arXiv subjects
Publications and source records attributed to Satoshi Mochizuki.
In this paper, we show local Gersten's conjecture for regular system of parameters. As its consequence we obtain Gersten's conjecture for a commutative regular local ring and smooth over a commutative discrete valuation ring.
In this short note, for a morphism of Waldhausen categories $f\colon \mathbb{A} = (\mathcal{A} ,w_{\mathbb{A}}) \to \mathbb{B} = (\mathcal{B},w_{\mathbb{B}})$, we will define $\operatorname{Cone} f$ to be a Waldhausen category. There exists the canonical morphism of Waldhausen categories $κ_f\colon \mathbb{B}\to \operatorname{Cone} f$. We will show that the sequence $\mathbb{A}\overset{f}{\to}\mathbb{B}\overset{κ_f}{\to}\operatorname{Cone}f$ induces fibration sequence of spaces $K(\mathbb{A})\overset{K(f)}{\to}K(\mathbb{B})\overset{K(κ_f)}{\to} K(\operatorname{Cone} f)$ on connective $K$-theory. Moreover we will define a notion of strictly derivable Waldhausen categories and define non-connective $K$-theory for strictly derivable Waldhausen categories.
In this article, we propose noncommutative versions of Tate conjecture and Hodge conjecture. If we consider these conjectures for a dg-category of perfect complexes over a certain schemes $X$, then they are equivalent to the classical Tate and Hodge conjectures for $X$ respectively. We also propose a strategy of how to prove these conjectures by utilizing a version of motivic Bass conjecture.
The purpose of this article is to show a version of dévissage theorem of non-connective $K$-theory. Our theorem contains Quillen's dévissage theorem, Waldhausen's cell filtration theorem and theorem of heart as special cases. In this sense, we give an affirmative answer to Thomason's problem in Thomason-Trobaugh's paper. We introduce the notions of cell structures and dévissage spaces and our main theorem states a structure of non-connective $K$-theory of dévissage spaces in terms of non-connective $K$-theory of heart of cell structures. A specific feature in our proof is 'motivic' in the sense that properties of $K$-theory which we will utilze to prove the theorem are only categorical homotopy invariance, localization and co-continuity. On the other hands, it is well-known that the analogue of the dévissage theorem for $K$-theory does not hold for Hochschild homology theory. In this point of view, we could say that dévissage theorem is not 'motivic' over dg-categories. To overcome this dilemma, the notion of dévissage spaces should not be expressed by the language of dg-categories. First three sections are devoted to the foundation of our model of stable $(\infty,1)$-categories which we will play on to give a description of dévissage spaces.
The purpose of this article is to clarify the question what makes motives $\mathbb{A}^1$-homotopy invariance. we give construction of the stable model category of nilpotent invariant motives $\mathcal{M}ot_{\operatorname{dg}}^{\operatorname{nilp}}$ and define the nilpotent invriant motives associated with schemes and relative exact categories. For a noetherian scheme $X$, there are two kind of motives associated with $X$ in the homotopy category $\operatorname{Ho}(\mathcal{M}ot^{\operatorname{nilp}}_{\operatorname{dg}})$, namely $M_{\operatorname{nilp}}(X)$ and $M_{\operatorname{nilp}}'(X)$. In general $M_{\operatorname{nilp}}(X)$ is not isomorphic to $M'_{\operatorname{nilp}}(X_{\operatorname{red}})$. But there exists a canonical isomorphism $M_{\operatorname{nilp}}'(X)\simeq M_{\operatorname{nilp}}'(X_{\operatorname{red}})$ and if $X$ is regular noetherian separated, $M(X)$ is canonically isomorphic to $M'(X)$.
This article is the extended notes of my survey talk of Gersten's conjecture given at the workshop "Bousfield classes form a set: a workshop in a memory of Tetsusuke Ohkawa" at Nagoya University in August 2015. In the last section, I give an explanation of my recent work of motivic Gernsten's conjecture.
The goal of this note is to give a variant of the generic fibration theorem for Waldhausen K-theory without assuming the factorization axiom.
The main theorem in this paper is that the base change functor from a noetherian abelian category to its noetherian polynomial category induces an isomorphism on K-theory. The main theorem implies the well-known fact that A^1-homotopy invariance of K'-theory for noetherian schemes.
In this paper, we give a sufficient condition which makes the total complex of a cube exact. This can be regarded as a variant of the Buchsbaum-Eisenbud theorem which gives a characterization of what makes a complex of finitely generated free modules exact in terms of the grade of the Fitting ideals of boundary maps of the complex.
We prove that negative K-groups of small abelian categories are trivial.
We introduce a delooping model of relative exact categories. It gives us a condition that the negative K-group of a relative exact category becomes trivial.
The main objective of this paper is to determine generators of the topological filtrations on the higher K-theory of a noetherian commutative ring with unit A. We introduce the concept of Koszul cubes and give a comparison theorem between the K-theory of Koszul cubes with that of topological filtrations.
The main objective of this paper is to propose a definition of non-connective K-theory for a wide class of relative exact categories which, in general, do not satisfy the factorization axiom and confirm that it agrees with the non-connective K-theory for exact categories and complicial exact categories with weak equivalences. The main application is to study the topological filtrations of non-connective K-theory of a noetherian commutative ring with unit in terms of Koszul cubes.
The main theorem in this paper is that the base change functor from an abelian category $\cA$ to its polynomial category in the sense of Schlichting $-\otimes_{\cA}\bbZ[t]:\cA \to \cA[t]$ induces an isomorphism on their $K$-theories if $\cA$ is noetherian and has enough projective objects. The main theorem implies the well-known fact that $\mathbb{A}^1$-homotopy invariance of $K'$-theory for noetherian schemes.
The main objective of this paper is to generalize a notion of Koszul resolutions and charcterizing modules which admits such a resolution. We turn out that for a noetherian ring $A$ and a coherent $A$ module $M$, $M$ has a two dimensional generalized Koszul resolution if and only if $M$ is a pure weight two module in the sense of \cite{HM09}.
We introduce a notion of quasi-weak equivalences associated with weak-equivalences in an exact category. It gives us a delooping for (idempotent complete) exact categories and a condition that the negative $K$-group of an exact category becomes trivial.
In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i:Y -> X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O_X-Modules supported on Y. The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh non-connected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting non-connected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes.
The purpose of this article is to prove that Gersten's conjecture for a commutative regular local ring is true. As its applications, we will prove the vanishing conjecture for certain Chow groups, generator conjecture for certain $K$-groups and Bloch's formula for absolute case.