Unstable Grothendieck–Witt Group

• Unstable Grothendieck–Witt group is an algebraic invariant that refines the classical GW ring by encoding nondegenerate symmetric bilinear forms with explicit determinant data.
• It is constructed as a fiber product over the determinant map and forms an additive abelian group, capturing non-virtual quadratic form data absent in the stable ring structure.
• Computational frameworks like the A1BrouwerDegrees package use algorithms based on Bézoutian and Newton matrices to compute local and global unstable A¹-degrees.

The unstable Grothendieck–Witt group is an algebraic invariant arising in the study of quadratic forms, Hermitian K-theory, and $\mathbb{A}^1$-homotopy theory, with applications to both abstract mathematics and computational frameworks such as the A1BrouwerDegrees package. This group serves as a refinement of the classical Grothendieck–Witt (GW) ring, encoding additional data on nondegenerate symmetric bilinear forms by supplementing them with explicit determinant information. The construction is fundamentally additive, expressed as a fiber product over the determinant map, and lacks the multiplicative structure present in the stable case. Its definition, algebraic properties, and computational algorithms link several branches of modern arithmetic geometry and stable homotopy theory, especially in contexts where orthogonal sum inversion is not performed and where local-to-global phenomena behave subtly.

1. Definition and Algebraic Structure

Given a field $k$ of characteristic not $2$, the unstable Grothendieck–Witt group $\GW^u(k)$ is defined as the fiber product

$\GW^u(k) = \{ (\beta, d) \in \GW(k) \times k^\times \mid \overline{d} = \det(\beta) \in k^\times/(k^\times)^2 \},$

where $\GW(k)$ denotes the stable Grothendieck–Witt ring (generated by isometry classes of nondegenerate symmetric bilinear forms), and $\det: \GW(k) \to k^\times/(k^\times)^2$ is the determinant map from quadratic forms to the square class group. The formal group law is

$$(\beta_1, d_1) + (\beta_2, d_2) = (\beta_1 + \beta_2, \, d_1 d_2).$$

The lack of multiplication in $\GW^u(k)$ differentiates it from the ring structure of $\GW(k)$; instead, $\GW^u(k)$ forms an abelian group. For finite étale algebras $L$, the same construction applies, replacing $k^\times$ and $\GW(k)$ by $L^\times$ and $\GW(L)$. This setup yields an exact sequence

$0 \longrightarrow W(k) \longrightarrow \GW^u(k) \longrightarrow k^\times \longrightarrow 0,$

where $W(k)$ is the Witt ring, i.e., the kernel of the determinant. A generator for $\GW^u(k)$ is given by a pair $(\langle a_1,\dots,a_n\rangle, d)$, with $\langle a_1,\dots,a_n\rangle$ a diagonal quadratic form and $d$ its determinant representative (Atherton et al., 16 Nov 2025).

2. Moduli Spaces and the Unstable/Stable Dichotomy

In abstract Hermitian K-theory, the unstable theory is realized as a sequence of moduli spaces $M_n(-)$ parametrizing nondegenerate $\epsilon$-hermitian forms of fixed rank $n$. For a Poincaré category $(C, \xi)$—where $C$ is a small stable $\infty$-category and $\xi$ a reduced quadratic functor—the moduli space $P_n(C, \xi)$ comprises pairs $(X, q)$ such that the adjoint $q^\sharp : X \to D_0 X$ is an equivalence, with $D_0$ the duality functor (Hebestreit et al., 2021). In classical settings, for a ring $R$ and form parameter $\Lambda$, one defines

$$M_n^X(R; \Lambda) \simeq P_n(DP(R), \xi^\Lambda)_{ht},$$

where $DP(R)$ denotes the perfect derived category. These spaces form an $E_\infty$-monoid under orthogonal direct sum with the canonical hyperbolic form $H$, but only impose "formal inverses" after group completion to produce the genuine GW spectrum: $GW(C, \xi) \simeq M(C, \xi)^{gp}.$ The unstable spaces $P_n$ retain non-virtual information about forms prior to sum inversion, interpolating between classical hermitian K-theory and L-theory. At the level of path-components, their $\pi_0$ recovers classical Witt and L-groups.

