# English translation or summary of “Relevements modulo $p^2$ et decomposition du complexe de de Rham. ”

I'm looking for either an English translation or summary of the article "Relevements modulo $p^2$ et decomposition du complexe de de Rham." by Deligne. I'm attempting to read this for background research, but I'm having a surprisingly hard time pinpointing the necessary portions- I'm not close to fluent in French, and I'm doing some readings where I think it might be very helpful to have this perspective.

Worst comes to worst, I guess I can always run it through google translate paragraph by paragraph, but I'd really like to know if there's an English translation or summary around so I can at least tell where I should concentrate my efforts. Thanks for your responses.

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I could not find Deligne & Illusie's 1987 translation in English online either. If you purchase or check out complete works sometimes they may have translation on one side of the book while original on the other. However the algebraic proof of Deligne and Illusie of vanishing theorem can be accessed on page 105 in Lectures on Vanishing Theorem

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You can find a quick overview of the theorem and proof here:

There seems to be no known freely accessible translation into English of that article, although it may be interesting for many advanced students in algebraic geometry to try it for the language requirements of their Ph.D. degrees. I just add here the reference of the original French article you talk about for those interested in it:

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I know this question is quite old, but I wanted to point out that Illusie has an article "Frobenius and Hodge Degeneration" attempting to explain the proof in the book Introduction to Hodge Theory by Bertin et al.. Demailly (one of the authors) has a copy of the book on his website so I think it's okay to link to it.

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