Takeshi Torii

Bio: Takeshi Torii is an academic researcher from Okayama University. The author has contributed to research in topics: Cohomology & Mathematics. The author has an hindex of 5, co-authored 25 publications receiving 82 citations. Previous affiliations of Takeshi Torii include Fukuoka University.

On degeneration of one-dimensional formal group laws and applications to stable homotopy theory

Abstract: In this note we study a certain formal group law over a complete discrete valuation ring F [[ u n -1 ]] of characteristic p 0 which is of height n over the closed point and of height n -1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H n -1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n -1 of H n -1 . We show that the automorphism group S n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S n -1 to the cohomology of S n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

Comparison of Morava E-theories

Abstract: We show that the nth Morava E-cohomology group of a finite spectrum with action of the nth Morava stabilizer group can be recovered from the (n + 1)st Morava E-cohomology group with action of the (n + 1)st Morava stabilizer group.

Topology of the moduli of representations with Borel mold

Abstract: We give descriptions of the moduli of representations with Borel mold for free monoids as fibre bundles over the configuration spaces. By using the associated Serre spectral sequences, we study the cohomology rings of the moduli. Also we calculate the virtual Hodge polynomials of them.

On quasi-categories of comodules and Landweber exactness

Abstract: In this paper we study quasi-categories of comodules over coalgebras in a stable homotopy theory. We show that the quasi-category of comodules over the coalgebra associated to a Landweber exact S-algebra depends only on the height of the associated formal group. We also show that the quasi-category of E(n)-local spectra is equivalent to the quasi-category of comodules over the coalgebra A\otimes A for any Landweber exact S_(p)-algebra A of height n at a prime p. Furthermore, we show that the category of module objects over a discrete model of the Morava E-theory spectrum in the K(n)-local discrete symmetric G_n-spectra is a model of the K(n)-local category, where G_n is the extended Morava stabilizer group.

Every K(n)-local spectrum is the homotopy fixed points of its Morava module

Abstract: Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).

Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2

Abstract: Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E∞-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E∞-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E∞-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava’s forms of K-theory into a sheaf of E∞-ring spectra.

Commutativity conditions for truncated Brown-Peterson spectra of height 2

Abstract: An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized trun- cated Brown-Peterson spectra of height 2 as E_\infty-ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level \Gamma_1(3)-structure.

Constructing the determinant sphere using a Tate twist

Abstract: Following an idea of Hopkins, we construct a model of the determinant sphere $S\langle det \rangle$ in the category of $K(n)$-local spectra. To do this, we build a spectrum which we call the Tate sphere $S(1)$. This is a $p$-complete sphere with a natural continuous action of $\mathbb{Z}_p^\times$. The Tate sphere inherits an action of $\mathbb{G}_n$ via the determinant and smashing Morava $E$-theory with $S(1)$ has the effect of twisting the action of $\mathbb{G}_n$. A large part of this paper consists of analyzing continuous $\mathbb{G}_n$-actions and their homotopy fixed points in the setup of Devinatz and Hopkins.