Categoria: Ring Theory

Michal Hbrek (Prague)

Some new results about the Telescope Conjecture in D(X)

Abstract: In the generality of a big tt-category, the Telescope Conjecture (TC) asks if every smashing ideal is compactly generated. This has been a conjecture in the case of the stable homotopy category of spectra until the announcement of the negative answer this year. For the derived category D(X) of a qcqs scheme, (TC) is a property which sometimes holds (namely, for noetherian schemes) and sometimes does not. Balmer and Favi showed that (TC) is an affine-local property, and thus the question reduces to affine schemes. With Hu and Zhu, we recently showed that (TC) is even stalk-local, and thus reduces to local rings.

In the present work (arXiv:2311.00601), we show a stronger locality proposition, which reduces (TC) to inspection of the definable ideal generated by the residue field of a local ring. This ties (TC) very strongly with the properties of the m-adic topology on the ring. We apply this to recover most known examples of validity or failure of (TC) in D(X), as well as to construct some new ones. Moreover, we show that certain restriction of (TC) can be characterized in terms of pseudoflat ring epimorphisms over R, yielding an interesting example of a non-surjective pseudoflat local ring morphism.

Pace Nielsen (Bigham Young University)

Connections between elementwise properties in rings

Abstract: We illustrate some recent connections (and disconnections) discovered between some standard properties of rings, and their elements. Of particular importance is the discovery of “better” inner inverses for von Neumann regular elements. This has some surprising consequences in module theory, regarding direct sum decompositions.

Roozbeh Hazrat (University of Western Sydney)

Sandpile Graphs and Graph Algebras

Abstract: We give a down to earth introduction to seemingly two very different topics, one about sandpile models (a model about spreading objects along networks) and the other is how to associate interesting algebras to graphs. We then relate these two topics, via the concept of monoids.

Manuel Saorín (Universidad de Murcia)

On an overlooked conjecture

Abstract: The concept of flat object can be defined in any Grothendieck category. In 2007 Juan Cuadra and Daniel Simson conjectured that any locally finitely presented Grothendieck with enough flats has enough projectives. Since by (an extended version of) Gabriel-Popescu’s theorem, any Grothendieck category is equivalent to the quotient $(\mathrm{Mod}-\mathcal{A})/\mathcal{T}$, where is $\mathcal A$ a preadditive category and $\mathcal T$ is a hereditary torsion class of $\mathrm{Mod}-\mathcal{A})$, in order to tackle the conjecture one needs to ask first what are the conditions on $\mathcal T$ for the mentioned quotient to: 1) be locally finitely presented; 2) have enough flats; 3) have enough projectives. In this talk we will identify those conditions in the particular case when $\mathcal T$ is also a torsion free class (i.e. it is a TTF class) in which case, due to a generalization of Jan’s bijection, there is a uniquely determined idempotent ideal $\mathcal I$ of $\mathcal A$ such that consists of the right $\mathcal A$-modules (=additive functors $\mathcal{A}^{op}\to\mathrm{Ab}$) that vanish on $\mathcal I$. It turns out that $(\mathrm{Mod}-\mathcal{A})/\mathcal{T}$ has enough projectives in this case if, and only, if $\mathcal I$ is the trace of a projective right $\mathcal A$-module. From this point of view, as the much more popular telescope conjecture of Ravenel, this restricted version of Cuadra-Simson’s conjecture is a particular case of the general question of when a given idempotent ideal $\mathcal I$ of a (pre)additive category, e.g. of a ring, is the trace of a (special type of) projective $\mathcal A$-module. We will give some partial positive answers to Cuadra-Simson’s conjecture when $\mathcal T$ is a TTF class.

Raimund Preusser (Nanjing University of Information Science and Technology)

Graded Bergman algebras

Abstract: This talk is about an ongoing research project with Roozbeh Hazrat and Huanhuan Li. Recall that for a (unital and associative) ring R, the V-monoid of R is the set of isomorphism classes of finitely generated projective left R-modules. It becomes an abelian monoid with direct sum. George Bergman has shown that any conical finitely generated abelian monoid with an order unit can be realised as the V-monoid of a hereditary algebra. We want to obtain a graded version of this result as follows. For an abelian group G, a G-monoid is an abelian monoid M together with an action of G on M via monoid homomorphisms. Our goal is to show that any conical finitely presented G-monoid with an order unit can be realised as the graded V-monoid of a G-graded algebra which is hereditary.

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.