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THE TRACE OF THE LOCAL A1-DEGREE THOMAS BRAZELTON, ROBERT BURKLUND, STEPHEN MCKEAN, MICHAEL MONTORO, AND MORGAN OPIE Abstract. We prove that the local A1-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A1-degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers, and by Kass and Wickelgren's work on the Scheja{Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja{Storch form and the local A1-degree. 1. Introduction The A1-degree, first defined by Morel [Mor04, Mor12], provides a foundational tool for 1 solving problems in A1-enumerative geometry. In contrast to classical notions of degree, 1 n n the local A -degree is not integer valued: given a polynomial function f : Ak ! Ak with 1 1 A isolated zero p, the local A -degree of f at p, denoted by degp (f), is defined to be an element of the Grothendieck{Witt group of the ground field. Definition 1.1. Let k be a field. The Grothendieck{Witt group GW(k) is defined to be the group completion of the monoid of isomorphism classes of symmetric non-degenerate bilinear forms over k. The group operation is the direct sum of bilinear forms. We may also give GW(k) a ring structure by taking tensor products of bilinear forms for our multiplication. The local A1-degree, which will be defined in Definition 2.9, can be related to other important invariants at rational points. The Scheja{Storch form (Definition 2.15) is an- other GW(k)-valued invariant defined via a duality on the local complete intersection cut out by the components of a given polynomial map (see Subsection 2.3 for details). Kass and Wickelgren show that the isomorphism class of the Scheja{Storch bilinear form [SS75] is equal to the local A1-degree at rational points [KW19]. Kass and Wickel- gren also show that at points with finite separable residue field, the Scheja{Storch form is given by taking the trace of the Scheja{Storch form over the residue field [KW17, Proposition 32]. Date: June 29, 2020. 2010 Mathematics Subject Classification. 14F42, 55P42, 55M25. 1 A1-enumerative geometry is the application of A1-homotopy theory to the study of enumerative geometry over arbitrary fields. For details, see the expository paper [WW19], as well as the exposition found in [KW17, Lev17, BKW18, SW18, KW19, LV19]. 1 2 BRAZELTON, BURKLUND, MCKEAN, MONTORO, AND OPIE In practice, one may need to consider the local A1-degree at non-rational points. This is the case of interest to us. At points whose residue field is a finite extension of the ground field, Morel's work on cohomological transfer maps [Mor12] suggests the following formula: the local A1-degree at a non-rational point should be computed by first taking the local A1-degree over the residue field, and then by post-composing with a field trace. This suggestion is supported by the aforementioned results of Kass{Wickelgren on the Scheja{Storch form [KW17, Proposition 32]. Our main result is to confirm this formula. We state our result precisely in Theorem 1.3, after introducing necessary terminology. Definition 1.2. Given a separable field extension L=k of finite degree, the trace TrL=k : GW(L) ! GW(k) is given by post-composing the field trace (which we also denote TrL=k). That is, if β : V × V ! L is a representative of an isomorphism class of symmetric non-degenerate bilinear forms over L, then its trace is the isomorphism class of the following symmetric bilinear form over k TrL=k TrL=kβ : V × V ! L −−−! k: n n When p is not a k-rational point, we can lift f to a function fk(p) : Ak(p) ! Ak(p) after fixing a choice of field embedding k ,−! k(p). Moreover, we may lift p to an isolated k(p)- 1 A rational zero p of fk(p), and we thus obtain the local degree deg (fk(p)) 2 GW(k(p)). e pe We can now state our main result. n n Theorem 1.3. Let k be a field, f : Ak ! Ak be an endomorphism of affine space, and n let p 2 Ak be an isolated zero of f such that k(p) is a separable extension of finite degree over k. Let pe denote the canonical point above p. Then 1 1 A A deg (f) = Trk(p)=k deg fk(p) p pe in GW(k). As a corollary, we strengthen Kass and Wickelgren's result relating the local A1-degree and the Scheja{Storch form [KW19] by weakening the requirement that the point be rational. Corollary 1.4. At points whose residue fields are finite separable extensions