Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the investigations of Ivan Prodanov in the theory of abstract spectra
131
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The results of Iv. Prodanov on abstract spectra and separative algebras were announced in the journal Trudy Mat. Inst. Steklova, 154, 1983, 200--208, but their proofs were never written by him in the form of a manuscript, preprint or paper. Since the untimely death of Ivan Prodanov withheld him from preparing the full version of this paper and since, in our opinion, it contains interesting and important results, we undertook the task of writing a full version of it and thus making the results from it known to the mathematical community. So, the aim of this paper is to supply with proofs the results of Ivan Prodanov announced in the cited above paper, but we added also a small amount of new results. The full responsibility for the correctness of the proofs of the assertions presented below in this work is taken by us; just for this reason our names appear as authors of the present paper.
rate research
Read More
The notions of a {em 2-precontact space}/ and a {em 2-contact space}/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connect
Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. Both of them imply easily the Tarski Duality Theorem, as well as two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.
Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal contact algebras with suprema-preserving Boolean homomorphisms which reflect the contact relation.
We give an exposition and generalization of Orlovs theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.
A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $C$, and under certain assumptions on the braiding (fulfilled if $C$ is symmetric), we construct a sequence for the Brauer group $BM(C;B)$ of $B$-module algebras, generalizing Beatties one. It allows one to prove that $BM(C;B) cong Br(C) times Gal(C;B),$ where $Br(C)$ is the Brauer group of $C$ and $Gal(C;B)$ the group of $B$-Galois objects. We also show that $BM(C;B)$ contains a subgroup isomorphic to $Br(C) times Hc(C;B,I),$ where $Hc(C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_HM$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $BM(K,H,R) times Hc({}_HM;B,K)$ is a subgroup of the Brauer group $BM(K,B times H,R),$ confirming the suspicion that a certain cohomology group of $B times H$ (second lazy cohomology group was conjectured) embeds into $BM(K,B times H,R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
