Non-Commutative Witt Vectors, Characteristic Map and Pairing

For a commutative ring \(A\), the ring of Witt vectors \(W(A)\) has been useful in number theory, algebraic topology, and algebraic K-theory. Classical results of Almkvist, Stienstra, Grayson, and Weibel demonstrate that the characteristic map bridges the ring \(\operatorname{End}_0(R)\) to its \(t\)-adic completion \(W(A)\), and the nil groups \(\operatorname{Nil}_0(A)\) are modules over \(W(A)\). Recently Dotto, Krause, Nikolaus, and Patchkoria (DKNP) gave an algebraic description of Witt vectors for a non-commutative ring \(R\). Generalizing this framework to non-commutative \(R\) strips away significant algebraic structure: \(\operatorname{End}_0(R)\) is reduced to an abelian group with only an external product, and the foundational arguments about the interaction of Frobenius, Verschiebung, and the pairings rely on the Cayley-Hamilton theorem, which breaks down in the non-commutative setting. This series will introduce these new definitions of non-commutative Witt vectors and explore an alternate approach that recovers their properties from the K-theoretic properties of \(\operatorname{End}_0(R)\).