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For the ordinary, commutative affine algebraic geometry, the basic object is the polynomial algebra in variables. In the noncommutative situation, we take the matrix polynomial algebra as our basic object. That is: Let be an -matrix. Then the matrix polynomial algebra is the matrix polynomial algebra generated by the idempotents together with the matrix variables for .Notice that we use the commutative polynomial -algebras on the diagonal. This is not the natural free object in the category, but we use it because it is simpler to give a (naive) geometric interpretation. Read more...
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