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Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
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Morphisms in K-theory: comparison of two pictures
It seems that by $M_n(A)$ you denote two different objects: actual $M_n(A)$ -- which is the same as $\text{Hom}(A^n, A^n)$ -- and $A^n$ itself.
One can show that any finitely generated projective $A$ …