Faithfully flat quotient morphisms by G-actions on factorial affine varieties

Consider a finite flat morphism [Formula: see text] between a reduced, projective, locally planar curve [Formula: see text] and a smooth projective curve [Formula: see text]. Associated to two generic polarizations [Formula: see text] and [Formula: see text] on [Formula: see text], one can construct the corresponding compactified Prym varieties [Formula: see text] and [Formula: see text]. With [Formula: see text] the rank of [Formula: see text], the finite group [Formula: see text] of [Formula: see text]-torsion line bundles acts on [Formula: see text] by tensorization. In this paper, we construct a Fourier–Mukai transform between the derived category of [Formula: see text] and the [Formula: see text]-equivariant derived category of [Formula: see text], providing a derived equivalence between the [Formula: see text]-Hitchin fiber and the dual [Formula: see text]-Hitchin fiber for a dense class of singular spectral curves. This is an extension of Fourier–Mukai transforms on compactified Jacobian varieties constructed by Arinkin and Melo–Rapagnetta–Viviani, which correspond to autoduality of [Formula: see text]-Hitchin fibers.

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc.

On anti-ample vector bundles and nef and big vector bundles

We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact p p -adic groups. Given a word w w in a free Lie algebra L r \mathcal {L}_{r} , it induces a word map φ w : g r → g \varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g} for every semisimple Lie algebra g \mathfrak {g} . Given two words w 1 ∈ L r 1 w_{1}\in \mathcal {L}_{r_{1}} and w 2 ∈ L r 2 w_{2}\in \mathcal {L}_{r_{2}} , we define and study the convolution of the corresponding word maps φ w 1 ∗ φ w 2 ≔ φ w 1 + φ w 2 : g r 1 + r 2 → g \varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g} . By introducing new degeneration techniques, we show that for any word w ∈ L r w\in \mathcal {L}_{r} of degree d d , and any simple Lie algebra g \mathfrak {g} with φ w ( g r ) ≠ 0 \varphi _{w}(\mathfrak {g}^{r})\neq 0 , one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O ( d 4 ) O(d^{4}) self-convolutions of φ w \varphi _{w} . Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ \ell becomes (FRS), locally around identity, after O ( ℓ 4 ) O(\ell ^{4}) self-convolutions, for every semisimple algebraic group G _ \underline {G} . We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word w 0 = [ X , Y ] w_{0}=[X,Y] , we show that φ w 0 ∗ 4 \varphi _{w_{0}}^{*4} is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact p p -adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form Z / p k Z \mathbb {Z}/p^{k}\mathbb {Z} . This allows us to relate them to properties of some natural families of random walks on finite and compact p p -adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p p -adic probabilistic Waring type problems.

For each prime integer p>0 , we construct a standard graded F -rational ring R , over a field K of characteristic p , such that R\otimes_{K}\overline{K} is not F -rational. By localizing, we obtain a flat local homomorphism (R, \mathfrak{m}) \to (S, \mathfrak{n}) such that R is F -rational, S/\mathfrak{m}S is regular (in fact, a field), but S is not F -rational. In the process, we also obtain standard graded F -rational rings R for which R\otimes_{K} R is not F -rational.

Saw varieties

It is a fundamental property of the Chow groups of algebraic schemes that they are contra-functorial with respect to flat morphisms between schemes. While the pullback homomorphism is easy to define at the level of algebraic cycles, the crucial step is to show that the pullback of cycles preserves rational equivalence, so that it descends to the Chow groups. The purpose of this note is to give a natural sheaf theoretic proof of the preservation of rational equivalence under flat pullback on cycles.

Exponentiable Grothendieck categories in flat algebraic geometry

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discuss in detail. We then give a catalogue raisonne of examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology). We discuss a weakly functorial aspect of this result. Splinters are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. We also investigate their mysterious fpqc analogs, and prove that in prime characteristic, they are all regular. This leads us to the problem of descent of regularity by (non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma.

