• Variation of GIT Computations

    I am currently reading Ed Segal’s Equivalences Between GIT Quotients of Landau-Ginzburg B-Models. In this post I want to do a computation from Section 1.1 of this paper regarding, in the notation of the paper, an isomorphism of cohomology groups \[ H^\ast_{\mathcal{X}}(\mathcal{O}(i)) \cong H^\ast_{X_\pm}(\mathcal{O}(i)),\ i \in[-1,1]. \] This is not too difficult and this post is largely for my personal understanding. Let’s recall the setup.

  • (No) Exceptional Bundles Over a Curve

    Today we will talk about exceptional objects, or lack thereof, in the derived category of a smooth and proper curve, say \(C\). Recall, an exceptional object \(E\) in a k-linear triangulated category \(\mathcal{T}\) is an object such that \[ \mathcal{T}(E,E[i])\cong k[0]. \]

  • Hochschild (Co)Homology of Curves and Projective Spaces

    Today we’ll talk about the Hochschild (Co)Homology of schemes and compute it in the case the scheme is either the projective space or a smooth and proper curve. Throughout X will be a smooth projective scheme over a field k. We assume k is algebraically closed and of characteristic zero. To any scheme we can associate the bounded derived category of coherent sheaves on X, denote it by D(X).

  • Purpose of the blog

    Hello and welcome to my blog. The main objective is to discuss computations and ideas that I have found interesting/found hard to understand/still don’t understand. My goal is a post a week for the forseable future. The types of posts will vary widely including but not limited to algebraic geometry, algebraic topology, group theory, and differential geometry.

  • New Site

    I’m currently working on my academic site. This site will be built using Jekyll. It seems like the easiest way to start blogging (about mathematics) with long term sustainability and full customization. For now, I just hope the teaching links work.