Thom–Sebastiani Theorem
In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ [math]\displaystyle{ f : (\mathbb{C}^{n_1 + n_2}, 0) \to (\mathbb{C}, 0) }[/math] defined as [math]\displaystyle{ f(z_1, z_2) = f_1(z_1) + f_2(z_2) }[/math] where [math]\displaystyle{ f_i }[/math] are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of [math]\displaystyle{ f }[/math] is isomorphic to the tensor product of those of [math]\displaystyle{ f_1, f_2 }[/math].[1] Moreover, the isomorphism respects the monodromy operators in the sense: [math]\displaystyle{ T_{f_1} \otimes T_{f_2} = T_f }[/math].[2] The theorem was introduced by Thom and Sebastiani in 1971.[3]
Observing that the analog fails in positive characteristic, Deligne suggested that, in positive characteristic, a tensor product should be replaced by a (certain) local convolution product.[2]
References
- ↑ Fu, Lei (30 December 2013). A Thom-Sebastiani Theorem in Characteristic p.
- ↑ 2.0 2.1 Illusie 2016, § 0.
- ↑ Sebastiani, M.; Thom, R. (1971). "Un résultat sur la monodromie". Inventiones Mathematicae 13 (1–2): 90–96. doi:10.1007/BF01390095. Bibcode: 1971InMat..13...90S.
- Illusie, Luc (24 April 2016). Around the Thom-Sebastiani theorem.
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