Playlist - Topological Cyclic Homology

This is a short overview of the main content of our course on Topological Cyclic Homology.

In this video, we give an important motivation for studying Topological Cyclic Homology, so called "trace methods".

In this video, we introduce classical Hochschild homology and discuss the HKR theorem.

In this video, we discuss the Connes operator on Hochschild homology.

In this video, we construct periodic and cyclic homology and compute examples.

In this video, we discuss the cotangent complex and give a proof of the HKR theorem (in its affine version).

We study the cotangent complex more in depth and explain its relation to obstruction theory. As an example we construct the Witt vectors of a perfect ring. This video is a slight digression from the rest of the lecture course and could be skipped.

We define and study Hochschild homology for schemes. This video is a slight digression from the rest of the lecture course and we assume familiarity with schemes.

In this video, we construct Hochschild homology in an arbitrary symmetric-monoidal ∞-category. The most important special case is the ∞-category of spectra, in which we get Topological Hochschild homology.

In this video, we give a proof of Bökstedt's fundamental result showing that THH of 𝔽_p is polynomial in a degree 2 class. This will rely on unlocking its relation to the dual Steenrod algebra and the fundamental fact, that the latter is free as an E₂-algebra.

In this video, we prove certain formal properties of THH, for example that it has a universal property in the setting of commutative rings. We also show base-change properties and use these to compute THH of perfect rings.

In this video we construct an action of the circle group 𝕊¹ = U(1) on the spectrum THH(R). We will see how this is the homotopical generalization of the Connes operator. The key tool will be Connes' cyclic category.

In this video, we explain how to compute THH of the integers. In order to do this we compute it first relative to the element p and then use a spectral sequence to deduce the final result.

In this video we define negative topological cyclic homology and compute it in the case of the ring 𝔽_p. Along the way we also give a more conceptual description of negative cyclic homology and study the homotopy fixed points spectral sequence.

In this video, we introduce another refinement of THH, the topological periodic homology TP. We see how it is an analogue of HP, how it is related to negative cyclic homology, and how to compute it for the field 𝔽_p.

In this video, we introduce the cyclotomic structure on THH. This is a map from THH to the Tate-C_p-construction of THH. This structure is specific to THH and does not exist on ordinary Hochschild homology.

In this video, we finally give the definition of topological cyclic homology. In fact, we will give two definitions: the first is abstract in terms of a mapping spectrum spectrum in cyclotomic spectra and then we unfold this to a concrete definition on terms of negative topological cyclic homology and topological periodic homology.

In this video, we compute TC of the field 𝔽_p with p-elements. As an application of this computation we deduce that THH of 𝔽_p-algebras is in a highly compatible fashion a module over Hℤ. This relates to fundamental work of Kaledin and has some subtle aspects to it, which we carefully discuss.

In this video, we compute TC, CT^- and TP of perfect rings of characteristic p. In order to do that we also have to discuss the Witt vectors and their universal property.

In this video, we "compute" TC of spherical group rings and more generally cyclotomic spectra with Frobenius lifts.