# pull back of hodge bundle via glueing map

Hi,

I need a precise reference for the following fact, which is certainly well known, but I do not find any.

I consider the natural glueing map of pointed curves $\overline{M}_{g_1,n}\times \overline{M}_{g_2,n} \rightarrow \overline{M}_{g_1+g_2+n-1}$. The pullback of the Hodge bundle such be equal to the direct sum of the two hodge bundles of the factors of the product plus n-1 trivial factors. where is this written properly?

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Suppose that we are given a curve $C \to S$ (not necessarily connected) and two disjoint sections, $D$ is the result of gluing these together, and $\nu \colon C \to D$ the natural map. Then there is an exact sequence $$0 \to O_D \to \nu_\ast O_C \to O_S \to 0$$ of sheaves on $D$. Push forward along $p \colon D \to S$. If $C$ has two components and the sections are on distinct components, we see that the first row in the long exact sequence is exact and $R^1 p_\ast O_D \cong R^1 p_\ast O_C$. If $C$ is connected we instead get a short exact sequence $0 \to O_S \to R^1p_\ast O_D \to R^1p_\ast O_C \to 0$. Dualizing gives the result for the Hodge bundle.