MOTIVIC HOCHSCHILD HOMOLOGY

Let S be a quasi-compact, quasi-separated scheme, such as the spectrum of a field k, and let SH(S) be Morel and Voevodski's stable motivic homotopy category over S. Let R be a motivic ring spectrum in SH(S); we define the motivic Hochschild homology of R via the derived tensor product: MHH(R)=R ∧_{R ∧ R^{op}} R. This spectrum can be considered the motivic analogue of topological Hochschild homology. The purpose of the first part of this thesis is to investigate the homotopy structure of motivic Hochschild homology of MZ/p- the Suslin-Voevodsky mod-p motivic cohomology ring spectrum for p any prime number. We will in particular consider the case S=Spec(k) is the spectrum of an algebraically closed field k of characteristic different from p. In this context, the homotopy groups of M Z/p are a polynomial ring over the field with p elements on a single generator τ in bidegree (0,-1). To do so, we introduce a tri-graded spectral sequence in motivic homotopy groups inspired by an analogue in stable homotopy theory defined by J. Greenlees in a 2016 paper. We will use our spectral sequence to investigate different spectra. As a motivation for the following computations, we first have a look at MHH(MZ/p) itself. Here we produce a first quadrant, multiplicative, tri-graded spectral sequence with E^2_{s,t,*}=π_{s,*}(MHH(MZ/p)) ⊗_{π_{0,*}(MZ/p)} π_{t,*}(A(p)) converging to π_{s+t,*}(MZ/p). A(p) here denotes the dual motivic Steenrod algebra. Notice that each page in our spectral sequences is tri-graded, meaning we can decompose the associated graded object as a direct sum of modules indexed on three natural numbers- the degrees. In this case, we will not provide a direct argument computing all the homotopy groups: we will simply use the spectral sequence to directly determine the structure in low degrees at the prime p=2. This allows us to perceive how abundant are elements of τ-torsion and motivates the following step: study a τ-inverted and mod-τ^{p-1} versions of MHH(MZ/p), which we are abe to determine via analogous spectral sequences. This strategy proves effective: by combining these two computations with some additional algebraic results, it is possible to reconstruct π_{*,*}(MHH(MZ/p)), as proven in a recent paper by Dundas, Hill, Ormsby and Oestvaer. The final part (chapter D) begins with the following observation: in the event the motivic ring spectrum R is E_∞, one has an equivalence MHH(R)=S^{1}_s ⊗ R, where the tensor symbol encodes an action of simplicial sets on Alg_{E_∞}(SH(S)) given by geometric realization along the associated simplicial object. This equivalence is proven as its classical analogue in topology via the equivalences using the bar complex resolution. In this part of the thesis, we extend the above action along the embedding of simplicial sets into motivic spaces, so that we are entitled to tensoring with more exotic motivic spheres, such as the multiplicative group scheme or the projective line. To do so, we tackle the problem from the more general perspective of left tensored ∞-categories. The construction of the extended action moreover allows us to prove that in the event R= R ∧ R is an idempotent E_∞ motivic ring spectrum, such as rational motivic cohomology, we have equivalences: S^1_s ⊗ R=G_m ⊗ R=P^1 ⊗ R=R. Chapters B, C and D already appeared on the arXiv.