely generated) of $M$ there exists a direct summand $D$ of $M$ such that $M = E + X$ holds if and only if $M = D + X$, for every submodule $X$ of $M$. In this paper we study factors, direct summands and direct sums of $emph{H}$-cofinitely supplemented modules. Let $M$ be an $emph{H}...

Example 1.1 (Example 1: The Grassmannian Functor.). Let S be a scheme, E a vector bundle on S and k a positive integer less than the rank of E. Let Gr(k, S, E) : {Schemes/S} � {sets} be the contravariant functor that associates to an S-scheme X subvector bundles of rank k of X ×S E. Example 1.2 (Example 2: The Hilbert Functor.). Let X � S be a projective scheme, O(1) a relatively ample line bun...

We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({\bf D})$, as well Eschmeier Langendorfer extension to unit ball ${\bf C}^n$. All our results are proved in more general setting poly-hyperballs D_n^m}(H)$, n,m}\in {\bf N}^k$, used characterize bounded free $k$-pluriharmonic fun...

Let $R$ be a commutative ring. We write $mbox{Hom}(mu_A, nu_B)$ for the set of all fuzzy $R$-morphisms from $mu_A$ to $nu_B$, where $mu_A$ and $nu_B$ are two fuzzy $R$-modules. We make$mbox{Hom}(mu_A, nu_B)$ into fuzzy $R$-module by redefining a function $alpha:mbox{Hom}(mu_A, nu_B)longrightarrow [0,1]$. We study the properties of the functor $mbox{Hom}(mu_A,-):FRmbox{-Mod}rightarrow FRmbox{-Mo...

For an arbitrary localic étale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of étendues.