Mes: març de 2023

Publicat a

Seminar (Ring Theory)

Guillem Quingles (Universitat Autònoma de Barcelona)

Finiteness properties of local cohomology modules

Abstract: Local cohomology modules were introduced by Grothendieck in 1961 and they quickly became an important tool in commutative algebra. They have been studied by a number of authors, but the structure of these modules is still quite unknown. When a local cohomology module $H^i(\Gamma_I(M*))$ is nonzero, it is rarely finitely generated, even if $M$ is. So it is not clear whether they satisfy finiteness properties that finitely generated modules do. Huneke proposed a list of problems on local cohomology which guided the study of local cohomology modules. One of the questions on the list asks the following: Is the number of associated primes of $H^i(\Gamma_I(R*))$ finite? Are all the Bass numbers of $H^i(\Gamma_I(R*))$ finite? Lyubeznik conjectured that the answer is affirmative when $R$ is a Noetherian regular commutative ring with unit. Substantial progress has been made on this conjecture. If the regular ring has prime positive characteristic $p$, then the conjecture was completely settled by Huneke and Sharp. Lyubeznik proved the conjecture for regular rings containing a field of characteristic zero. For complete unramified regular local rings of mixed characteristic, the conjecture was also settled by Lyubeznik. The finiteness of associated primes of local cohomology was also proved by Bhatt, Blickle, Lyubeznik, Singh and Zhang for smooth $Z$-algebras. The conjecture is still open when $R$ is a ramified regular local ring of mixed characteristic.

In this talk I will explain the concepts and tools needed to understand the problem of the finiteness of the set of associated primes and Bass numbers of local cohomology modules $H^i(\Gamma_I(M*))$, and the techniques that lead to the proof of the cases where $R$ has positive characteristic and where $R$ is a $K$-algebra, with a field of characteristic 0.

Publicat a

Seminar (Operator Algebras)

Martin Mathieu (Queen’s University Belfast)

A contribution to Kaplansky’s problem

Abstract: A Jordan homomorphism between two unital, complex algebras A and B is a linear mapping T such that $T(x^2)=(Tx)^2$ for all $x\in A$. Equivalently, T preserves the Jordan product $xy+yx$. Every surjective unital Jordan homomorphism preserves invertible elements. In 1970, Kaplansky asked whether the following converse is true: Suppose $T\colon A\to B$ is a unital surjective invertibility-preserving linear mapping between unital (Jacobson) semisimple Banach algebras A and B. Does it follow that T is a Jordan homomorphism?

In the past 50 years a lot of progress has been made towards a positive solution to Kaplansky’s problem, however, as it stands, it is still open. We will report on some recent joint work with Francois Schulz (University of Johannesburg, SA) which gives a positive answer if B is a C*-algebra with faithful tracial state. Until recently, the existence of traces had been a major obstacle to a solution. Moreover, in our approach, no assumption on the existence of projections (such as real rank zero) is necessary.

I will further discuss a sharpening of Kaplansky’s problem in which the assumption on T is reduced to the preservation of the spectral radius only (a spectral isometry).

Publicat a

Seminar (Operator Algebras)

Laurent Cantier (Universitat Autònoma de Barcelona)

The Cu$_1$-semigroup as an invariant for K$_1$-obstruction cases

Abstract: The aim of this talk is to explicitly shows that the unitary Cuntz semigroup, defined using the Cuntz semigroup and the K$_1$ group, strictly contains more information than the latter invariants alone. To that end, we construct two C*-algebras, distinguished by their unitary Cuntz semigroup, whose K-Theory and Cu-semigroup are isomorphic. Both A and B, constructed as inductive limits of NCCW 1-algebras, are non-simple unital separable C∗-algebras of stable rank one with K$_1$-obstructions. This shows that a likewise invariant is necessary in order to extend classification results of C*-algebras by means of Cuntz semigroup to the non trivial K$_1$ group case.