The Norm map between Jacobians and the Prym scheme are widely used in the description of spectral data of G-Higgs pairs, for G varying among classical Lie group. The aim of this paper is to generalize the Norm map to compactified Jacobians and provide a compactification for the Prym scheme. Given a finite, flat morphism between curves (e.g. embeddable noetherian schemes of pure dimension 1), we first define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly reducible, non-reduced) projective curves over a field and the codomain curve is smooth, we introduce the compactified Jacobians parametrizing torsion-free rank-1 sheaves and then we study the Norm and inverse image maps between compactified Jacobians. Finally, we introduce and study the Prym stack defined as the kernel of the Norm map.

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then the locus in $X$ over which $f$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$-rationality, and the ‘Cohen–Macaulay and $F$-injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.

We give an explicit formula for the Deligne pairing for proper and flat morphisms f:Xrightarrow S of schemes, in terms of the determinant of cohomology. The whole construction is justified by an analogy with the intersection theory on non-singular projective algebraic varieties.

We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of Voevodsky's cancellation theorem.

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic morphisms $\varphi:X\rightarrow G$ and $\psi:Y\rightarrow G$, we define their convolution $\varphi*\psi:X\times Y\to G$ by $\varphi*\psi(x,y)=\varphi(x)\cdot\psi(y)$. We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism $\varphi:X\rightarrow G$ which is dominant when restricted to each absolutely irreducible component of $X$, by convolving it with itself finitely many times, one can obtain a flat morphism with reduced fibers of rational singularities, generalizing the main result of our previous paper. Uniform bounds on families of morphisms are given as well. Moreover, as a key analytic step, we also prove the following result in motivic integration; if $\{f_{\mathbb{Q}_{p}}:\mathbb{Q}_{p}^{n}\rightarrow\mathbb{C}\}_{p\in\mathrm{primes}}$ is a collection of functions which is motivic in the sense of Denef-Pas, and $f_{\mathbb{Q}_{p}}$ is $L^{1}$ for any $p$ large enough, then in fact there exists $\epsilon>0$ such that $f_{\mathbb{Q}_{p}}$ is $L^{1+\epsilon}$ for any $p$ large enough.

We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.

In this paper, (higher) Gorenstein flat phantom morphisms over rings will be introduced and studied. To study their relationship, a characterization of Gorenstein flat objects in morphism category is given.Communicated by Alberto Facchini

A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).

The pro-\'etale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual \'etale fundamental group $\pi_1^{\mathrm{et}}$ defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes, generalizing finite \'etale covers. We prove exactness of the general homotopy sequence for the pro-\'etale fundamental group, i.e. that for a geometric point $\bar{s}$ on $S$ and a flat proper morphism $X \rightarrow S$ of finite presentation whose geometric fibres are connected and reduced, the sequence $$ \pi_1^{\mathrm{proet}}(X_{\bar{s}}) \rightarrow \pi_1^{\mathrm{proet}}(X) \rightarrow \pi_1^{\mathrm{proet}}(S) \rightarrow 1 $$ is "nearly exact". This generalizes a theorem of Grothendieck from finite \'etale covers to geometric coverings. We achieve the proof by constructing an infinite (i.e. non-quasi-compact) analogue of the Stein factorization in this setting.

Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in $\operatorname{Spec}R$ using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate between the classes of locally free and flat sheaves, and has serious technical advantages over both. In this paper we show that very flat modules and sheaves are ubiquitous in algebraic geometry: if $S$ is a finitely presented commutative $R$-algebra which is flat as an $R$-module, then $S$ is a very flat $R$-module. This proves a conjecture formulated in the February 2014 version of the long preprint arXiv:1209.2995. We also show that the (finite) very flatness property of a flat module satisfies descent with respect to commutative ring homomorphisms of finite presentation inducing surjective maps of the spectra.