 # Homological Properties of Hopf Algebras

Lee, Ji-Hyang (2000) Homological Properties of Hopf Algebras. MSc(R) thesis, University of Glasgow.

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## Abstract

The main aim of this dissertation is to prove a version of the result [Bro98, Proposition 2.3], following the outline suggested in that paper. This result has a distinctly homological flavour, and unsurprisingly relies quite heavily on homological algebra for its proof. We have also drawn upon a wider variety of mathematical techniques, mostly ring theory and Hopf algebraic methods in our discussion. As by-products of the proof, we get a condition for Galois extensions and Frobenius extensions to be equivalent, and also a generalisation of a well-known theorem by Larsson and Sweedler. We discuss this in more detail below. We state the proposition: Proposition. We let H he a Noetherian k-Hopf algebra, where k is an algebraically dosed field. Let K he a central affine sub-Hopf algebra of H with inj.dimK(K) = Krull dim(K) = m. Suppose further that H is a finitely generated K-module. Then inj.dimK(K) = Krull dim(K) = m. Throughout this thesis, inj.dim refers to the injective dimension of the module (defined in Definition 3.2) and Krull dim is the Krull dimension of a commutative Noetherian ring which we also define in Definition 3.2. We also note the fact that if a commutative Noetherian ring has finite injective dimension, then inj.dim(-) = Krull dim(-), as above. The proof is split into four parts, which we summarise briefly here. In the first part, we show that for any ring R which is a Frobenius extension over a subring S the injective dimension of S as a module over itself is equal to the injective dimension of R as an 5-module. Proof of this is obtained from Nakayama and Tsuzuku's fundamental paper ([NT60]) and some basic facts about projective modules. In the second part, we prove that, in the notation above, H is Frobenius over K. This requires that we show H to be a Galois extension over K, which requires substantial preparation as discussed in Chapter 2. The key results come from Kreimer and Takeuchi's paper [KT81] and a paper by Schneider [Sch93]. This step also generalises the Larsson and Sweedler result mentioned before, which states that any finite-dimensional Hopf algebra is Erobenius over any sub-Hopf algebra. The third part shows that if is a Gorenstein ring. The fourth part uses some simple facts on projective modules to place the required restriction on the injective dimension of H as an H-module. These steps, taken together, prove the proposition. This proof is contained in the second section of Chapter 4. Chapter 1 is concerned with the basic definition of a Hopf algebra and discusses some of their basic properties, including comodules, invariants and coinvariants, and smash products. We also introduce Sweedler's sigma notation and use it to describe many Hopf algebraic properties. As indicated above. Chapter 2 contains the majority of the results needed to prove the proposition. We begin by defining and discussing normal sub-Hopf algebras and establish two key results which give an if and only if condition for a sub-Hopf algebra to be normal. This forms part of the proof of the proposition. The main point of the chapter, however, is to show that under certain conditions, Galois extensions are equivalent to Frobenius extensions. A key tool in proving this result is the notion of faithful flatness. We are interested in when a Hopf algebra is flat, faithfully flat, or free over a sub-Hopf algebra. There has been a substantial amount of work done in this area, some of which we discuss in detail, especially results by Schneider [Sch93]. This discussion forms the backbone of the chapter and establishes the crucial fact that the conditions in the proposition imply that H is faithfully flat over K. Finally, we discuss a result from Kreimer and Takeuchi's paper, which gives the condition for equivalence between Galois and Frobenius extensions that we require.

Item Type: Thesis (MSc(R)) Masters Adviser: Kenneth Brown Mathematics 2000 Enlighten Team glathesis:2000-76464 Copyright of this thesis is held by the author. 19 Nov 2019 14:18 19 Nov 2019 14:18 http://theses.gla.ac.uk/id/eprint/76464 View Item