Related papers: A uniform treatment of Grothendieck's localization…

In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a…

We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism…

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then…

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…

In this paper, we present a general methodology for investigating the linear stability of localized solutions in PDEs and nonlocal equations on $\mathbb{R}^m$. More specifically, we control the spectrum of the Jacobian…

It is proved that for any Grothendieck site $X$, there exists a coreflection (called $\mathbf{cosheafification}$) from the category of precosheaves on $X$ with values in a category $\mathbf{K}$, to the full subcategory of cosheaves,…

We show that the existence of locally finite stability conditions on the bounded derived category $\mathbf{D}^{b}(X)$ of coherent sheaves on an affine Noetherian scheme $X$ is equivalent to $\dim X=0$. We also study the spaces of stability…

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov--Fokker--Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the…

We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat…

In the paper, for the Cauchy problem on the non-cutoff Boltzmann equation in torus, we establish the global-in-time Gevrey smoothness in velocity and space variables for a class of low-regularity mild solutions near Maxwellians with the…

We prove geometric and cohomological stabilization results for the universal smooth degree $d$ hypersurface section of a fixed smooth projective variety as $d$ goes to infinity. We show that relative configuration spaces of the universal…

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a…

To, say, a proper algebraic or holomorphic space $X/S$, and a coherent sheaf ${\mathcal F}$ on $X$ we identify a functorial ideal, the fitted flatifier, blowing up sequentially in which leads to a flattening of the proper transform of…

Let $(M,g)$ be a smooth connected Riemannian manifold. We show an improvement of flatness theorem for hypersurfaces of $M$ of bounded nonlocal mean curvature in the viscosity sense. It implies local $ C^{1,\alpha}$ regularity of these…

We realise Buchweitz and Flenner's semiregularity map (and hence a fortiori Bloch's semiregularity map) for a smooth variety $X$ as the tangent of a generalised Abel--Jacobi map on the derived moduli stack of perfect complexes on $X$. The…

We show that any homomorphism between Noetherian $F$-finite rings can be factored into a regular morphism between Noetherian $F$-finite rings followed by a surjection. This result establishes an analog of the 'smooth-by-surjective'…

Fulton and MacPherson introduced the notion of bivariant theories and Grothendieck transformations related to Riemann-Roch-theorems. But there are many situations, where such a bivariant theory or a corresponding Grothendieck transformation…

We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to…

Let $G$ be a connected reductive group over a finite extension of $\mathbb{Q}_p$. We show that for each $b \in B(G)$, the strongly regular locus of the inertia stack of $\operatorname{Bun}_G^b$ is open in the inertia stack of…

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and…