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Graded Rings and Graded Grothendieck Groups Graded Rings and Graded Grothendieck Groups R. Hazrat Centre for Research in Mathematics University of Western Sydney Australia October 13, 2013 ... inside of a book is there whether you read it or not. Even if nobody ever reads it, it’s there, doing its work. The Ground Beneath Her Feet Salman Rushdie 1 Contents 1 Graded Rings And Graded Modules 7 1.1 Graded rings ........................................ 7 1.2 Graded modules ...................................... 20 1.3 Grading on matrices .................................... 32 1.4 Graded division rings .................................... 40 1.5 Strongly graded rings and Dade’s theorem ........................ 46 1.6 Grading on graph algebras ................................. 50 1.7 The graded IBN and graded type ............................. 60 1.8 The graded stable rank .................................. 62 2 Graded Morita Theory 64 2.1 First instance of the graded Morita equivalence ..................... 64 2.2 Graded generators ..................................... 66 2.3 General graded Morita equivalence ............................ 68 3 Graded Grothendieck Groups 75 gr 3.1 The graded Grothendieck group K0 ........................... 76 gr 3.2 K0 from idempotents ................................... 82 gr 3.3 Relative K0 -group ..................................... 88 gr 3.4 K0 of non-unital rings ................................... 90 gr 3.5 K0 is a pre-ordered module ................................ 92 gr 3.6 K0 of graded division rings ................................ 94 gr 3.7 K0 of graded local rings .................................. 98 gr 3.8 K0 of Leavitt path algebras ................................100 3.9 Symbolic dynamics .....................................106 gr 3.10 K1 -theory .........................................109 4 Graded Picard Groups 112 4.1 Picgr of a graded commutative ring ............................112 4.2 Picgr of a graded noncommutative ring ..........................115 2 gr 5 Graded Ultramatricial Algebras, Classification via K0 120 5.1 Graded matricial algebras .................................120 gr 5.2 Graded ultramatricial algebras, classification via K0 ..................124 6 Graded Versus Non-graded (Higher) K-Theory 129 gr 6.1 K∗ of positively graded rings ...............................130 6.2 The fundamental theorem of K-theory ..........................136 gr 6.3 Relating K∗ to K∗ .....................................142 3 Introduction A bird’s eye view of the graded module theory over a graded ring gives an impression of the module theory with the added adjective “graded” to all its statements. Once the grading is considered to be trivial, the graded theory reduces to the usual module theory. So from this perspective, the graded module theory can be considered as an extension of the module theory. However, one aspect that could be easily missed from such a panoramic view is that, the graded module theory comes equipped with a shifting, thanks to being able to partition the structures and rearranging these par- titions. This adds an extra layer of structure (and complexity) to the theory. An sparkling example of this is the theory of graded Grothendieck groups which is the main focus of this monograph. The construction of the Grothendieck group of a ring is easy. Starting from a ring A with identity, the isomorphism classes of finitely generated projective (right) A-modules equipped with the direct sum forms an abelian monoid, denoted by V(A). The group completion of this monoid is denoted by K0(A) and is called the Grothendieck group of A. For a Γ-graded ring A, the graded isomorphism classes of graded finitely generated projective modules equipped with the direct sum forms an abelian monoid, denoted by Vgr(A). The shifting of graded modules (see §1.2.1) induces gr gr a Γ-module structure on V (A). The group completion of this monoid is denoted by K0 (A) and is called the graded Grothendieck group of A which has, consequently, a natural Z[Γ]-module structure. As we will see throughout this note, this extra structure carries a substantial information about the graded ring A. Starting from a graded ring A, the relation between the category of graded A-modules, Gr-A, the category of A-modules, Mod- A, and the category of modules over the ring of the zero homogeneous part, Mod- A0, is one of the main forces to study the graded theory. In the same way, the relations gr between K0 (A), K0(A) and K0(A0) are of interest. The motivation to write this note came from the recent activities which adopt the graded Grothendieck group as an invariant to classify the Leavitt path algebras [46, 47, 48, 76]. Surprisingly, not much is recorded about the graded version of the Grothendieck group in the literature, despite the fact that K0 has been used in many occasions as a crucial invariant, and there