Perverse coherent sheaves and derived equivalences

This thesis focuses on the study of Van den Bergh’s paper, ``Three-dimensional flops and noncommutative rings''. We consider a projective birational morphism $f: Y \to X$ between Noetherian equidimensional schemes over a field $k$, assuming the existence of a point $p \in X$ such that the fiber $f^{-1}(p) = C$ is a curve contained in $Y$, with $f$ being an isomorphism outside $C$. Additionally, we assume that $\mathbf{R} f_* \mathcal{O}_Y = \mathcal{O}_X$. A prototypical example of this scenario is the resolution of rational singularities of surfaces. Direct images of the morphism $f$ are used in the definition of two abelian categories, $\Per^0(Y/X)$ and $\Per^{-1}(Y/X)$, whose objects are referred to as perverse coherent sheaves. The primary result is an equivalence between $\Per^p(Y/X)$, for $p = 0, -1$, and the category of sheaves on $X$ that are coherent over a certain sheaf of finite-dimensional $k$-algebras. This is accomplished by establishing a derived equivalence on the bounded derived category $\mathsf{D}^b(Y)$. In the affine case, this corresponds to the category of finitely generated modules over a finite-dimensional algebra. Such algebras can be studied through their associated quivers, effectively transforming the original geometric problem into a combinatorial one. The first chapter of this thesis reviews key results on tilting in abelian categories, which are central to constructing the category of perverse sheaves. This discussion is primarily based on the work of Happel, Reiten, and Smalø, ``Tilting in Abelian Categories and Quasitilted Algebras'', which builds upon the foundational theory introduced by Beilinson, Bernstein, and Deligne, ``Faisceaux Pervers''. The second chapter revisits classical results related to Morita theory for finite-dimensional algebras. The core idea is to study not the algebras themselves but their module categories. Specifically, two algebras are considered Morita equivalent if their categories of finitely generated modules are equivalent. The combinatorial aspects of Morita theory are particularly evident through the study of quivers associated with these algebras. The third and final chapter contains the main body of this thesis, where we closely follow Van den Bergh’s work. We provide detailed proofs of the aforementioned equivalence in both the affine and general cases, and we explore specific applications to specific examples.