Introduction to the Almost Suslin Matrices-The Key Lemma

Title

Introduction to the Almost Suslin Matrices-The Key Lemma

Author

1. Dr. Reema Khanna, Professor, K.J. Somaiya College of Science and Commerce, Maharashtra, India

Abstract

The Suslin matrices exhibit interesting properties which lead to the Local Global Principle and other important results. In this paper, we follow the Suslin construction by defining a variant of the Suslin matrices which are named the Almost Suslin Matrices S_r^(λ,μ) (v,w), w.r.t. a pair of parameters λ,μ∈R. The Almost Suslin matrices, despite the parameters, satisfy properties which can be used to derive stronger results. We define a special product ⟨⟨v,w⟩⟩ for v,w∈M_1(r+1)  (R). This helps us to derive results which finally leads us to our goal of proving the Key Lemma.

Keywords

Conclusion

We have proved the Key Lemma which describes the action of the Almost Special Unimodular Vector group using the top bottom matrices. We will be using the Key Lemma to prove a Fundamental Property for Almost Suslin matrices, parallel to the one proved for Suslin Matrices. This will lead us to the equivalence of the Key Lemma and the Fundamental Property.

Author Contrubution

The author handled all aspects of the study, including its design, data collection, analysis, and manuscript preparation.

Funding

No specific financial support from public, commercial, or non-profit funding agencies was received for this research.

Conflict of Interest

There are no conflicts of interest to report from any of the authors.

Data Sharing Statement

Not applicable

Software And Tools Use

Not applicable

Acknowledgements

I am grateful for the expertise and help provided by all who contributed to this study and manuscript, and for the comments from anonymous reviewers.