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In a manner analogous to the non-graded case, one may define the graded Grothendieck group K i gr ( A ) for each i ≥ 0. Each of these groups becomes a Z [ x, x - 1 ] -module, via the suspension operation.
So Hazrat's conjecture, slightly rephrased, asserts that the graded K 0 (viewed with the Z [ x, x - 1 ] -module structure induced by the suspension operation), together with the natural order and position of the regular module, is a complete graded isomorphism invariant for the collection of all Leavitt path algebras over row-finite graphs.
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