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Вестник Челябинского государственного университета. 2015. № 3 (358). Математика. Механика. Информатика. Вып. 17. С. 135-139.
УДК 512.58 ББК В152.4
A NOTE ON THE GROTHENDIECK GROUP OF AN ADDITIVE CATEGORY*
D. E. V. Rose
There are two abelian groups which can naturally be associated to an additive category A: the split Grothendieck group of A and the triangulated Grothendieck group of the homotopy category of (bounded) complexes in A. We prove that these groups are isomorphic. Along the way, we deduce that the 'Euler characteristic' of a complex in A is invariant under homotopy equivalence, a result which has implications for (de)categorification.
Keywords: Grothendieck group,additive category,categorification.
1. Introduction
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 [1] for the abelian case. In certain categorifications of quantum invariants of tangles, 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 tangle invariant (see [2] and [3]). The categorified tangle invariant decategorifies to give the original tangle invariant by taking the 'Euler characteristic' of the complex, the alternating sum of the terms in 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 in an additive (but not necessarily abelian) category invariant under homotopy equivalence? We answer both these questions in the affirmative:
Theorem 1. Let A be an additive category and K,h(A) denote the homotopy category of bounded complexes in A. The split Grothendieck group of A is isomorphic to the triangulated Grothendieck group of Kh(A). Theorem 2. Let AB' he homotopy equivalent complexes in Kh(A), then
£ (-iy(A') =£ i-iy{B'),
where (■) denotes the corresponding element in the split Grothendieck group of A.
Of course, this result is not surprising; indeed, in the case that A is abelian, the analog of Theorem 2 is an easy exercise in homological algebra. Nevertheless, the proof presented here is unexpectedly non-trivial and the general result is of interest to the categorification community as non-abelian additive categories arise naturally in this field. Furthermore, this result seems to have been implicitly assumed in the categorification literature, while a proof has up to now not appeared.
We present the relevant background on additive categories and Grothendieck groups in Section 2. In Section 3 we prove Theorems 1 and 2 and in Section 4 we mention a slight generalization of Theorem 2 which is used in [4].
* The author was partially supported by NSF grant DMS-0846346 during the completion of this work.
2. Background
Let A be an additive category. Recall that this means that A has a zero object, finite bi-products, and that HomA(A4, A2) is an abelian group for any objects A1,A2 in A with addition distributing over composition.
Definition 1. The split Grothendieck group of A, denoted Kffi(A), is the abelian group generated by isomorphism classes (A) of objects in A modulo the relations <A1 © A2) = <A1> + (A2) for all objects A1, A2 in A.
Recall that the Grothendieck group of an abelian category is the abelian group generated by isomorphism classes (A) of objects modulo the relations (A2) = (A1) + (A3) for every short exact sequence 0—A4—A2—A3—0 in A. We can think of Definition 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 0—>Aj—>Aj©A2—>A2—►O.
Suppose now that C is not only additive, but triangulated.
Definition 2. The triangulated Grothendieck group, denoted KA(C), is the abelian group generated by isomorphism classes (C) of objects in C quotiented by the relation (C2) = (C1) + (C3) for all distinguished triangles C1— C2— C3.
Again, we think of distinguished triangles as the analogs of short exact sequences in C.
3. Grothendieck Groups of Additive Categories
Now fix an additive category A. Let Kb(A) denote the homotopy category of bounded (co-chain) complexes in A. Let A' = (Ak ••• Al) 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 A' — 0 — A[-1]' gives that
(At-m = -(A*) (1)
and the triangle Ak = (Ak+1 ^^ ••• Al) — A[-k- 1]; shows (via induction) that
(A*) = x(A') (2)
in KA(Kb(A)). Here x(A') := ^ (-l)^A^ and Ai is shorthand for the complex with the object
i = —'X>
Ai in degree zero and all other terms zero. From this, we see that KA(Kb(A)) and K©(A) are generated by the same elements.
Given complexes A; and A, the distinguished triangle A; — A; © A — A shows that
((A1 © A2)-) = (a;) + (A2). (3)
It follows that there is a surjective map K©(A) — KA(Kb(A)).
To prove Theorem 1, it suffices to show that this map is injective or equivalently that there are no additional relations imposed on KA(Kb(A)) other than those given in equations (1), (2) and (3). Given a map A1 -— A2, these equations show that
(cone(f )•) = £(<-0 + (-1);+1 (A1)) = (AO ^A)
j = 'X>
so distinguished triangles of the form
A4 A2 — cone(/); (5)
contribute no new relations. Since all distinguished triangles are isomorphic to those of the form (5) and isomorphism in K,b(A) is homotopy equivalence, it suffices to prove Theorem 2.
To this end, suppose that 9: A; — A is a homotopy equivalence. The following result from [5] is given in the context of the category of abelian groups, but the proof sketched there carries over to arbitrary additive categories.
