In complex analysis, a branch of mathematics, the Thom–Sebastiani theorem states: given the germ f : ( C n 1 + n 2 , 0 ) → ( C , 0 ) {\displaystyle 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 ) {\displaystyle f(z_{1},z_{2})=f_{1}(z_{1})+f_{2}(z_{2})} where f i {\displaystyle f_{i}} are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of f {\displaystyle f} is isomorphic to the tensor product of those of f 1 , f 2 {\displaystyle f_{1},f_{2}} . Moreover, the isomorphism respects the monodromy operators in the sense: T f 1 ⊗ T f 2 = T f {\displaystyle T_{f_{1}}\otimes T_{f_{2}}=T_{f}} . The theorem was introduced by Thom and Sebastiani in 1971. 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
Illusie, Luc (24 April 2016). "Around the Thom-Sebastiani theorem". arXiv:1604.07004. {{cite journal}}: Cite journal requires |journal= (help)