3. Computational Frameworks and Algorithms

The A1BrouwerDegrees Macaulay2 software extends the manipulation and computation of unstable Grothendieck–Witt classes, distinguishing between stable (GrothendieckWittClass) and unstable (UnstableGrothendieckWittClass) types. Key algorithmic interfaces include:

• Construction via makeGWuClass(M, d) yielding $(\beta, d)\in \GW^u(L)$ where $M$ is a Gram matrix and $d$ its determinant.
• Comparison of classes through `isIsomorphicForm(α1, α2)$$, requiring both isometry of the GW factors and equality of determinants.</li> <li>Decomposition (<code>getSumDecomposition</code>) and recombination, including divisorial sums of local degrees via `addGWuDivisorial({α_i}, {r_i})$$.
• Explicit computational routines for unstable $\mathbb{A}^1$-Brouwer degrees, utilizing Bézoutian and Newton matrices for global and local degree formulas, respectively.

Worked examples demonstrate correct identification and summation of unstable degrees in explicit rational function cases, validating the divisorial sum formula against Bézoutian-based computations (Atherton et al., 16 Nov 2025).

4. Homotopical and K-theoretic Contexts

The unstable GW-groups naturally appear as low-degree homotopy groups of moduli spaces:

• $\pi_0 P_n Q(C, \xi) \cong L_{n-1}(C, \xi)$, with $L$-groups as path-components.
• In the derived context for rings $R$, one has $$\pi_0 \mathrm{Unimod}^X_0(R; \Lambda) \cong W^X(R; \Lambda)$$ and $$\pi_0 \mathrm{Unimod}^X_1(R; \Lambda) \cong L_0(R; \Lambda)$$.
• Higher homotopy groups of the L-space coincide with those in the split-exact/zero-dimensional case via weight-structure arguments.

A Bott–Genauer fibre sequence relates unstable and stable objects: $$GW(C, \xi[-1]) \rightarrow K(C) \rightarrow GW(C, \xi),$$ allowing calculations of spectral invariants to be transferred between $\mathrm{K}$-theory and GW-theory.

5. Formulas and Relations

In concrete applications, the unstable $\mathbb{A}^1$-degree of a morphism $f/g:P^1_k\rightarrow P^1_k$ is expressed as

$\deg^u(f/g) = (\mathrm{Bez}(f/g), \det\,\mathrm{Bez}(f/g)) \in \GW^u(k),$

where the Bézoutian $\mathrm{Bez}(f/g)$ captures the form and its determinant the unit part. At a $k$-rational zero $r$ of multiplicity $m$, the local degree is

$\deg^u_r(f/g) = (\mathrm{Nwt}_r(f/g), \det\,\mathrm{Nwt}_r(f/g)) \in \GW^u(k),$

with the Newton matrix determined by the principal part of the Laurent expansion (Atherton et al., 16 Nov 2025). The Poincaré–Hopf formula splits the global degree as a sum of local degrees, incorporating an additional divisorial correction factor: $$\deg^u(f/g) = \left( \bigoplus_i \beta_i, \prod_i d_i \cdot \prod_{i<j}(r_i - r_j)^{2 m_i m_j} \right).$$ This contrasts with the stable setting, which lacks the divisorial term.

6. Connections to Milnor–Witt K-theory and Homology Stability

There exists a close relationship between the unstable GW-groups and Milnor–Witt K-theory, especially over local commutative rings with infinite residue field of $\operatorname{char}\neq2$. For $n=2,3$, the symbol map

$\mathrm{K}^{\!MW}_n(R) \longrightarrow \GW_n(R)$

is an isomorphism, as established by Schlichting–Sarwar (Schlichting et al., 2021), relying on improved homology stability for symplectic groups (notably, for all $n\geq 3$, map $H_3(\Sp_{2n-2}(R))\to H_3(\Sp_{2n}(R))$ is an isomorphism). Explicit formulas mirror the relations:

• Multiplicativity: $[ab]=[a]+[b]$
• Steinberg relation: $[a]\cup[1-a]=0$
• Anticommutativity: $[a]\cup[b]+[b]\cup[a]=0$ This suggests a robust parallelism between the structural properties of $\GW^u$ and Milnor–Witt K-theory in the relevant degrees.

7. Remarks, Significance, and Computational Implications

The unstable Grothendieck–Witt group captures non-virtual quadratic form data essential in algebraic and arithmetic geometry, especially where summing invariants does not invert orthogonal addition. Its absence of ring structure and requirement to track determinants explicitly impact both the theory and practical computation, as evidenced by the necessity of handling both form and unit in software frameworks. Over non-perfect fields, symbolic computation of Bézoutians and expansions may require analytic or resultant methods. The moduli-space perspective from Hebestreit–Steimle and the transfer and divisorial correction phenomena in unstable $\mathbb{A}^1$-degrees furnish deep connections to both classical and motivic approaches to Hermitian K-theory (Hebestreit et al., 2021, Atherton et al., 16 Nov 2025, Schlichting et al., 2021).