A Note on the Grothendieck Group of an Additive Category
aa r X i v : . [ m a t h . C T ] S e p A Note on the Grothendieck Group of anAdditive Category
David E. V. Rose
Mathematics Department,Duke University, Durham, NC 27708-0320, USA email: [email protected]
Abstract
There are two abelian groups which can naturally be associated to anadditive category A : the split Grothendieck group of A and the triangu-lated Grothendieck group of the homotopy category of (bounded) com-plexes in A . We prove that these groups are isomorphic. Along the way,we deduce that the ‘Euler characteristic’ of a complex in A is invariantunder homotopy equivalence. A categorification of an algebraic structure is typically given by an additivecategory (often possessing additional structure) from which the original struc-ture 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 whichcategorifies an algebraic structure and then passing to the homotopy categoryof complexes to give the categorification of the knot invariant (see [1] and [3]).The categorified knot invariant decategorifies to give the original knot invari-ant by taking the ‘Euler characteristic’ of the complex, the alternating sum ofterms of the complex, viewed as an element of the split Grothendieck group ofthe additive category. Since the homotopy category is triangulated, the nat-ural decategorification of this category is its triangulated Grothendieck group.This posits the question, are these two Grothendieck groups isomorphic? Thisquestion can equivalently be stated: is the Euler characteristic of a complexinvariant 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 homotopycategory of bounded complexes in A . The split Grothendieck group of A isisormophic to the triangulated Grothendieck group of K b ( A ) . heorem 1.2. Let A · ≃ B · be homotopy equivalent complexes in K b ( A ) , then ∞ X i = −∞ ( − i h A i i = ∞ X i = −∞ ( − i h B i i , where h−i denotes the corresponding element in the split Grothendieck group of A . We present the relevant background on additive categories and Grothendieckgroups in Section 2. In Section 3 we prove Theorems 1.1 and 1.2 and discuss aslight generalization of Theorem 1.2 which is used in [4].
Acknowledgments:
I would like to thank Ezra Miller for a helpful conversa-tion and Scott Morrison for useful correspondence. I would also like to thankmy advisor Lenny Ng for his continued guidance. The author was partiallysupported 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 , A ) is an abelian group for any objects A , A in A with addition distributing over composition. Definition 2.1.
The split Grothendieck group of A , denoted K ⊕ ( A ) , is theabelian group generated by isomorphism classes h A i of objects in A modulo therelations h A ⊕ A i = h A i + h A i for all objects A , A in A . Recall that the Grothendieck group of an abelian category is the abeliangroup generated by isomorphism classes h A i of objects modulo the relations h A i = h A i + h A i for every short exact sequence0 → A → A → A → A . We can think of Definition 2.1 as the analog of this notion in an additivecategory where we impose relations corresponding to the only notion of exactsequence that makes sense, the split exact sequences0 → A → A ⊕ A → A → . Suppose now that C is not only additive, but triangulated. Definition 2.2.
The triangulated Grothendieck group , denoted K △ ( C ) , is theabelian group generated by isomorphism classes h C i of objects in C quotiented bythe relation h C i = h C i + h C i for all distinguished triangles C → C → C . Again, we think of distinguished triangles as the analogs of short exact se-quences in C . 2 Grothendieck Groups of Additive Categories
Now fix an additive category A . Let K b ( A ) denote the homotopy categoryof bounded complexes in A (we apologize for the confusing, but somewhatstandard, notation).Let A · = (cid:18) A k d k −→ · · · d l − −→ A l (cid:19) be a bounded complex and let A [ m ] · denotethe complex shifted up by m in homological degree. We will underline theterm in homological degree zero when it is not clear from the context. Thedistinguished triangle A · → → A [ − · gives that h A [ − · i = −h A · i (3.1)and the triangle A k → (cid:18) A k +1 d k +1 −→ · · · d l − −→ A l (cid:19) → A [ − k − · shows (via induction) that h A · i = χ ( A · ) (3.2)in K △ ( K b ( A )). Here χ ( A · ) := P ∞ i = −∞ ( − i h A i i and A i is shorthand for thecomplex with the object A i in degree zero and all other terms zero. From thiswe see that K △ ( K b ( A )) and K ⊕ ( A ) are generated by the same elements.Given complexes A · and A · , the distinguished triangle A · → ( A ⊕ A ) · → A · shows that h ( A ⊕ A ) · i = h A · i + h A · i . (3.3)It follows that there is a surjective map K ⊕ ( A ) → K △ ( K b ( A )).To prove Theorem 1.1, it suffices to show that this map is injective or equiva-lently that there are no additional relations imposed on K △ ( K b ( A )) other thanthose given in equations (3.1), (3.2), and (3.3). Given a map A f → A , theseequations show that h cone( f ) · i = ∞ X j = ∞ (cid:16) ( − j h A j i + ( − j +1 h A j i (cid:17) (3.4)= h A · i − h A · i so distinguished triangles of the form A · f → A · → cone( f ) · (3.5)contribute no new relations. Since all distinguished triangles are isomorphic tothose of the form (3.5) and isomorphism in K b ( A ) is homotopy equivalence, itsuffices to prove Theorem 1.2. 3o this end, suppose that ϕ : A · → A · is a homotopy equivalence. Thefollowing 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. Weprovide the details of the proof for completeness. Lemma 3.1.
