Proto-exact categories of matroids, Hall algebras, and K-theory
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Mathematische Zeitschrift
Proto-exact categories of matroids, Hall algebras, and K-theory Chris Eppolito1 · Jaiung Jun2 · Matt Szczesny3 Received: 9 May 2019 / Accepted: 23 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper examines the category Mat• of pointed matroids and strong maps from the point of view of Hall algebras. We show that Mat• has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory K ∗ (Mat• ) of Mat• via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections πns (S) → K n (Mat• ) from the stable homotopy groups of spheres for all n. Finally, we show that the Hall algebra of Mat• is a Hopf algebra dual to Schmitt’s matroid-minor Hopf algebra. Keywords Matroid · Matroid strong maps · Matroid-minor Hopf algebra · Hall algebra · Proto-exact category · K-theory Mathematics Subject Classification Primary 18D99; Secondary 05B35 · 16T30 · 19A99 · 19D99
1 Introduction In this paper we examine the category of pointed matroids and strong maps from the perspective of Hall algebras. This perspective sheds new light on certain combinatorial Hopf algebras built from matroids and opens the door to defining algebraic K-theory of matroids. This introduction is devoted to introducing the main actors.
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Matt Szczesny [email protected] Chris Eppolito [email protected] Jaiung Jun [email protected]
1
Department of Mathematical Sciences, Binghamton University, Binghamton, NY, USA
2
Department of Mathematics, SUNY New Paltz, New Paltz, NY, USA
3
Department of Mathematics and Statistics, Boston University, 111 Cumminton Mall, Boston, USA
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C. Eppolito et al.
1.1 Hall algebras of Abelian and exact categories The study of Hall algebras is now well-established, with several applications in representation theory and algebraic geometry; see [22] for an overview. We now recall the most basic version of this construction. Given an abelian category C such that both Hom(M, M ) and Ext1 (M, M ) are finite sets for any pair of objects M, M ∈ C (i.e. so that C is finitary), one may define the Hall algebra of C as follows. As a vector space HC = Qc [Iso(C )], where the right-hand side denotes the space of Q–valued functions on isomorphism classes in C with finite support. The (associative) multiplication on HC is given by ( f • g)([R]) =
f ([R/Q])g([Q]),
(1)
Q⊆R
where [R] denotes the isomorphism class of R ∈ C and the sum is over all sub-objects Q ⊆ R. The Hall algebra HC is spanned by δ-functions δ[M] , [M] ∈ Iso(C ) supported on individual isomorphism classes. The product (1) can then be explicitly written δ[M] • δ[N ] =
R P M,N δR ,
(2)
R∈Iso(C )
where R P M,N := # {L ⊆ R | L N , R/L M} .
The number R | Aut(M)|| Aut(N )| P M,N
counts the isomorphism classes of short exact sequences of the form 0 → N → R → M → 0,
(3)
where Aut(M) is the automorphism group of M. Thus, the product in HC encodes
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