is a substantial amount of information about the graded version of other invariants such as (co)homology groups, Brauer groups, etc. The other surge of interest on this group stems from the recent activities on (graded) representation theory of Hecke algebras. In particular for a quiver Hecke algebra, its graded Grothendieck group is closely related to its corresponding quantised enveloping algebra. For this line of research see the survey [56]. This note tries to fill this gap, by systematically developing the theory of graded Grothendieck groups. In order to do this, we have to carry over and work out the details of known results in non-graded case to the graded setting, and gather important results scattered in research papers on the graded theory. The emphasis is on using the graded Grothendieck groups as invariants for classifications. 4 ∗ The group K0 has been successfully used in the theory of C -algebras to classify certain class ∗ of C -algebras. Building on the work of Brattelli, Elliott in [34] used the the pointed ordered K0- groups (called dimension groups) as a complete invariant for AF C∗-algebras. Another cornerstone of using K-groups for the classifications of a wider range of C∗-algebras was the work of Kirchberg and Phillips [77], who showed that K0 and K1-groups together are complete invariant for a certain type of C∗-algebras. The Grothendieck group considered as a module induced by a group action was used by Handelman and Rossmann [43] to give a complete invariant for the class of direct limits of finite dimensional, representable dynamical systems. Krieger [57] introduced (past) dimension groups as a complete invariant for the shift equivalence of topological Markov chains (shift of finite types) in symbolic dynamics. Surprisingly as we will see, Krieger’s groups can be naturally expressed by graded Grothendieck groups (§3.9). For two reasons we develop the theory for rings graded by abelian groups rather than an arbitrary groups, although most of the results could be carried over to non-abelian graded rings. One reason is that, by using the abelian grading, the presentation and proofs are much more transparent. Furthermore, in most applications of graded K0, the ring has an abelian grading (often times Z-grading). A brief outline. Let Γ be an (abelian) group and A be a Γ-graded ring. For any subgroup Ω ⊆ Γ, one can consider the natural Γ/Ω-grading on A (see Example 1.1.8). In the same manner, any Γ-graded A-module can be considered as Γ/Ω-graded module. These induce functors (see §1.2.8) U : GrΓ-A −→ GrΓ/Ω-A, Γ (−)Ω : Gr -A −→ Mod- AΩ, where GrΓ-A is the category of Γ-graded (right) A-modules, GrΓ/Ω-A is the category of Γ/Ω-graded A-modules and Mod- AΩ is the category of AΩ-modules. One aspect of the graded theory of rings is to investigate how these categories are related, and what properties can be lifted from one category to another. In particular when the subgroup Ω happen to be Γ or trivial group, respectively, we obtain the (forgetful) functors U : GrΓ-A −→ Mod- A, Γ (−)0 : Gr -A −→ Mod- A0. One aspect of this note is to study the interrelation between these categories. This has been done in Chapters 1 and 2. The rest of the note is devoted to the Grothendieck group, denoted by K0, of these category. The category of graded finitely generated projective A-modules, PgrΓ-A, is an exact category. Thus using Quillen’s machinery of K-theory [79], one can define graded K-groups, gr Γ Ki (A) := Ki(Pgr -A), for i ∈ N. Since the shifting of modules induces an auto-equivalence functor on GrΓ-A (§1.2.2), these K-groups carry a Γ-module structure. This extra structure is an emphasis of this note. In many important examples, in fact this shifting is all the difference between the graded Grothendieck group and the usual Grothendieck group, i.e., gr ∼ K0 (A)/h[P ] − [P (α)]i = K0(A), where P is a graded projective A-module, α ∈ Γ and P (α) is a shift module (§6, see Corollary 6.3.2). 5 gr In Chapter 3 we compute K0 for certain graded rings, such as graded local rings and (Leavitt) gr path algebras. We study the pre-ordering available on K0 and determine the action of Γ on this group. In Chapter 4 the graded Picard groups are studied and in Chapter 5 we prove that for the so called graded ultramatricial algebras, the graded Grothendieck group is a complete invariant. gr Finally in Chapter 6, we explore the relations between (higher) Ki and Ki, for the class of Z- graded rings. We describe a generalisation of the Quillen and van den Bergh theorems. The latter theorem uses the techniques employed in the proof of the fundamental theorem of K-theory, where the graded K-theory appears. For this reason we present a proof of the fundamental theorem in this chapter. Conventions. Throughout this note, unless it is explicitly stated, all rings have identities, ho- momorphisms preserve the identity and all modules are unitary.
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