A chain map ϕ : A · → A · is a homotopy equivalence iff cone( ϕ ) · is null-homotopic.Proof. Let ϕ : A · → A · be a homotopy equivalence, so there exists a chain map ψ : A · → A · so that ϕ j ψ j − id j = d j − H j + H j +12 d j and ψ j ϕ j − id j = d j − H j + H j +11 d j for maps H j : A j → A j − and H j : A j → A j − . We now construct maps H j : cone( ϕ ) j → cone( ϕ ) j − so that id j cone( ϕ ) = d j − H j + H j +1 d j where d j = (cid:18) − d j +11 − ϕ j +1 d j (cid:19) . Let H j = (cid:18) H j +11 + ψ j H j +12 ϕ j +1 − ψ j ϕ j H j +11 − ψ j H j H j +12 ϕ j +1 − H j ϕ j H j +11 − H j (cid:19) and denote M j = d j − H j − H j +1 d j . We now compute the entries M j ( kl ) of thismatrix: M j (11) = − d j H j +11 − d j ψ j H j +12 ϕ j +1 + d j ψ j ϕ j H j +11 − H j +21 d j +11 − ψ j +1 H j +22 ϕ j +2 d j +11 + ψ j +1 ϕ j +1 H j +21 d j +11 + ψ j +1 ϕ j +1 = id j +11 − ψ j +1 ϕ j +1 − d j ψ j H j +12 ϕ j +1 + d j ψ j ϕ j H j +11 − ψ j +1 H j +22 ϕ j +2 d j +11 + ψ j +1 ϕ j +1 H j +21 d j +11 + ψ j +1 ϕ j +1 = id j +11 − ψ j +1 ( d j H j +12 + H j +22 d j +12 ) ϕ j +1 + ψ j +1 ϕ j +1 ( d j H j +11 + H j +21 d j +11 )= id j +11 − ψ j +1 ( ϕ j +1 ψ j +1 − id j +12 ) ϕ j +1 + ψ j +1 ϕ j +1 ( ψ j +1 ϕ j +1 − id j +11 )= id j +11 , M j (12) = d j ψ j − ψ j +1 d j = 0 , j (21) = − ϕ j H j +11 − ϕ j ψ j H j +12 ϕ j +1 + ϕ j ψ j ϕ j H j +11 + d j − H j H j +12 ϕ j +1 = − d j − H j ϕ j H j +11 − H j +12 H j +22 ϕ j +2 d j +11 + H j +12 ϕ j +1 H j +21 d j +11 + H j +12 ϕ j +1 = ( id j + d j − H j − ϕ j ψ j ) H j +12 ϕ j +1 + ( ϕ j ψ j − d j − H j − id j ) ϕ j H j +11 − H j +12 H j +22 ϕ j +2 d j +11 + H j +12 ϕ j +1 H j +21 d j +11 = − H j +12 d j H j +12 ϕ j +1 + H j +12 d j ϕ j H j +11 − H j +12 H j +22 ϕ j +2 d j +11 + H j +12 ϕ j +1 H j +21 d j +11 = − H j +12 ( d j H j +12 + H j +22 d j +12 ) ϕ j +1 + H j +12 ϕ j +1 ( d j H j +11 + H j +21 d j +11 )= − H j +12 ( ϕ j +1 ψ j +1 − id j +12 ) ϕ j +1 + H j +12 ϕ j +1 ( ψ j +1 ϕ j +1 − id j )= 0 , M j (22) = ϕ j ψ j − d j − H j − H j +12 d j = id j . This shows that cone( ϕ ) · ≃ ϕ ) · be null-homotopic then id j cone( ϕ ) = (cid:18) − d j − ϕ j d j − (cid:19) (cid:18) h j +111 h j h j +121 h j (cid:19) + (cid:18) h j +211 h j +112 h j +221 h j +122 (cid:19) (cid:18) − d j +11 − ϕ j +1 d j (cid:19) which gives the equations − d j h j + h j +112 d j = 0 ,id j +11 = − d j h j +111 − h j +211 d j +11 − h j +112 ϕ j +1 , and id j = − ϕ j h j + d j − h j + h j +122 d j for maps h j : A j → A j , h j : A j → A j − , h j : A j → A j − , and h j : A j → A j − . This shows that ϕ is a homotopy equivalence with inverse the (chain!)map − h · .Still assuming ϕ is a homotopy equivalence, consider the distinguished tri-angle A · ϕ → A · → cone( ϕ ) · . Lemma 3.1 gives that cone( ϕ ) ≃ Proposition 3.2.
