Derived direct summands -- Bhargav Bhatt, April 16, 2010

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A variety satisfies the direct summand condition if its structure
sheaf is a summand of that of a finite cover; in characteristic 0, this
condition reduces to normality. We will discuss a derived category
enrichment of this condition. The two conditions diverge in characteristic 0
(the latter characterises rational singularities), but turn out to be very
closely related in positive and mixed characteristic. Using this relation, I
will also discuss some applications, including extensions of standard
cohomological vanishing results beyond the ample cone in positive
characteristic algebraic geometry.