arXiv subjects
Publications and source records attributed to Nobuaki Yagita.
Algebraic Morava K-theories are defined by Sechin,Vishik and others as quotients of algebraic cobordisms. On the other hand, the author had defined them as some (two degrees) cohomology theories. In this paper, we compare these theories.
We give many examples of non stable rationalities for projective approximations of classifying spaces. Here we use the new invariant by Benoit-Ottem, and also use the classical unramified cohomology..
In this paper, we study etale cohomologies of quadrics over R. An element in the etale cohomology is called algebraic, if it is in the image of the cycle map from the Chow ring. In this paper, we compute the etale cohomology of norm quadrics, and give examples which have many non-algebraic elements.
Let $G_k$ be a split algebraic group over $k$. Given a non-trivial $G_k$-torsor $G$, we consider the (twisted) flag variety $F=G/B_k$ for the Borel subgroup $B_k$ containing in $G_k$. The purpose of this paper is the study how to compute the Chow rings $CH^*(F)$ for various $F$ over some extension fields $K$ of $k$.
Let BG be the classifying space of an algebraic group over the complex field C. We compute a new stable rational invariant defined by the difference of two coniveau filtrations (by Benoist and Ottem) of a (projective) approximation for BG.
In this note we construct approximations by smooth projective varieties of some Eienberg-MacLane spaces in the $A^1$-homotopy category. Using these, we study the cycle maps from Chow rings to etale cohomology rings.
Let $X$ be an algebraic variety over $k$ such that $\bar X=X\otimes_k\bar k$ is cellular. We study torsion elements in the Chow ring $CH^*(X)$ which corresponds to $v_iy$ in the algebraic cobordism $Ω^*(\bar X)$ where $0\not=y\in CH^*(\bar X)/p$ and $v_i$ is the generator of $BP^*$ with $|v_i|=-2(p^i-1).$ In particular, we try to compute $CH^*(X)$ from $Ω^*(\bar X)$ when $X$ are twisted complete flag varieties.
Let $G$ be a compact Lie group and $T$ its maximal torus. In this paper, we try to compute $gr_γ^*(G/T)$ the graded ring associated with the gamma filtration of the complex $K$-theory $K^0(G/T)$ for $G=Spin(n)$. In particular, we give a counterexample for a conjecture by Karpenko when $G=Spin(17)$. The arguments for $E_7$ in $§11$ of the old version were not correct, and they are deleted in this version.
Let $G$ be a compact Lie group and $T$ its maximal torus. The composition of maps $ H^*(BG)\to H^*(BT) \to H^*(G/T)$ is zero for positive degree, while it is far from exact. We change $H^*(G/T)$ by Chow ring $CH^*(X)$ for $X$ some twisted form of $G/T$, and change $H^*(BG)$ by $CH^*(BG)$. Then we see that it becomes near to exact but still not exact, in general. We also see that the difference for exactness relates to the generalized Rost motive in $X$.
In this paper, I correct errors in my paper (Kodai Math. J. (2015)) about gamma filtrations for classifying spaces for abelian p-groups which are not elementary. We also extend Chetard's results for such 2-groups to p-groups for odd prime.
In this paper, we study the graded ring gr(X) defined by the K-theory of twisted flag variety X. In particular, Kunneth map from gr(R')(\otimes)gr(R') to gr(R) is studed explicitly for an original Rost motive R' and a generalized Rost motive R. Using this, we give an example that T(X)^2 is nonzero for the torsion ideal T(X) in the Chow ring CH(X).
Let G be a simply connected compact Lie group and T its maximal torus. We compute the graded ring gr_{gamma}(G/T) associated with the gamma filtration of the complex K-theory K(G/T). We use the Chow ring of the corresponding versal flag variety.
In this paper, we compute Chow rings of generically twisted (versal) complete flag varieties corresponding to simple Lie groups
Let BP be the p-completed classifying space of a p-group P with p-rank 2. For an odd prime p, by using stable homotopy splitting of BP, we study the decomposition of even dimensional parts of the integral cohomology and the Chow ring of BP.
Let G be an algebraic group corresponding to a compact Lie group. We study the restiction map from the Chow ring CH(BG) to the Wely group invariants CH(BT)^W. For example, we note that if G is simply connect and H(G) has torsion, then the restriction map is always not surjective.
We write down the mod 2 Chow ring of the classifying space of the central product of two dihedral groups of order 8. This Chow ring has nilpotent elements.
Let E be the extraspecial p-group of order p^3 and exponent p where p is an odd prime. We consider the mod p cohomology of summands in the stable splitting of p-completed classifying space BE. Moreover, we consider the stable splitting for some finite groups with Sylow p-subgroup E.
In this note we discuss some examples of non torsion and non algebraic cohomology classes for varieties over finite fields. The approach follows the construction of Atiyah-Hirzebruch and Totaro.