Isambard Goodbody

Strong Generation of the Derived Category by Injectives

The notion of strong generation can be used to extract properties of a geometric or algebraic object from its derived category. For example, the global dimension of a Noetherian ring is equal to the minimum number of steps it takes to generate every object from the regular module. A result of Neeman shows that a Noetherian separated scheme is regular and of finite dimension if and only if its perfect derived category admits a strong generator. However, unlike the algebraic situation, there is no candidate generator playing the role of the regular module. This means that the problem of extracting the precise dimension of a scheme using this technique remains open. Orlov has conjectured that the dimension of a smooth projective variety is equal to the smallest generation time ranging across all possible generators. We take a different approach, motivated from the algebraic situation and a result of Rickard, by taking our candidate generator to be all of the indecomposable injectives. We’ll present some results showing that one can bound the dimension of the scheme in this way.