Let A · be a null-homotopic complex in K b ( A ) , then χ ( A · ) =0 in K ⊕ ( A ) . roof. We may assume that A · = A d −→ A d −→ · · · d k −→ A k +1 contains all of the non-zero terms of A · . It suffices to show that k M i =0 A i ∼ = k M i =0 A i +1 which we shall do by explicitly writing down the matrices giving the isomor-phism.Since A · is null-homotopic there exist maps A j h j → A j − so thatid j = d j − h j + h j +1 d j . Using these equations, we can deduce the relations h j h j +1 · · · h j +2 l +1 = d j − h j − h j · · · h j +2 l +1 + h j · · · h j +2 l +1 h j +2 l +2 d j +2 l +1 . For instance, we can compute h j h j +1 = h j id j h j +1 = h j d j − h j h j +1 + h j h j +1 d j h j +1 = h j h j +1 − d j − h j − h j h j +1 + h j h j +1 − h j h j +1 h j +2 d j +1 and h j h j +1 h j +2 h j +3 = h j h j +1 id j +1 h j +2 h j +3 = h j h j +1 d j h j +1 h j +2 h j +3 + h j h j +1 h j +2 d j +1 h j +2 h j +3 = h j h j +1 h j +2 h j +3 − h j d j − h j h j +1 h j +2 h j +3 + h j h j +1 h j +2 h j +3 − h j h j +1 h j +2 h j +3 d j +2 h j +3 = d j − h j − h j h j +1 h j +2 h j +3 + h j h j +1 h j +2 h j +3 h j +4 d j +3 . and similar computations (or induction on l ) show the result in general.Consider now the maps R : k M i =0 A i → k M i =0 A i +1 and L : k M i =0 A i +1 → k M i =0 A i R = d α h α h h h α h · · · h · · · α k − h · · · h k d α h α h h h · · · α k − h · · · h k d α h · · · α k − h · · · h k d · · · α k − h · · · h k ... ... ... ... . . . ...0 0 0 0 · · · d k and L = α h α h h h α h · · · h α h · · · h · · · α k h · · · h k +1 d α h α h h h α h · · · h · · · α k − h · · · h k +1 d α h α h h h · · · α k − h · · · h k +1 d α h · · · α k − h · · · h k +1 ... ... ... ... . . . ...0 0 0 0 · · · α h k +1 where { α k } are integers defined by the recursion α = 1, α = −
1, and α k = − k − X j =0 α j α k − − j . It is easy to see that in fact α k = ( − k c k where c k is the k th Catalan number.We now compute the entries of the matrices RL and LR . For i < j we have( RL ) ij = α j − i d i − h i − · · · h j − + α α j − i − h i · · · h j − + · · · + α j − i − α h i · · · h j − + α j − i h i · · · h j d j − = α j − i ( d i − h i − · · · h j − + h i · · · h j d j − − h i · · · h j − )= 0and ( RL ) ij = 0 for i > j . We also compute( RL ) jj = α ( d j − h j − + h j d j − ) = id j − which shows that RL = id . Similarly, for i < j we have( LR ) ij = α j − i d i − h i − · · · h j − + α α j − i − h i − · · · h j − + · · · + α j − i − α h i − · · · h j − + α j − i h i − · · · h j − d j − = α j − i ( d i − h i − · · · h j − + h i − · · · h j − d j − + h i − · · · h j − )= 0and ( LR ) ij = 0 for i > j . We also see that( LR ) jj = α ( d j − h j − + h j − d j − ) = id j − so LR = id . 7e can slightly extend Proposition 3.2 to the category K + ( A ) of boundedbelow complexes in A . If A · is such a complex and is null-homotopic, the infinitestable limit as k → ∞ of the matrices R and L gives an isomorphism ∞ a i = −∞ A i ∼ = ∞ a i = −∞ A i +1 . If the category A is such that we can define a notion of Euler characteristic,this shows that null-homotopic complexes have zero Euler characteristic. Inparticular, this fact is used in [4]. 8 eferences [1] Dror Bar-Natan, Khovanov’s homology for tangles and cobordisms , Geom.Topol. (2005), 1443–1499 (electronic). MR 2174270 (2006g:57017)[2] Mikhail Khovanov, Volodymyr Mazorchuk, and Catharina Stroppel, A briefreview of abelian categorifications , Theory Appl. Categ. (2009), No. 19,479–508. MR 2559652 (2010m:18001)[3] Scott Morrison and Ari Nieh, On Khovanov’s cobordism theory for su knothomology , J. Knot Theory Ramifications (2008), no. 9, 1121–1173. MR2457839 (2009j:57006)[4] David E. V. Rose, A categorification of quantum sl projectors and the sl Reshetikhin-Turaev invariant of tangles , arXiv:1109.1745v1 [math.GT].[5] Edwin H. Spanier,