The Grothendieck monoid of an extriangulated category
Haruhisa Enomoto
This talk is based on joint work with Shunya Saito (Nagoya University). The Grothendieck group is theclassical and basic invariant for both a triangulated category and an exact category. For exact categories,
the Grothendieck monoid, a natural monoid version of the Grothendieck group, has been recently studied
by several authors [1, 2, 6].
In the representation theory of algebras, we often consider extension-closed subcategories of a triangulated category which are not exact nor triangulated. An extriangulated category introduced byNakaoka–Palu [5] is a convenient framework to consider such subcategories. Extriangulated categories
unify both exact categories and triangulated categories, and have the notion of conflations, which generalize conflations (short exact sequences) in an exact category and triangles in a triangulated category.
We can naturally define the Grothendieck monoid M(C) of an extriangulated category C using conflations.
In this talk, we give several results about it.
The first result is about the classifications of several classes of subcategories, which extends [6] and [7]respectively.
Theorem 1. Let C be an extriangulated category. Then we have the following two bijections.
(1) A bijection between the set of Serre subcategories of C and the set of faces of M(C).
(2) A bijection between the set of dense 2-out-of-3 subcategories and the set of cofinal subtractive
submonoids of M(C).
The second result is about the localization of an extriangulated category. For a nice subcategory N ofan extriangulated category C, Nakaoka–Ogawa–Sakai [4] constructed the exact localization C/N , which
generalizes the Verdier quotient of a triangulated category and the Serre quotient of an abelian category.
We show that under some conditions, this commutes with the Grothendieck monoid :Theorem 2. Let C be an extriangulated category and N a subcategory of C satisfying some conditions.
Then we have an isomorphism of monoids
M(C/N ) ∼= M(C)/MN ,where the right hand side is the monoid quotient by MN := {[N ] | N ∈ N }. This can be applied to theVerdier quotient of a triangulated category, the stable category of a Frobenius category, and the Serrequotient of an abelian category.
As a toy example, we consider an intermediate subcategory of the derived category D(A) of an abeliancategory A, which is a subcategory C closed under extensions and direct summand satisfying A ⊆ C ⊆A[1] ∗ A. We show that an intermediate subcategory is precisely a subcategory of the form F[1] ∗ A fora torsionfree class F of A, and then compute its Grothendieck group, classify Serre subcategories, andstudy the exact localization.
References[1] A. Berenstein, J. Greenstein, Primitively generated Hall algebras, Pacific J. Math. 281 (2016), no. 2, 287–331.
[2] H. Enomoto, The Jordan-Hölder property and Grothendieck monoids of exact categories, Adv. Math, 396 (2022).
[3] H. Matsui, Classifying dense resolving and coresolving subcategories of exact categories via Grothendieck groups, Algebr.
Represent. Theory 21 (2018), no. 3, 551–563.
[4] H. Nakaoka, Y. Ogawa, A. Sakai, Localization of extriangulated categories, arXiv:2103.16907.
[5] H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom.
Différ. Catég. 60 (2019), no. 2, 117–193.
[6] S. Saito, The spectrum of Grothendieck monoid: a new approach to classify Serre subcategories, arXiv:2206.15271.
[7] R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1–27.

Graduate School of Mathematics
Osaka Metropolitan University
1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
Email: henomoto@omu.ac.jp–1–