Lemma 1. A chain map 9: A[ — A is a homotopy equivalence iff cone(^)' is null-homotopic. The distinguished triangle — A'2 — cone(^)* together with (4) then show that Theorem 2 (and hence Theorem 1) follows from the next result.
Proposition 1. Let A' be a null-homotopic complex in Kb(A), then x(A') = 0 when viewed as an element of Kffi(A).
Proof. We may assume that A' = A0 terms of A*. It suffices to show that
A1
A2k+1 contains all of the non-zero
0 a2i ^ 0 A2i+1
i=0 i=0
in A, which we shall do by explicitly writing down the matrices giving the isomorphism.
Since A' is null-homotopic there exist maps imply the relations hjhj+1... hj+2l+1 = dj-2hl- 1 hj... hj +21+1 + hj ... hj +2l+ 1 hj +21+2 dj+21+1. Consider now the maps
R : 0 a21 ^ 0 A2l+1, L : 0 a21 ^ 0 a21
i=0
1=0
1=0
1=0
given by the matrices in (6) and (7), where {aj are integers defined by the recursion a0 = 1, a1 = -1,
1-1
and a{ = -^ajai-1-j. It is easy to see that in fact a{ = (~1)lci where c{ is the ith Catalan number.
j=0
R =
\ o
d0 a0h2 a1h2h3h4 a2h2...h6 . . ak- _ih2.. .h2k
0 d02 a0h4 a1h4h5h6 . . ak- ■ft.. .h2k
0 0 d4 a0h6 . . ak- -sh6.. hk
0 0 0 d6 . . ak- CO • ■ hk
0 0 0 0. d2k
\
(6)
L =
/ a0hl a1h1h2h
d01 a0h3
0 d03
0 0
0 0
a2h\..h7
a1h3h4h5
5
a0h d5
a3h\..h7 a2h\..h7 a1h5h6h7 a0h7
akhi...h2k+i \
ak-1h3...h2k+1 ak 2h5...h2k+1
ak_^h7...h2k+1
a0h
2k+1
(7)
0
0
We now compute the entries of the matrices RL and LR. For i < j we have
(RL)ij = aj_id2i-2h2i- 1 ... h2- 1 + a0aj_i-1h2i ... h2j- 1 + ... + aj_i-1a0h2i ... h2j- 1 +
+ a., h2 ... h2jd2j- 1 = a.. .(d2i-2h2i- 1 ... h2j- 1 + h2i ... h2jd2j- 1 - h2i ... h2j- 1) = 0
j i j i
and (RL). = 0 for i > j. We also compute (RL). = a0(d2j-2h2j-i + h2jd2j-i) = id2j-1 which shows that RL = id. Similarly, for i < j we have
(LR). = ahid2i - h- 2 ... h2j - 2 + a0aj_i-1h2i - 1 ... h2j - 2 + ... + + aj_i-1a0h2i - 1 ... h2j - 2 + aj_ih2i - 1 ... h2j - 1d2j - 2 = = a. .(d2i-3h2i-2 ... h2j-2 + h2i-1 ... h2j-1 d2j-2 + h2i-1 ... h2j-2) = 0 and (LR). = 0 for i > j. We also see that (LR). = a0(d2j~3h2j-2 + h2j-id2j-2) = id2j_2 so LR = id. □
4. An extension to K+(A)
In the categorification of colored link invariants, one has to additionally consider the ho-motopy category of semi-infinite complexes in an additive category, see [4; 6; 7] (or [8-10] for the abelian/derived case). We can extend Proposition 1 to the category K(A) of bounded below complexes in A. If A' is such a complex and is null-homotopic, the infinite stable limit as k ^ <x> of the matrices R and L gives an isomorphism
jj a2 -ij a2 +1
i=-c» i=- c»
where U denotes the categorical coproduct (a similar result holds for the category of bounded above complexes). If the category A is such that we can define a notion of Euler characteristic, this can be used to show that null-homotopic complexes have zero Euler characteristic; for example, see Section 5.2 of [11] for complete details and Section 3.3.1 of [10] for the analogous result in the setting of the derived category of an abelian category.
References
1. Khovanov M., Mazorchuk V., Stroppel C. A brief review of abelian categorifications. Theory Appl. Categ., 2009, vol. 22, no. 19, pp. 479-508.
2. Bar-Natan D. Khovanov's homology for tangles and cobordisms. Geom. Topol., 2005, vol. 9, pp. 1443-1499.
3. Morrison S., Nieh A. On Khovanov's cobordism theory for su knot homology. J. Knot Theory Ramifications, 2008, vol. 17, no. 9, pp. 1121-1173.
