Susumu Oda
The Cancellation Problem for Affine Spaces is settled affirmatively, that is, it is proved that : Let $ k $ be an algebraically closed field of characteristic zero and let $n, m \in \mathbb{N}$. If $R[Y_1,..., Y_m] \cong_k k[X_1,..., X_{n+m}]$ as $k$-algebras, where $Y_1,..., Y_m, X_1,..., X_{n+m}$ are indeterminates, then $R \cong_k k[X_1,..., X_n]$.
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In this article, we shall discuss the solution to the Zariski Cancellation Problem in positive characteristic, various approaches taken so far towards the possible solution in characteristic zero, and several other questions related to this problem.
We show that the Cancellation Conjecture does not hold for the affine space A^3_k over any field k of positive characteristic. We prove that an example of T. Asanuma provides a three-dimensional k-algebra A for which A is not isomorphic to k[X_1,X_2,X_3] although A[T] is isomorphic to k[X_1, X_2, X_3, X_4].
Algebraic families of higher dimensional $\mathbb{A}^{1}$-contractible affine varieties non-isomorphic to affine spaces
该研究在特征零域上构造了维数不小于4的、与仿射空间不同构的高维$\mathbb{A}^1$-可缩仿射簇代数族，证明其是Zariski消去问题及广义消去问题的反例。
Suslin's cancellation conjecture on smooth real affine varieties with few real points
本文研究实轨迹为空或上同调维数小的光滑实仿射簇上的Suslin消去猜想。
In this article we extend a cancellation theorem of D. Wright to the case of affine normal domains. We shall show that if $A$ is an algebra over a Noetherian normal domain $R$ containing a field $k$ and if $A[T ] = R^{[3]}$, then $A = R^{[2]}$ if and only if $A[T]$ has a variable of the form $bT^n - a$ for some $a, b \in A$ with $n \ge 2$ and $ch(k) \nmid n$.
If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{θ_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{θ_{0}} (a_{0},a_{R})$ for the generalized Vaserstein symbol whenever $R$ is a smooth affine algebra over a perfect field $k$ with $char(k) \neq 2$ such that $-1 \in {k^{\times}}^{2}$. This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension $d \geq 4$ over algebraically closed fields of characteristic $\neq 2$. As a consequence, we prove that any projective module of rank $3$ with trivial Chern classes o…
The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained.
We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.
Suslin's cancellation conjecture in the smooth case
该研究在代数闭域k且d!可逆的条件下，证明了光滑d维仿射簇上秩d-1的稳定同构向量丛为同构，解决了Suslin的经典消去猜想。
Let A be a ring of dimension d and let P be a projective A-module of rank d. We prove that if for every finite extension R of A, R^d is cancellative, then P is cancellative. This gives an alternate proof of Bhatwadekar's result: every projective module of rank d over an affine algebra of dimension d over a C_1-field of characteristic 0 is cancellative. Let P be a projective module of rank d-1 over an affine agebra of dimension d over an algebraically closed field. Then, it is not known if P is cancellative. We prove some results in this direction also.
We have proved the following Problem:{\it Let $R$ be a $\mathbb{C}$-affine domain, let $T$ be an element in $R \setminus \mathbb{C}$ and let $i : \mathbb{C}[T] \hookrightarrow R$ be the inclusion. Assume that $R/TR \cong_{\mathbb{C}} \mathbb{C}^{[n-1]}$ and that $R_T \cong_{\mathbb{C}[T]} \mathbb{C}[T]_T^{[n-1]}$. Then $R \cong_{\mathbb{C}} \mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {\it Let $A:=\mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be…
In this paper we shall show that when k is a field of positive characteristic the affine space A^n_k is not cancellative for any n greater than 2.
We prove that cancellation of reflexive modules over affine rings holds under some restrictions. We construct examples to show that this is false even over polynomial rings without the extra assumptions.
We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove that stably free $R$-modules of rank $2$ are free if and only if a Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ is trivial.
Let $K$ be an arbitrary field of characteristic 0, and $\Aff^n$ the $n$-dimensional affine space over $K$. A well-known cancellation problem asks, given two algebraic varieties $V_1, V_2 \subseteq \Aff^n$ with isomorphic cylinders $V_1 \times \Aff^1$ and $V_2 \times \Aff^1$, whether $V_1$ and $V_2$ themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of $\Aff^{n+1}$) cylinders $V_1 \times \Aff^1$ and $V_2 \times \Aff^1$, are $V_1$ and $V_2$ equivalent under an automorphism of $\Aff^n$? We call this stable equivalenc…
An Algebraic characterization of the affine three space in arbitrary characteristic
本文给出任意特征代数闭域上仿射三维空间的代数刻画，并借此研究一类正特征环是否同构于多项式环，同时利用ML及ML*不变量证明了二维多项式环强消去的一个特殊情形。
Let $K$ be an algebraically closed field of arbitrary characteristic. Let $A$ be an affine domain over $K$ with transcendence degree 1 which is not isomorphic to $K[x]$, and let $B$ be a domain over $K$. We show that the AK invariant distributes over the tensor product of $A$ by $B$. As a consequence, we obtain a generalization of the cancellation theorem of S. Abhyankar, P. Eakin, and W. Heinzer.
Generalized Zariski cancellation for Brieskorn--Pham varieties
本文证明复 Brieskorn–Pham 簇在乘以任意具有光滑点的分离复簇后同构则原簇同构，结合唯一奇点簇的消去定理与指数刚性定理。
Methods in complete intersections in corank one
本文探讨了在余秩一情况下，如何将完全交集理论的方法应用于稳定自由模的消去定理，指出需要改进以使该方法有效，并提出了相关假设及其后果。
On Generalised Danielewski Surfaces over fields of arbitrary characteristic
本文研究任意特征域上广义Danielewski曲面的指数映射，刻画其Makar-Limanov不变量与Derksen不变量，完整分类曲面间同构，并构造消去问题的反例子族。
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