| 研究概要 |
I visited the University of Copenhagen in the spring semester again this year and continued my collaboration with Professor Ib Madsen on real algebraic K-theory. The main focus of our research was on laying the foundations of the theory; in particular, finding the appropriate notions of simplicial categories with duality and their real realization turned out to be rather time-consuming. In the fall semester, Marco Schlichting visited Nagoya for one month and gave four 90 minutes lectures on his theory of higher Grothendieck-Witt groups of exact categories with duality. Using his additivity theorem for Grothendieck-Witt theory, the real additivity theorem for real algebraic K-theory proved by myself and Madsen last year, and a new group-completion result proved by Madsen's student Krisitian Moi, we were able to prove that for every exact category with duality, Schlichting's higher Grothendieck-Witt group in degree p is canonically isomorphic to the real algebraic K-group in bidegree (p,0). This is surprising result, which begs for an explanation. Indeed, the analogous result for the corresponding direct sum versions of the two theories is generally false, unless the exact category with duality is enriched in Z[1/2]-modules. In addition, I also revised my paper "The big de Rham-Witt complex" and resubmitted it for publication in Acta Mathematica.
The results mentioned above were presented in numerous invited lectures.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
The level of achievement is very satisfactory. Real algebraic K-theory is receiving much attention, in particular from research groups in the U.S., U.K., and Germany.
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| 今後の研究の推進方策 |
In the next year, I hope to complete two long joint papers with Madsen, the first of which will contain the definition and fundamental theorems for real algebraic K-theory, including the real additivity theorem, and the second of which will contain the proof of our theorem that, for every split-exact category with duality, the real direct sum K-theory and the real algebraic K-theory agree. The proof of the latter theorem, which does not require the assumption that the category be enriched in Z[1/2]-modules, is rather intricate and is based on Quillen's proof of the analogous statement for algebraic K-theory of split-exact categories.
I further intend to begin to extend real algebraic K-theory from exact categories with duality to exact infinity-categories with duality, and I plan to investigate this extension in joint work with David Gepner at Purdue University.
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