A Note on the Grothendieck Group of an Additive Category

A categorification of an algebraic structure is typically given by an additive category (often possessing additional structure) from which the original structure can be recovered by taking the Grothendieck group; see for instance [2] for the abelian case. In certain categorifications of quantum knot invariants, the categorification is accomplished by first finding an additive category which categorifies an algebraic structure and then passing to the homotopy category of complexes to give the categorification of the knot invariant (see [1] and [3]). The categorified knot invariant decategorifies to give the original knot invariant by taking the 'Euler characteristic' of the complex, the alternating sum of terms of the complex, viewed as an element of the split Grothendieck group of the additive category. Since the homotopy category is triangulated, the natural decategorification of this category is its triangulated Grothendieck group. This posits the question, are these two Grothendieck groups isomorphic? This question can equivalently be stated: is the Euler characteristic of a complex invariant under homotopy equivalence?

We answer both these questions in the affirmative:

Theorem 1.1. Let A be an additive category and K b (A) denote the homotopy category of bounded complexes in A. The split Grothendieck group of A is isormophic to the triangulated Grothendieck group of K b (A).

wheredenotes the corresponding element in the split Grothendieck group of A.

We present the relevant background on additive categories and Grothendieck groups in Section 2. In Section 3 we prove Theorems 1.1 and 1.2 and discuss a slight generalization of Theorem 1.2 which is used in [4]. Acknowledgments: I would like to thank Ezra Miller for a helpful conversation and Scott Morrison for useful correspondence. I would also like to thank my advisor Lenny Ng for his continued guidance. The author was partially supported by NSF grant DMS-0846346 during the completion of this work.

Let A be an additive category. Recall that this means that A has a zero object, finite biproducts, and that Hom A (A 1 , A 2 ) is an abelian group for any objects A 1 , A 2 in A with addition distributing over composition.

Definition 2.1. The split Grothendieck group of A, denoted K ⊕ (A), is the abelian group generated by isomorphism classes A of objects in A modulo the relations

Recall that the Grothendieck group of an abelian category is the abelian group generated by isomorphism classes A of objects modulo the relations A 2 = A 1 + A 3 for every short exact sequence

We can think of Definition 2.1 as the analog of this notion in an additive category where we impose relations corresponding to the only notion of exact sequence that makes sense, the split exact sequences

Suppose now that C is not only additive, but triangulated. Definition 2.2. The triangulated Grothendieck group, denoted K △ (C), is the abelian group generated by isomorphism classes C of objects in C quotiented by the relation

Again, we think of distinguished triangles as the analogs of short exact sequences in C.

Now fix an additive category A. Let K b (A) denote the homotopy category of bounded complexes in A (we apologize for the confusing, but somewhat standard, notation).

-→ A l be a bounded complex and let A[m] • denote the complex shifted up by m in homological degree. We will underline the term in homological degree zero when it is not clear from the context. The distinguished triangle

and the triangle

and A i is shorthand for the complex with the object A i in degree zero and all other terms zero. From this we see that K △ (K b (A)) and K ⊕ (A) are generated by the same elements.

Given complexes A • 1 and A • 2 , the distinguished triangle

To prove Theorem 1.1, it suffices to show that this map is injective or equivalently that there are no additional relations imposed on K △ (K b (A)) other than those given in equations (3.1), (3.2), and

contribute no new relations. Since all distinguished triangles are isomorphic to those of the form (3.5) and isomorphism in K b (A) is homotopy equivalence, it suffices to prove Theorem 1.2.

To this end, suppose that ϕ :

2 is a homotopy equivalence. The following result from [5] is given in the setting of the category of abelian groups, but the proof sketched there carries over to arbitrary additive categories. We provide the details of the proof for completeness.

2 be a homotopy equivalence, so there exists a chain map

1 and H j 2 : A j 2 → A j-1 2 . We now construct maps H j : cone(ϕ) j → cone(ϕ) j-1 so that id j cone(ϕ) = d j-1 H j + H j+1 d j where

and denote M j = d j-1 H j -H j+1 d j . We now compute the entries M j (kl) of this matrix:

Conversely, let cone(ϕ) • be null-homotopic then

2 . This shows that ϕ is a homotopy equivalence with inverse the (chain!) map -h • 12 .

Still assuming ϕ is a homotopy equivalence, consider the distinguished triangle

• . Lemma 3.1 gives that cone(ϕ) ≃ 0 so equation (3.4) shows that The

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