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Splitting vector bundles outside the stable range and \(\mathbb A^1\) – homotopy sheaves of punctured affine spaces. (English) Zbl 1329.14045
The authors establish the following theorem: if \(k\) is an algebraically closed field of characteristic different from \(2\), \(X\) a smooth affine \(4\)-fold over \(k\) and \(E\) a rank three vector bundle over \(X\), then \(E\) splits off a trivial summand if and only if the third Chern class of \(E\) (in Chow theory) vanishes. This establishes a low-dimensional case of a conjecture of Murthy.
The proof is a crafty application of \(\mathbb{A}^1\)-algebraic topology, as initiated by Morel. In analogy with classical topology, there is a “space” (simplicial sheaf) \(\mathrm{BGL}_n\) such that for a smooth affine variety \(X\), the set of weak \(\mathbb{A}^1\)-homotopy classes \([X, \mathrm{BGL}_n]_{\mathbb{A}^1}\) is in natural bijection with the set of isomorphism classes of rank \(n\) vector bundles over \(X\). There is a map \(\mathrm{BGL}_{n-1} \to \mathrm{BGL}_n\) corresponding to adding a trivial bundle, and so the question of splitting off a trivial bundle is converted into a lifting problem. This can be tackled by obstruction theory. Again in analogy with classical topology, there is an appropriate type of fibre sequence \(\mathbb{A}^n \setminus 0 \to \mathrm{BGL}_{n-1} \to \mathrm{BGL}_n\). Thus the splitting problem is intimately related to the unstable homotopy sheaves of \(\mathbb{A}^n \setminus 0 \simeq S^{n-1} \wedge \mathbb{G}_m^{\wedge n}\). As established by F. Morel [\(\mathbb A^1\)-algebraic topology over a field. Lecture Notes in Mathematics 2052. Berlin: Springer (2012; Zbl 1263.14003)], one has \(\underline{\pi}_d^{\mathbb{A}^1} (\mathbb{A}^n \setminus 0) = 0\) for \(d < n-1\) and \(\underline{\pi}_{n-1}^{\mathbb{A}^1} (\mathbb{A}^n \setminus 0) = \underline{K}_n^{MW}\). This is where the primary (Euler class) obstruction lives. The authors explicitly compute \(\underline{\pi}_{3}^{\mathbb{A}^1} (\mathbb{A}^3 \setminus 0)\) (and a host of other unstable homotopy sheaves), which allows them to completely control the question of rank three vector bundles over affine four-folds (since the higher homotopy sheaves are irrelevant for dimension reasons). This way their theorem is reduced to a (not entirely trivial) computation.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
55S35 Obstruction theory in algebraic topology
13C10 Projective and free modules and ideals in commutative rings
19A13 Stability for projective modules
19D45 Higher symbols, Milnor \(K\)-theory
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