Periodic Reporting for period 1 - K-theory (Algebraic K-theory -- Arithmetic and Topology)
Période du rapport: 2020-10-01 au 2022-09-30
Algebraic K-theory was invented by Grothendieck in his formulation of the Riemann--Roch theorem in algebraic geometry. Since then, the ideas involved in its definition have lead to numerous related invariants such as topological K-theory, higher algebraic K-theory, Milnor K-theory, and hermitian K-theory. These invariants play a role in many areas of mathematics: operator algebras, homotopy theory, and algebraic and arithmetic geometry. As such, both the understanding of the structural behaviour, as well as explicit computations of these invariants have seen many breakthroughs in the area over decades of mathematical research.
The purpose of the EU funded project ``Algebraic K-theoy: Arithmetic and Topology'' (K-theory) was to build on recent advances in algebraic and hermitian K-theory and to study both structural and computational results, relating K-theory to questions in arithmetic and algebraic geometry on the one hand and to topology on the other hand.
The objectives in the project were three-fold:
- The first objective was to find a formula for a ring spectrum, called the circle-dot ring, the ER had found in earlier work with Georg Tamme, which is more amenable for computations that its definition. Having such a formula, together with earlier work of Tamme and the ER, gives new methods for computations of K-theories of classical objects.
- Generally, algebraic K-theory is a complicated object to compute. However, the situation becomes more tractable when one simplifies the algebraic K-theory, for instance by neglecting torsion, or considering only p-primary information for a prime number p. Chromatic homotopy theory is a systematic study of various (more drastic) ways of simplifying objects such as algebraic K-theory. The purpose of the second objective was to study chromatic simplifications of algebraic K-theory both from structural and computational points of views.
- The last objective was about relating recently developed fibre sequences in the theory of hermitian K-theory to know long exact sequences describing bordism theories of Poincar\'e duality complexes (rather than manifolds): Both of the sequences contain a common term, and the purpose was to find a general explanation for this common term to appear.
The purpose of the EU funded project ``Algebraic K-theoy: Arithmetic and Topology'' (K-theory) was to build on recent advances in algebraic and hermitian K-theory and to study both structural and computational results, relating K-theory to questions in arithmetic and algebraic geometry on the one hand and to topology on the other hand.
The objectives in the project were three-fold:
- The first objective was to find a formula for a ring spectrum, called the circle-dot ring, the ER had found in earlier work with Georg Tamme, which is more amenable for computations that its definition. Having such a formula, together with earlier work of Tamme and the ER, gives new methods for computations of K-theories of classical objects.
- Generally, algebraic K-theory is a complicated object to compute. However, the situation becomes more tractable when one simplifies the algebraic K-theory, for instance by neglecting torsion, or considering only p-primary information for a prime number p. Chromatic homotopy theory is a systematic study of various (more drastic) ways of simplifying objects such as algebraic K-theory. The purpose of the second objective was to study chromatic simplifications of algebraic K-theory both from structural and computational points of views.
- The last objective was about relating recently developed fibre sequences in the theory of hermitian K-theory to know long exact sequences describing bordism theories of Poincar\'e duality complexes (rather than manifolds): Both of the sequences contain a common term, and the purpose was to find a general explanation for this common term to appear.
First, the ER worked on establishing the formula for the new ring spectrum, the circle-dot ring, which appeared in earlier work of the ER together with Tamme. Again, the ER collaborated with Tamme. First, they found a generalisation of their initial result and were then able to give a concrete formula for the new ring spectrum in many cases (including many for which the indicated generalisation is crucial). The first objective then contained a number of Milestones of which the ER was able to solve most, among them exhibiting relations between tensor algebras and square zero extensions, establishing a formula for the new ring for (mixed characteristic) coordinate axes and calculations of K-theories for coordinate axes over special base rings from this perspective, as well as the study of the K-theory of certain coconnective ring spectra. The described findings of this objective will appear in a paper whose writing is currently in progress.
Second, the ER worked on chromatic localisations of K-theory. Based on results obtained shortly prior to the beginning of the funding period, the ER, in collaboration with Mathew, Meier and Tamme, proves a very general purity result for chromatic localisations of K-theory. There is an infinite family of chromatic localisations, one for each height n (which can be any natural number), and loosely speaking, the result is that the height-n-localised K-theory of a ring spectrum depends only on the height-(n-1)- and height-n-localisation of said ring spectrum. All Milestones proposed in this project are immediate special cases of this result and therefore fully resolved. A preprint about these results appeared on the arXiv during the first part of the funding period and is currently submitted for publication.
The ER presented his findings at several research seminars (Melbourne, Berlin, Warwick, MIT, Chicago). Due to the pandemic, further plans for public outreach were unfortunately cancelled.
The ER terminated the project 8 months prior to the original plan in order to start a new job at LMU Munich and has then not worked on the third objective of the project.
Second, the ER worked on chromatic localisations of K-theory. Based on results obtained shortly prior to the beginning of the funding period, the ER, in collaboration with Mathew, Meier and Tamme, proves a very general purity result for chromatic localisations of K-theory. There is an infinite family of chromatic localisations, one for each height n (which can be any natural number), and loosely speaking, the result is that the height-n-localised K-theory of a ring spectrum depends only on the height-(n-1)- and height-n-localisation of said ring spectrum. All Milestones proposed in this project are immediate special cases of this result and therefore fully resolved. A preprint about these results appeared on the arXiv during the first part of the funding period and is currently submitted for publication.
The ER presented his findings at several research seminars (Melbourne, Berlin, Warwick, MIT, Chicago). Due to the pandemic, further plans for public outreach were unfortunately cancelled.
The ER terminated the project 8 months prior to the original plan in order to start a new job at LMU Munich and has then not worked on the third objective of the project.
During the fellowship, new insight was gained in regards to computations in algebraic K-theory using the circle-dot ring, as well as to chromatic localisations of algebraic K-theory. Prior to the fellowship, only very little was known in these directions: Except for some very special cases, formulas for the circle-dot ring in many examples of interest were not available, and beyond the height 1 case, nothing was known about the question to what extent chromatic localisation of K-theory depend only on chromatic localisations of the rings of which one takes the K-theory. Now, this question is fully answered, and a formula for the circle-dot ring (which is indeed useable) is available in many (but not all) instances. Many more calculations than the ones proposed in the project can now be attacked using these new methods. The results about chromatic localisations of K-theory have subsequently been used by other researchers in regards to the question to what extend algebraic K-theory increases chromatic complexity.