Thom–Sebastiani Theorem

In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ f : (\mathbb{C}^{n_1 + n_2}, 0) \to (\mathbb{C}, 0) defined as f(z_1, z_2) = f_1(z_1) + f_2(z_2) where f_i are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of f is isomorphic to the tensor product of those of f_1, f_2.{{cite journal |last1=Fu |first1=Lei |title=A Thom-Sebastiani Theorem in Characteristic p |date=30 December 2013 |arxiv=1105.5210 }} Moreover, the isomorphism respects the monodromy operators in the sense: T_{f_1} \otimes T_{f_2} = T_f.{{harvnb|Illusie|2016|loc=§ 0.}}

The theorem was introduced by Thom and Sebastiani in 1971.{{cite journal |last1=Sebastiani |first1=M. |last2=Thom |first2=R. |title=Un résultat sur la monodromie |journal=Inventiones Mathematicae |date=1971 |volume=13 |issue=1–2 |pages=90–96 |doi=10.1007/BF01390095 |bibcode=1971InMat..13...90S |s2cid=121578342 }}

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.

References

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  • {{cite journal |last1=Illusie |first1=Luc |title=Around the Thom-Sebastiani theorem |date=24 April 2016 |arxiv=1604.07004 }}

Category:Theorems in complex analysis

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