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Grothendieck Group of Projective modules Let A be a ring and M be a module

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GROTHENDIECK CONSTRUCTION

All rings in this note have identity 1 and all the modules are left unitary modules.

1. Grothendieck Group of Projective modules

Let A be a ring and M be a module. M is said to be a free module if M ∼= AI for some index set I. A module P is said to be finitely generated projective if there exists an A-module Q so that P ⊕ Q ∼= An for some n ≥ 1. The set of isomorphism classes of finitely generated projective A modules is denoted by P(A) whose elements are denoted by [P ].

On P(A), we define

[P ] + [P0] = [P ⊕ P0].

Then (P(A), +) becomes an abelian semigroup.

Definition 1.1. The Grothendieck group of A denoted by K0(A) is defined to be K(P(A)), where K is the Grothendieck construction defined below.

Let S be an abelian semigroup. Consider a category S whose objects and morphisms are given as follows. Objects of S are semigroup homomorphisms ψ : S → G where G is an abelian group. A morphism from ψ : S → G to ψ0 : S → G0 consists of a group homomorphism f : G → G0 so that ψ0 = f ◦ ψ. The universal object (if exists) of the category S is denoted by ψS : S → K(S) and called the Grothendieck construction of S.

The Grothendieck construction is unique up to isomorphisms.

Let us use two different ways to construct ψS : S → K(S).

Let F (S) be the free abelian group generated by elements of P(S) and R(S) be the subgroup of F (S) generated by s + s0− (s ⊕Ss0). The quotient group F (S)/R(S) is denoted by K(S). We define ψ : S → K(S) to be the canonical map s 7→ [s], where [s] is the equivalent class of s.

Lemma 1.1. The construction ψS : S → K(S) given above is the Grothendieck construc- tion of S.

Proof. Let ϕ : S → G be a semigroup homomorphism with G a group. Given x =P

inisi ∈ F (S), we define u(x) =P

iniϕ(si). Then u : F (S) → G is a group homomorphism so that u(s) = ϕ(s) for all s ∈ S. Let s, s0, s +Ss0 be elements of S. Using the fact that ϕ is a semigroup homomorphism, we have

u(s + s0− (s +Ss0)) = u(s) + u(s0) − u(s +Ss0)

= ϕ(s) + ϕ(s0) − ϕ(s +Ss0)

= ϕ(s) + ϕ(s0) − (ϕ(s) + ϕ(s0))

= 0.

This shows that u(R(S)) = {0}. Hence u induces a well-defined group homomorphism f : K(S) → G

sending [s] → u(s). We see f ◦ ψS(s) = f ([s]) = ϕ(s) for all s. Then f ◦ ψS= ϕ.

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2 GROTHENDIECK CONSTRUCTION

Assume that f0 : K(S) → G is another group homomorphism so that f0◦ ψS = ϕ. Then f0([s]) = ϕ(s) for all s ∈ S. This shows that f0([s]) = f ([s]) for all s ∈ S. Since {[s] : s ∈ S}

generates K(S), we find f0 = f. 

Now, let us gives another construction. Let

∆ : S → S × S be the diagonal map s 7→ (s, s). Define

K(S) = (S × S)/ Im ∆.

Denote elements of K(S) by [s, t]. Let us define an addition [s, t] + [s0, t0] = [s + s0, t + t0].

We can check (K(S), +) forms a group; [s, s] is the (additive) identity element of K(S) and [t, s] is the (additive) inverse of [s, t]. Therefore elements of K(S) can be represented by [s, 0] − [t, 0]. Define αS : S → K(S) by s 7→ [s, 0].

Lemma 1.2. The semigroup homomorphism αS : S → K(S) is the Grothendieck construc- tion of S.

Proof. Let ϕ : S → G be a semigroup homomorphism with G an abelian group. We define f : K(S) → G

by f ([s, t]) = ϕ(s) − ϕ(t). Let us check this map is well-defined.

If (s, t) − (s0, t0) ∈ Im ∆, then s − s0= t − t0. Hence s + t0 = t + s0. Since ϕ is a semigroup homomorphism, ϕ(s) + ϕ(t0) = ϕ(t) + ϕ(s0). Then ϕ(s) − ϕ(t) = ϕ(s0) − ϕ(t0). This shows that f ([s, t]) = f ([s0, t0]) whenever [s, t] = [s0, t0].

For each s ∈ S, f ◦ αS(s) = f ([s, 0]) = ϕ(s). Hence f ◦ αS = ϕ.

If f0 : K(S) → G is another group homomorphism so that f0◦ αS = ϕ, then f0([s, 0]) = ϕ(s) for all s ∈ S. Hence f0([s, t]) = ϕ(s) − ϕ(t) = f ([s, t]) for all [s, t] ∈ K(S). We find f0 = f.

We conclude that αS : S → K(S) is the universal object in the category S. Hence

αS : S → K(S) is the Grothendieck construction. 

Let X be a topological space and C(X) be the ring of complex-valued continuous functions on X. The space of global sections Γ(E) of a vector bundle π : E → X is a C(X)-module.

If X is compact Hausdorff and E has finite rank, Γ(E) is a finitely generated projective C(X)-module. In fact, we have the following theorem:

Theorem 1.1. (Swan) The global section functor

(1.1) Γ : Vect(X) → PC(X)

is an equivalence of category, where Vect(X) is the category of complex vector bundles of finite ranks on X and PC(X) is the category of finitely generated projective modules over C(X). The global section functor Γ is defined as follows. A finite rank vector bundle E is sent to its global section Γ(E) and a morphism ϕ : E → F of vector bundles is sent to Γ(ϕ) : Γ(E) → Γ(F ), where Γ(ϕ)(s) = ϕ ◦ s.

Hence it is very natural to define the K-theory of the compact Hausdorff space X by K0(X) = K0(C(X)).

Let Vect(X) be the set of isomorphism classes of complex vector bundles of finite ranks on a topological space X and define a binary operation on Vect(X) by [E] + [F ] = [E ⊕ F ].

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GROTHENDIECK CONSTRUCTION 3

Then (Vect(X), +) is an abelian semigroup. If X is compact Hausdorff, by the equivalence of categories (1.1), we obtain an isomorphism of abelian semigroups

Γ : Vect(X) → P(C(X)), [E] 7→ [Γ(E)].

This implies that

K0(X) = K(Vect(X)),

where K(Vect(X)) is the Grothendieck construction of the abelian semigroup Vect(X). This observation leads us to define K-theory on any topological space X as follows.

Definition 1.2. Let X be a topological space and Vect(X) be the set of isomorphism classes of complex vector bundles of finite rank on X. Define the (topological) K0-group of X by

K0(X) = K(Vect(X)).

In algebraic geometry, we have similar consideration. Let A be the coordinate ring of an affine variety X. The category of finitely generated projective modules over A is denoted by PA and the category of algebraic vector bundles (the locally free sheaves of OX-modules) of finite ranks on X is denoted by V(X).

Theorem 1.2. (Serre) We have an equivalence of categories

(1.2) Γ : V(X) → PA

where Γ is the global section functor on X.

Similarly, we set K0(X) = K0(A). If we denote V(X) the isomorphism classes of objects in V(X), then V(X) has an abelian semigroup structure so that it is isomorphic to the abelian semigroup PA. We find K0(X) = K(V(X)).

In general, let X be a scheme and V(X) be the category of locally free sheaves of finite ranks on X. Denote V(X) the semigroup consisting of isomorphism classes of objects in V(X).

Definition 1.3. The Grothendieck group of a scheme X is defined to be K0(X) = K(V(X)).

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