On the Surjectivity of Certain Maps III: The Unital Set Condition

Abstract

In this article, for generalized projective spaces with any weights, we prove four main theorems in three different contexts where the Unital Set Condition USC (Definition $2.8$) on ideals is further examined. In the first context we prove, in the first main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal $\mathcal{I}=\underset{i=1}{\overset{k}{\prod}}\mathcal{I}_k$ with a given factorization into mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq k$ where $\mathcal{I}$ satisfies the USC, using the key concept of choice multiplier hypothesis (Definition $4.10$) which is satisfied. In the second context, for a positive $k$, we prove in the second main Theorem $\Lambda$, the surjectivity of the reduction map $SP_{2k}(\mathcal{R})\rightarrow SP_{2k}(\frac{\mathcal{R}}{\mathcal{I}})$ of strong approximation type for a ring $\mathcal{R}$ quotiented by an ideal $\mathcal{I}$ which satisfies the USC. In the third context, for a positive integer $k$, we prove in the thrid main Theorem $\Omega$, the surjectivity of the map from special linear group of degree $(k+1)$ to the product of generalized projective spaces of $(k+1)$-mutually co-maximal ideals $\mathcal{I}_j,0\leq j\leq k$ associating the $(k+1)$-rows or $(k+1)$-columns, where the ideal $\mathcal{I}=\underset{j=0}{\overset{k}{\prod}}\mathcal{I}_j$ satisfies the USC. In the fourth main Theorem $\Sigma$, for a positive integer $k$, we prove the surjectivity of the map from the symplectic group of degree $2k$ to the product of generalized projective spaces of $(2k)$-mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq 2k$ associating the $(2k)$-rows or $(2k)$-columns where the ideal $\mathcal{I}=\underset{j=1}{\overset{2k}{\prod}}\mathcal{I}_j$ satisfies the USC. The answers to Questions [1.1,1.2,1.3] in a greater generality are not known.