A Bicategorical Version of Masuoka’s Theorem (original) (raw)

Some exact sequences associated with adjunctions in bicategories. Applications

Transactions of the American Mathematical Society, 2019

We prove that the classical result asserting that the relative Picard group of a faithfully flat extension of commutative rings is isomorphic to the first Amitsur cohomology group stills valid in the realm of symmetric monoidal categories. To this end, we built some group exact sequences from an adjunction in a bicategory, which are of independent interest. As a particular byproduct of the evolving theory, we prove a version of Hilbert's theorem 90 for cocommutatvie coalgebra coextensions (=surjective homomorphisms).

Invertible unital bimodules over rings with local units, and related exact sequences of groups, II

Journal of Algebra, 2012

Let R be a ring with a set of local units, and a homomorphism of groups Θ : G → Pic(R) to the Picard group of R. We study under which conditions Θ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension R ⊆ S with the same set of local units and assuming that Θ is induced by a homomorphism of groups G → Inv R (S) to the group of all invertible R-sub-bimodules of S, then we construct an analogue of the Chase-Harrison-Rosenberg seven terms exact sequence of groups attached to the triple (R ⊆ S, Θ), which involves the first, the second and the third cohomology groups of G with coefficients in the group of all R-bilinear automorphisms of R. Our approach generalizes the works by Kanzaki and Miyashita in the unital case.

The Brauer Group of Azumaya Corings and the Second Cohomology Group

K-Theory, 2000

Let R be a commutative ring. An Azumaya coring consists of a couple (S, C), with S a faithfully flat commutative R-algebra, and an Scoring C satisfying certain properties. If S is faithfully projective, then the dual of C is an Azumaya algebra. Equivalence classes of Azumaya corings form an abelian group, called the Brauer group of Azumaya corings. This group is canonically isomorphic to the second flat cohomology group. We also give algebraic interpretations of the second Amitsur cohomology group and the first Villamayor-Zelinsky cohomology group in terms of corings.

On the Picard Group: Torsion and the Kernel Induced by a Faithfully Flat Map

Journal of Algebra, 1996

For a homomorphism f : A → B of commutative rings, let D(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) is finitely generated. (In positive characteristic, we need resolution of singularities to prove this.) For an arbitrary field k of positive characteristic p, we find that modulo p-power torsion, D(A, B) is finitely generated. [2]: Suppose B = A ⊗ k k sep. We find that D(A, B) is finite. [3]: Suppose B = A ⊗ k L, where L is a finite, purely inseparable extension. We give examples to show that D(A, B) may be infinite. [4]: Assuming resolution of singularities, we show that if K/k is any algebraic extension, there is a finite extension E/k contained in K/k such that D(A ⊗ k E, A ⊗ k K) is trivial. The remaining results are absolute finiteness results for Pic(A). [5]: For every n prime to the characteristic of k, Pic(A) has only finitely many elements of order n. [6]: Structure theorems are given for Pic(A), in the case where k is absolutely finitely generated. In the body of the paper, all of these results are stated in a more general form, valid for schemes.

Ring extension problem, Shukla cohomology and Ann-category theory

Arxiv preprint arXiv:0706.0315, 2007

Every ring extension of A by R induces a pair of group homomorphisms L * : R → End Z (A)/L(A); R * : R → End Z (A)/R(A), preserving multiplication, satisfying some certain conditions. A such 4-tuple (R, A, L * , R * ) is called a ring pre-extension. Each ring pre-extension induces a R-bimodule structure on bicenter K A of ring A, and induces an obstruction k, which is a 3-cocycle of Z-algebra R, with coefficients in R-bimodule K A in the sense of Shukla. Each obstruction k in this sense induces a structure of a regular Ann-category of type (R, K A ). This result gives us the first application of Anncategory in extension problems of algebraic structures, as well as in cohomology theories.

A bimodule structure for the bounded cohomology of commutative local rings

Journal of Algebra, 2019

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring R with residue field k, stable cohomology modules Ext n R (k, k), defined for n ∈ Z, have been studied by Avramov and Veliche. Stable cohomology carries a structure of Z-graded k-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology Ext R (k, k) and a shift of Hom k (Ext R (k, k), k). We use this information to characterize the rings R for which stable cohomology is graded-commutative. Stable cohomology is connected through an exact sequence to bounded cohomology. We use this connection to understand the algebra structure of Ext R (k, k) by investigating the structure of bounded cohomology Ext R (k, k) as a graded Ext R (k, k)-bimodule.