of the ground field, the local A1-degree coincides with the isomorphism class of the Scheja{ Storch form. In this paper we utilize the machinery of stable A1-homotopy theory, initially developed by Morel and Voevodsky [MV99], as well as the six functors formalism in this setting [Ayo07, CD19]. We also rely heavily on results of Hoyois [Hoy15b] to prove our main result. After working in the stable A1-homotopy category, we apply Morel's A1-degree to obtain the desired equality in GW(k). Conventions 1.5. Throughout, we adopt the following conventions: 1 THE TRACE OF THE LOCAL A -DEGREE 3 • We will use k to denote a general field. If p is a point of a k-scheme, with residue field k(p) such that k(p)=k a separable extension of finite degree, we call p a finite separable point. We may also say that p has a finite separable residue field in this context. We remark that all such points are closed points. • Whenever a closed point p of a k-scheme X is chosen, we denote by ρ : Speck(p) ! p Speck the composite of the morphism Speck(p) −! X defining the point p, and the structure map X ! Speck. This fixes a field embedding k ,−! k(p). • Given a scheme X over k, we denote the base change X ×Spec k Speck(p) by Xk(p), and given a morphism of k-schemes f : X ! Y , we denote its base change by fk(p) : Xk(p) ! Yk(p). n • The structure map ρ allows us to define the canonical k(p)-rational point in Ak(p) sitting above p, which we denote byp ~. 1.1. Acknowledgements. We would like to thank Kirsten Wickelgren for her excellent guidance during this project. We would also like to thank Matthias Wendt and Jesse Kass for their helpful insights during the 2019 Arizona Winter School. We are grateful to an anonymous referee for comments and suggestions that greatly improved the paper. Finally, we would like to thank the organizers of the Arizona Winter School, as this work is the result of one of the annual AWS project groups. The first and fifth authors are supported by NSF Graduate Research Fellowships under grant numbers DGE-1845298 and DGE-1144152, respectively. 2. Preliminaries In this section, we introduce the main notions necessary to state and prove Theorem 1.3. We begin in Subsection 2.1 by defining the local A1-degree. In Subsection 2.2, we highlight key properties of the stable motivic homotopy category in the form that we will need them. Finally, in Subsection 2.3, we discuss the Scheja{Storch form. We will assume some familiarity with motivic homotopy theory. For more detail about 1 A the category of motivic spaces Spck and the unstable motivic homotopy category H(k), we refer the reader to the excellent expository articles [AE16, WW19]. For the construc- tion of the stable motivic homotopy category SH(k), we refer the reader to [Mor04]. Notation 2.1. We denote by SH(k) the stable motivic homotopy category over the scheme Speck. The sphere spectrum in this category will be denoted by 1k. We will also use [−; −] to denote 1-weak equivalence classes of maps between two motivic spaces, A1 A by which we mean a hom-set in the homotopy category H(k). 1 n n 2.1. The local A -degree. Given an endomorphism of affine space f : Ak ! Ak with an isolated zero at a point p, we describe how to obtain an endomorphism of the sphere spectrum in the stable motivic homotopy category SH(k), following the exposition of [KW19, pp. 438{439]. We remind the reader of Conventions 1.5, which we use in what follows. 4 BRAZELTON, BURKLUND, MCKEAN, MONTORO, AND OPIE n Since p is an isolated zero of f, we may find an open neighborhood U ⊆ Ak for which f −1(0)\U = fpg, that is, an open neighborhood containing no other zeros of f. Viewing n n U ⊆ Ak ⊆ Pk as an open subset of projective space via a standard affine chart, we may 1 A take a Nisnevich-local pushout diagram in Spck : n U r fpg Pk r fpg p n U Pk : This induces an A1-weak equivalence on cofibers n U ∼ Pk −! n : U r fpg Pk r fpg We now appeal to the purity theorem [MV99, Theorem 2.23], a fundamental result in A1-homotopy, which we record for future use. While the purity theorem holds for smooth schemes over a sufficiently nice base scheme, we only need the result for smooth schemes over a field. Theorem 2.2 (Morel{Voevodsky). Let Z ! X be a closed embedding of smooth schemes over a field k.
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