Graded rings and graded Grothendieck groups (Q2814155)

As the author explains, the motivation to write this book came from a recent direction of research, where the graded Grothendieck group was used as an invariant to classify Leavitt path algebras. The graded version of the Grothendieck group associated with a graded ring is discussed and studied, and then it is applied to certain classification problems.NEWLINENEWLINEThe book contains six chapters: 1. Graded rings and graded modules. 2. Graded Morita theory. 3. Graded Grothendieck groups. 4. Graded Picard groups. 5. Graded ultramatricial algebras, classification via \(K_0^{\mathrm{gr}}\). 6. Graded versus nongraded (higher) \(K\)-theory.NEWLINENEWLINEIn Chapter 1 basic results on graded rings are reviewed. The author deals only with gradings by abelian groups. Several examples are presented, including the Leavitt path algebras. In Chapter 2 the author discusses equivalences between categories of graded modules over two graded rings. Also, the relation with the equivalence of the associated categories of (non-graded) modules is studied. In Chapter 3 it is constructed and investigated the graded Grothendieck group \(K_0^{\mathrm{gr}}(A)\) of a \(\Gamma\)-graded ring \(A\), where \(\Gamma\) is an abelian group. The construction is performed as a completion of the monoid of isomorphism classes of finitely generated projective graded \(A\)-modules. It is showed that \(K_0^{\mathrm{gr}}(A)\) has a structure of a \(\mathbb{Z}[\Gamma]\)-module, with the \(\Gamma\)-action induced by the shift on graded modules. Explicit computations of the graded Grothendieck group are done in the case where \(A\) is a graded division ring, a graded local ring, or a Leavitt path algebra. In Chapter 4 it is considered the (graded version) of the Picard group of a graded ring. The commutative case is considered first, and several results are extended from the non-graded to the graded case. Extending concepts and results known in the non-graded case, the author defines graded ultramatricial algebras over a graded field in Chapter 5, and uses the graded Grothendieck group to classify them. In Chapter 6, the \(K\)-group \(K_i^{\mathrm{gr}}(A)\) for any \(i\geq 0\) are defined, and their connection to the ungraded \(K\)-groups is investigated for certain graded rings graded by \(\mathbb{Z}\).NEWLINENEWLINEThe book is useful to researchers in ring theory and to graduate students.