GAGA Theorems For Singularity Categories

In this master thesis,we present several GAGA type results about singularity categories.Firstly,for a complex projective variety,we show that its(bounded)singularity category is naturally equivalent to that of its analytificaton.Secondly,we define the torsion singularity category of a formal scheme,and then show that under Orlov's condition(ELF),for the formal completion of a Noetherian scheme along a closed subset,its torsion singularity category is equivalent to the singularity of the original scheme with support in the closed subset.Lastly,using Artin Approximation Theorem,we prove that for a Noetherian local ring with isolated singularity,its singularity category is equivalent,up to direct summands,to the singularity category of its Henselization,while the latter is equivalent to the singularity category of its completion.