No Arabic abstract
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 the polynomials in $mathbb{C}[u]$. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} Consequently our result in this short paper guarantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem, the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
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 $R cong_k k[X_1,..., X_n]$.
We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.
The proximinality of certain subspaces of spaces of bounded affine functions is proved. The results presented here are some line
Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal $S$ resolution of $R$ at degree $i$. In this paper we prove that $t_nleq t_1+T_{n-1}$, for all $n$ and as a consequence, we show that for Gorenstein algebras of codimension $h$, the subadditivity of maximal shifts $T_i$ in the minimal resolution holds for $i geq h-1$, i.e, we show that $T_i leq T_a+T_{i-a}$ for $igeq h-1$.
We consider a smooth Poisson affine variety with the trivial canonical bundle over complex numbers. For such a variety the deformation quantization algebra A_h enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A_h intrinsic to A_h. We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra A_h is free if and only if the corresponding Poisson structure is unimodular.