Remark: I’ve rewritten the material in this post into a new blog post, Algebraic K-Theory revisited; the Grothendieck ring of Projective Space, with more detail. However, for a first encounter, the material below is also fine.
In this post, we view the Grothendieck group of a scheme over a field , as a functor (from the category of schemes over to the category of abelian groups) applied to the scheme . Our goal is to see calculations for the simplest examples of the Grothendieck group, and what machinery is used throughout. Citation for this machinery will be included, as well as precise statements for the results used throughout but, we often omit proof with our goal in mind.
Several motivations can be given as to why one would study the Grothendieck group. The original motivation, I think, being: it aids in studying extensions of vector bundles over a scheme .
Remark: this definitely was not the original motivation.
Recall there is an equivalence of categories between vector bundles over , and locally free sheaves of finite rank on (the equivalence is given as so: for every vector bundle we can associate the sheaf given by the presheaf where, for each open , we define – it can be checked this is a locally free sheaf of finite rank – and conversely for every locally free sheaf we construct the vector bundle as the relative Spec of the symmetric algebra of – note that we could construct the vector bundle using instead). We can define an extension of a sheaf by a sheaf to be a sheaf which fits in a short exact sequence
Any two extensions which admit a map from one to the other, commuting with the maps from and to , give a commutative diagram which we may apply the 5-lemma or the Snake lemma to show . One can hope to classify all such extensions by classifying each distinct short exact sequence. Sounds like a big task, but a start is with the
Definition 1: the functor applied to a scheme is called the Grothendieck group of . For any scheme , the group is defined to be the free abelian group on isomorphism classes of locally free sheaves of finite rank on quotiented by the subgroup generated by the relations for every short exact sequence .
Immediate from this definition is that and . We remark the Grothendieck group satisfies a, slightly irrelevant for our purposes, equivalent characterization as the object satisfying the following
Universal Mapping Property 2: for any functor which factors additively through the category of locally free sheaves of finite rank over a scheme, and for every scheme , there is a unique morphism of abelian groups .
We could similarly define by replacing the category of locally free sheaves of finite rank by coherent sheaves over a scheme . Then for any scheme , there is a homomorphism which maps the class of a locally free sheaf to itself . For smooth, quasi-projective schemes this is in fact an isomorphism (the inverse given by for a locally free finite rank resolution ). Since all of the schemes we will consider below are smooth and quasi-projective, we can work with either group.
Below we compute, for a field , in proposition 3, in proposition 6 proposition 10 and proposition 12, and in proposition 8.
Proposition 3: .
Proof. Since any locally free sheaf over is a finite dimensional vector space, and any two finite dimensional vectorspaces are isomorphic if and only if they have the same dimension, we can define an isomorphism directly by .
Our next goal will be to compute the Grothendieck group of all higher dimensional affine spaces, which we do in three distinct ways throughout this post. The following lemma of Serre gives a useful characterization of vector bundles over an affine scheme which we will employ later.
Lemma 4 (Serre-Swan): Let be a coherent sheaf on a connected affine variety . Then the following are equivalent:
i) is a projective -module
ii) is locally free
iii) is isomorphic to the sheaf of germs of sections of a vector bundle over .
Reference. This result can be found in [Faisceaux Algébriques Cohérents – Serre, Chapter 1, Section 4]. There is an english translation available via Google at the moment.
Lemma 5 (Quillen-Suslin): Every finitely-generated projective -module with a field is free of finite rank.
Reference. Aside from the original papers, an exposition of this result aimed towards students is given in [Introduction to Commutative Algebra and Algebraic Geometry – Ernst Kunz, Chaper 4, Section 3, Theorem 3.15]. The necessary tools for the proof are also developed in this chapter.
Proposition 6: for every .
Proof. By Lemma 4, a locally free sheaf of finite rank over is canonically isomorphic with the associated sheaf, , of a finitely generated projective -module. Lemma 5 implies every class (or its additive inverse) in can be represented by for some . The function defined by is an isomorphism, which completes the proof.
Before proving proposition 6, again, we calculate the Grothendieck group of projective space. The way we’ll do it is by considering a theorem, which is very useful, about projective bundles over a scheme . For any locally free sheaf of rank , say , we can construct a vector bundle, as stated in the introduction, by constructing . The related projective bundle associated to , is constructed as the relative Proj of the symmetric algebra of .
Lemma 7: Let be a connected, smooth, quasi-projective scheme and a locally free sheaf of rank over . Then where is the th exterior power sheaf of . Further, in the Grothendieck group.
Reference(s): Two references are provided – [Lectures on the K-Functor in Algebraic Geometry – Yuri Manin, Theorem 4.5] and [Riemann Roch Algebra – Fulton and Lang, Chapter 5, Section 2, Theorem 2.3].
Proposition 8: .
Proof. Note for the scheme , a locally free sheaf of rank is a dimensional vector space, and the associated projective bundle is isomorphic with . It was shown in Proposition 3 that by the dimension map. Thus, the class is associated with the dimension of this vector space, which is . By Lemma 7 this yields the isomorphism with but this latter polynomial factors as and the proposition follows.
There are other ways to calculate the Grothendieck group of affine space. We include an alternative calculation which uses the localization exact sequence of -theory as its main tool. One interesting question raised by the appearance of this sequence was whether or not the localization exact sequence could naturally be extended (to the left) as a long exact sequence. An answer was given (yes) in the work of higher algebraic -theory by Daniel Quillen. Using some of the results given by his generalization of this theory, we provide a third calculation of . Along with some final remarks, this will conclude our post.
Lemma 9 (Localization Exact Sequence): For any closed subscheme of a scheme , there is an exact sequence of abelian groups where the first map is given by and the second by for the canonical inclusions .
Proof. See [Le Théorème de Riemann–Roch – Borel and Serre, Proposition 7]. This is also shown in [Riemann Roch Algebra – Fulton and Lang, Chapter 6, Section 3, Theorem 3.2].
Proposition 10: .
Proof. Since . Using Lemma 9 we obtain an exact sequence . The result will follow if we show the generators of map to distinct elements in . (Moreover since is on the right side of this sequence, it follows that surjects onto a summand of . Currently, I can’t see where the basis elements go to but, I’m gonna mark it done anyways. If I can figure out how it splits on the left I’ll update it).
We end mentioning one result often titled the homotopy invariance property.
Lemma 11: For any map which is flat and has fibres isomorphic with affine space, there is an isomorphism .
Reference. This is [Higher Algebraic K-Theory: I – Quillen, Propsoition 4.1]. One should note the results Quillen derives are in a much more general framework than may be alluded to by this post.
Proof. Apply lemma 11 to the projection and the result is then immediate from proposition 3.
Much more is true using the results provided above. For instance, lemma 11 implies the -theory of a vector bundle is the same as the -theory of its base space. There are also many other results one can find along the way. However, my main goal for writing this was to see if I could calculate some simple examples of the Grothendieck group (which I would say is pretty damned difficult from the definitions).