For a homomorphism f : A → B of commutative rings, letD(A,B) denote the kernel of the map Pic(A) → Pic(B). Let k be a field and assume that A is a finitely generated k-algebra. We prove a number of finiteness results for D(A,B). Here are four of them. [1]: Suppose B is a finitely generated and faithfully flat A-algebra which is geometrically integral over k. If k is perfect, we find that D(A,B)...

The aim of this paper is to lay the groundwork for a theory of quadratic forms over several significant, and quite extensive, classes of preordered rings. By “quadratic forms” we understand, here, diagonal quadratic forms with unit coefficients; and “ring” stands for commutative unitary ring where 2 in invertible. We achieve this by the use, in the ring context, of our abstract theory of quadra...

In the theory of schemes, faithfully flat descent is a very powerful tool. One wants a descent theory not only for quasi-coherent sheaves and morphisms of schemes (which is rather elementary), but also for geometric objects and properties of morphisms between them. In rigid-analytic geometry, descent theory for coherent sheaves was worked out by Bosch and Görtz [BG, 3.1] under some quasi-compac...

We study the property of a normal scheme, that the complement of every hypersurface is an affine scheme. To this end we introduce the affine class group. It is a factor group of the divisor class group and measures the deviation from this property. We study the behaviour of the affine class group under faithfully flat extensions and under the formation of products, and we compute it for differe...

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as quantization gauge groupoid classical bundle. show that group is isomorphic to bisections bialgebroid, and we give crossed module structure for automorphisms bialgebroid. Examples illustrating these constructions include: Galois objects Taft algebras, monopole over quantum spheres not faithfully flat Hopf--G...

We study and classify faithfully balanced modules for the algebra of triangular n by matrices more generally Nakayama algebras. The theory extends known results about tilting modules, which are classified binary trees, counted with Catalan numbers. number is a 2-factorial number. Among them n! indecomposable summands, can be interleaved trees or increasing trees.

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...

Calcium ion is the ubiquitous messenger in cells and plays a key role in neuronal signaling and fusion of synaptic vesicles. These vesicles are typically ∼20-50 nm in diameter, and thus their interaction with calcium ions cannot be modeled faithfully with a conventional flat membrane bilayer setup. Within our newly developed molecular dynamics simulations setup, we characterize here interaction...