Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.

We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over an algebraic closure of a finite field if the variety admits a normal projective compactification with boundary locus of codimension greater than or equal to $2$.

Let $\pi :X\,\to \,S$ be a finite type morphism of noetherian schemes. A smooth formal embedding of $X$ (over $S$) is a bijective closed immersion $X\,\subset \,\mathfrak{X}$ , where $\mathfrak{X}$ is a noetherian formal scheme, formally smooth over $S$. An example of such an embedding is the formal completion $\mathfrak{X}={{Y}_{/X}}$ where $X\,\subset \,Y$ is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham(co)homology.

Our main application is an explicit construction of the Grothendieck residue complex when $S$ is a regular scheme. By definition the residue complex is the Cousin complex of ${{\pi }^{!}}{{O}_{S}}$, as in $[\text{RD}]$. We start with $\text{I-C}$. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf ${{\oplus }_{q}}K_{X/S}^{q}.$ We then use smooth formal embeddings to obtain the coboundary operator $\delta :K_{X/S}^{q}\to K_{X/S}^{q+1}.$ We exhibit a canonical isomorphism between the complex $K_{X/S}^{\cdot },\delta $ and the residue complex of $[\text{RD}]$. When $\pi $ is equidimensional of dimension $n$ and generically smooth we show that ${{\text{H}}^{-n}}K_{X/S}^{\cdot }$ is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi $[\text{KW}]$.

Another issue we discuss is Grothendieck Duality on a noetherian formal scheme $\mathfrak{X}$ . Our results on duality are used in the construction of $K_{X/S}^{\cdot }$.