Cohen-Macaulay Ring/Positive Charakteristik/Kohomologisches Kriterium für tight closure/Fakt/en/Beweis

Beweis

We consider the short exact sheaf sequences

on coming from the resolution for ( is just the structure sheaf). Because all these sheaves are locally free, taking the absolute Frobenius (and all its iterations) is exact, therefore we get short exact sequences

and cohomology pull-backs . Note also that for and we get

so the image of this map inside is exactly . By the universal property of the absolute Frobenius and of the connecting homomorphisms in cohomology we have

and also

Because of the injectivity of in the given range we have that belongs to the ideal if and only if if and only if .