Valuation rings and perfectoid rings are examples of (usually non-Noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-Noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck’s definition of formal smoothness as in the Noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the Noetherian to the non-Noetherian case, as well as generalizing some recent results in the non-Noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.

For a finitely dominated Poincaré duality space $M$, we show how the first author's total obstruction $\mu _M$ to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ in [17] relates to the Reidemeister trace of the identity map of $M$. We then apply this relationship to show that $\mu _M$ vanishes when suitable conditions on the fundamental group of $M$ are satisfied.

In this paper we compute the topological Hochschild homology of quotients of discrete valuation rings (DVRs). Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of $\operatorname {THH}$ (respectively, logarithmic $\operatorname {THH}$) of complete DVRs originally due to Lindenstrauss and Madsen (respectively, Hesselholt and Madsen).

We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$, where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$. In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$. Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$-modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$, its descent properties and the notion of $\mathbb{H}$-affineness. We then prove the $\mathbb{H}$-affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$-category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$-category of modules for $\mathbb{H}({\mathcal{Y}})$, the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$, then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$, where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$. As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$, thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$.

The $\text{SL}\left( 2,\mathbb{C} \right)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth $\text{SL}\left( 2,\mathbb{C} \right)$-representation varieties of the one-holed torus.

In order to develop the foundations of derived logarithmic geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log-étale maps, and use them to define derived log stacks.

Generalizing a conjecture of Deligne, Kontsevich proposed that there should be a notion of Hochschild cohomology of algebras over the little cube operad (or its chain complex) which in a natural way generalizes Hochschild cohomology of associative algebras. He moreover conjectured that the Hochschild cohomology, in this new sense, of an algebra over the little k-cube operad is an algebra over the little (k + 1)-cube operad. In this paper, we precisely state and prove this conjecture.

We define a support variety for a finitely generated module over an artin algebra $\Lambda$ over a commutative artinian ring $k$, with $\Lambda$ flat as a module over $k$, in terms of the maximal ideal spectrum of the algebra ${\rm HH}^*(\Lambda)$ of $\Lambda$. This is modelled on what is done in modular representation theory, and the varieties defined in this way are shown to have many of the same properties as those for group rings. In fact the notions of a variety in our sense and those for principal and non-principal blocks are related by a finite surjective map of varieties. For a finite-dimensional self-injective algebra over a field, the variety is shown to be an invariant of the stable component of the Auslander–Reiten quiver. Moreover, we give information on nilpotent elements in ${\rm HH}^*(\Lambda)$, give a thorough discussion of the ring ${\rm HH}^*(\Lambda)$ on a class of Nakayama algebras, give a brief discussion of a possible notion of complexity, and make a comparison with support varieties for complete intersections.

For an arbitrary field $F$ we study the double scissors congruence groups $A_n(F)$ generated by admissible pairs of simplices in the projective $n$-space ${\mathbb P}_F^n$ over $F$. Let $(L, M)$ be an admissible pair of simplices whose faces are defined as hyperplanes in ${\mathbb P}_F^n$ such that they do not have common faces of the same dimension. When $L$ and $M$ are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology $H^{\ast}({\mathbb P}_F^n\setminus L, M\setminus L; {\mathbb Q})$. The motivic cohomology provides a simple explanation of why the generic part of $A_\bullet$ forms a Hopf algebra with well-defined coproduct. When $n = 2$ we define explicitly the coproduct on $A_2$ which simplifies the approach of Beilinson et al. We also complete the same task for $A_3$ and $A_4$ which enables us to develop further results in another paper connecting Aomoto trilogarithms to the classical ones.

Let $A$ be a commutative noetherian local ring, $I$ an ideal of $A$, and $B\,{=}\,A/I$. Assume that the André-Quillen homology functors $H_n (A,B,-) = 0$ for all $n \,{\ge}\, 3$. Then $A$ is Gorenstein if and only if $B$ is.

The behaviour of the Hasse-Schmidt algebra of higher derivations under localization is studied using André cohomology. Elementary techniques are used to describe the Hasse-Schmidt derivations on certain monomial rings in the nonmodular case. The localization conjecture is then verified for all monomial rings.

In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete.

We consider a positively graded noetherian domain R = ⊕n∈NoRn for which R0 is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the natural morphism π: Y = Proj(R) → Y0 = Spec(R0) is geometrically connected, geometrically normal and of dimension > 1. Then we give bounds on the “ranks” of the n-th homogeneous part H2(R)n of the second local cohomology module of R with respect to R+:= ⊕m>0Rm for n < 0. If Y is in addition normal, we shall see that the R0-modules H2R+ (R)n are torsion-free for all n < 0 and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

Soient $A$ un anneau noethérien, $B$ un anneau régulier essentiellement de type fini sur $A$. Si la cohomologie d’André-Quillen ${{H}^{q}}\left( A,\,B,\,B \right)\,=\,0$ pour tout $q\,\ge \,2$ alors $A$ est un anneau régulier.