I enjoy writing expository notes, although usually with a heavy bias towards algebraic topics. If you have any question or found any typos, feel free to contact me. I do not claim any originality.

• Geometric and Formal Smoothness
  Abstract: The main goal of this note is to prove that the coordinate ring of a smooth affine k-variety is a formally smooth k-algebra, hence satisfies the infinitesimal lifting property. In order to keep the exposition mostly self-contained, we introduce, in reasonable detail, first, the theory of derivations and algebraic differential forms, second, the geometric notion of smoothness for varieties, and, third, the algebraic considerations about locally free and projective modules needed. Throughout, we highlight the similarities to—as well as the intuition derived from—the classical differential-geometric picture. The final section is dedicated to a brief discussion of Hochschild homology and the Hochschild–Kostant–Rosenberg theorem, hinting at the flavour of the noncommutative theory.

• Diagram Lemmata
  Abstract: The aim of this note is to provide a concise overview of the standard diagram lemmata. To this end, we briefly recall the fundamental idea underlying exact sequence. We then thoroughly introduce the concept of diagram chase and indicate common use cases for the lemmata at hand. Moreover, we cover the Salamander Lemma which provides a formal framework for all diagram chase techniques used throughout.

• Extensions and Ext
  Abstract: Let R be a ring and let M,N be R-modules. This article proves, in detail, the correspondence between equivalence classes of extensions of M by N and Ext_R^1(M,N).  Special attention is paid towards proving that any kind of choices made along the way are negligible.

• The Exponential Sheaf Sequence
  Abstract: We introduce the formalism of sheaves, though concentrating on elementary properties of sheaf morphisms. This is used as starting point to introduce the exponential sheaf sequence on the complex numbers. We then explore the relation between exact sequences of sheaves and their respective restrictions to global section, especially focusing on the preservation of left- or right-exactness in certain cases.