4. Rose D. E. V. A categorification of quantum sl3 projectors and the sl3 Reshetikhin-Turaev invariant of tangles. Quantum Topol., 2014, vol. 5, no. 1, pp. 1-59.
5. Spanier E. H. Algebraic topology. New York, Springer-Verlag Publ., 1966.
6. Cooper B., Krushkal V. Categorification of the Jones-Wenzl projectors. Quantum Topol., 2012, vol. 3, no. 2, pp. 139-180. (arXiv:1005.5117).
7. Rozansky L. An infinite torus braid yields a categorified Jones-Wenzl projector. Available at: http://arxiv.org/abs/1005.3266.
8. Frenkel I., Stroppel C., Sussan J. Categorifying fractional Euler characteristics, Jones-Wenzl projectors and 3/-symbols. Quantum Topol., 2012, vol. 3, no. 2, pp. 181-253. (arXiv:1007.4680).
9. Achar P. N., Stroppel C. Completions of Grothendieck groups. Available at: http://arxiv. org/abs/1105.2715.
10. Cautis S. Clasp technology to knot homology via the affine Grassmannian. Available at: http: //arxiv.org/abs/1207.2074.
11. Rose D. E. V. Categorification of quantum sl3 projectors and the sl3 Reshetikhin-Turaev invariant of framed tangles. Ph.D. thesis. New York, Duke University Publ., 2012.
About the author
David E. V. Rose, professor, Department of Mathematics of University of Southern California, Los Angeles, USA. davidero@usc.edu.
Bulletin of Chelyabinsk State University. 2015. № 3 (358). Mathematics. Mechanics. Informatics. Issue 17. Р. 135-139.
О ГРУППЕ ГРОТЕНДИКА АДДИТИВНЫХ КАТЕГОРИЙ
Д. Е. В. Роуз
Есть две абелевых группы, которые могут быть естественным образом ассоциированы с аддитивной категорией Л: расщепленная группа Гротендика категории Л и триангулированная группа Гротендика гомотопической категории (ограниченных) комплексов в Л. Доказывается, что эти группы изоморфны. Попутно получается, что «Эйлерова характеристика» комплекса в Л является инвариантом относительно гомотопической эквивалентности. Этот результат имеет значение для (де)категорификации.
Ключевые слова: группа Гротендика, аддитивная категория, категорификация.
Список литературы
1. Khovanov, M. A brief review of abelian categorifications / M. Khovanov, V. Mazorchuk, C. Stroppel // Theory Appl. Categ. — 2009. — Vol. 22, № 19. — P. 479-508.
2. Bar-Natan, D. Khovanov's homology for tangles and cobordisms / D. Bar-Natan // Geom. Topol. — 2005. — Vol. 9. — P. 1443-1499.
3. Morrison, S. On Khovanov's cobordism theory for su knot homology / S. Morrison, A. Nieh // J. Knot Theory Ramifications. — 2008. — Vol. 17, № 9. — P. 1121-1173.
4. Rose, D. E. V. A categorification of quantum sl3 projectors and the sl3 Reshetikhin-Turaev invariant of tangles / D. E. V. Rose // Quantum Topol. — 2014. — Vol. 5, № 1. — P. 1-59.
5. Spanier, E. H. Algebraic topology / E. H. Spanier. — 2nd ed. — New York : Springer-Verlag, 1966.
6. Cooper, B. Categorification of the Jones-Wenzl projectors / B. Cooper, V. Krushkal // Quantum Topol. — 2012. — Vol. 3, № 2. — P. 139-180.
7. Rozansky, L. An infinite torus braid yields a categorified Jones-Wenzl projector [Электронный ресурс] / L. Rozansky. — URL : http://arxiv.org/abs/1005.3266.
8. Frenkel, I. Categorifying fractional Euler characteristics, Jones-Wenzl projectors and 3/-symbols / I. Frenkel, C. Stroppel, J. Sussan // Quantum Topol. — 2012. — Vol. 3, № 2. -P. 181-253. (arXiv:1007.4680).
9. Achar, P. N. Completions of Grothendieck groups [Электронный ресурс] / P. N. Achar, C. Stroppel. — URL : http://arxiv.org/abs/1105.2715.
10. Cautis, S. Clasp technology to knot homology via the affine Grassmannian [Электронный ресурс] / S. Cautis. — URL : http://arxiv.org/abs/1207.2074.
11. Rose, D. E. V. Categorification of quantum sl3 projectors and the sl3 Reshetikhin-Turaev invariant of framed tangles / D. E. V. Rose // Ph.D. thesis. — USA : Duke University, 2012. (arXiv:1109.1745v2 [math.GT]).
Сведения об авторе
Рoуз Дэвид Е. В., профессор, математический факультет Университета Южной Калифорнии, Лос-Анджелес, США. davidero@usc.edu.
