Theory Of Sheaves Research Articles

Local, colocal, and antilocal properties of modules and complexes over commutative rings

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles.

On the minimal model program for projective varieties with pseudo-effective tangent sheaf

In this paper, we develop a theory of pseudo-effective sheaves on normal projective varieties. As an application, by running the minimal model program, we show that projective klt varieties with pseudo-effective tangent sheaf can be decomposed into Fano varieties and Q-abelian varieties.

The d-critical structure on the Quot scheme of points of a Calabi–Yau 3-fold

The Artin stack [Formula: see text] of zero-dimensional sheaves of length [Formula: see text] on [Formula: see text] carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack [Formula: see text], another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points [Formula: see text], which is a global critical locus for every [Formula: see text], and also carries a derived-in-flavor d-critical structure besides the one induced by the potential [Formula: see text]. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on [Formula: see text], where [Formula: see text] is a locally free sheaf of rank [Formula: see text] on a projective Calabi–Yau [Formula: see text]-fold [Formula: see text]. Finally, we prove that the perfect obstruction theory on [Formula: see text] induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential [Formula: see text].

Ramification theory of reciprocity sheaves, II Higher local symbols

We construct a theory of higher local symbols along Paršin chains for reciprocity sheaves. Applying this formalism to differential forms, gives a new construction of the Paršin–Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato’s higher local class field theory in the geometric case. The higher local symbols satisfy various reciprocity laws. The main result of the paper is a characterization of the modulus attached to a section of a reciprocity sheaf in terms of the higher local symbols.

Constructible sheaves on schemes

We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived ∞-category of proétale sheaves, while constructible sheaves are those that are lisse on a stratification.We show that constructible sheaves satisfy proétale descent. We also establish a t-structure on constructible sheaves in a wide range of cases. We finally provide a toolset to manipulate categories of constructible sheaves with respect to the choices of coefficient rings, and use this to prove that our notions reproduce and extend the various approaches to, say, constructible ℓ-adic sheaves in the literature.

Cyclic forms on DG-Lie algebroids and semiregularity

Given a transitive DG-Lie algebroid (\mathcal{A},\rho) over a smooth separated scheme X of finite type over a field \mathbb{K} of characteristic zero, we define a notion of connection \nabla\colon\mathbf{R}\Gamma(X,\operatorname{Ker}\rho)\to\mathbf{R}\Gamma(X,\Omega_X^1[-1]\otimes\operatorname{Ker}\rho) and construct an L_\infty -morphism between DG-Lie algebras f\colon \mathbf{R}\Gamma(X,\operatorname{Ker}\rho) \rightsquigarrow\mathbf{R}\Gamma(X,\Omega_X^{\leq1}[2]) associated to a connection and to a cyclic form on the DG-Lie algebroid. In this way, we obtain a lifting of the first component of the modified Buchweitz–Flenner semiregularity map in the algebraic context, which has an application to the deformation theory of coherent sheaves on X admitting a finite locally free resolution. Another application is to the deformations of (Zariski) principal bundles on X .

Deformation theory of orthogonal and symplectic sheaves

We show that the deformation functor of an orthogonal (resp. symplectic) sheaf over a smooth projective scheme admits a miniversal pro-family, identifying its space of first-order deformations with the first hypercohomology space of a complex which is naturally constructed out of the orthogonal (resp. symplectic) sheaf. We also provide an obstruction theory of these objects whose target is the second hypercohomology space of this complex.

Ramification theory of reciprocity sheaves, I: Zariski–Nagata purity

Abstract We prove a Zariski–Nagata purity theorem for the motivic ramification filtration of a reciprocity sheaf. An important tool in the proof is a generalization of the Kato-Saito reciprocity map from geometric global class field theory to all reciprocity sheaves. As a corollary we obtain cut-by-curves and cut-by-surfaces criteria for various ramification filtrations. In some cases this reproves known theorems, in some cases we obtain new results.

O-minimal GAGA and a conjecture of Griffiths

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.

Equivariant sheaves for profinite groups

We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We resolve a serious deficit in the existing theory by constructing a good notion of equivariant presheaves, with a suitable equivariant sheafification functor. Using equivariant sheafification, we develop the general theory of equivariant sheaves of modules over a ring, give explicit constructions of infinite products and introduce an equivariant analogue of skyscraper sheaves.These results underlie recent work by the authors which proves that there is an algebraic model for rational G-spectra in terms of equivariant sheaves over profinite spaces. That model is constructed in terms of Weyl-G-sheaves over the space of closed subgroups of G, where the term Weyl indicates that the stalk over H is H-fixed. In this paper, we prove that Weyl-G-sheaves of R-modules form an abelian category with enough injectives and is a coreflective subcategory of equivariant sheaves of R-modules.We end the paper with a structural result that provides another way to conveniently build equivariant sheaves from simpler data. We prove that a G-equivariant sheaf over a profinite base space X is a colimit of equivariant sheaves over finite discrete spaces Xi with actions of finite groups Gi, where X is the limit of the Xi and G is the limit of the Gi.

KLR and Schur Algebras for Curves and Semi-Cuspidal Representations

Abstract Given a smooth curve $C$, we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on $C$. In particular, they provide a geometric realization for certain affinized symmetric algebras. When $C={\mathbb{P}}^1$, a version of curve Schur algebra turns out to be Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver in any characteristic. As a consequence, we argue that one should not expect to have a reasonable theory of parity sheaves for affine quivers.

The six functors for Zariski-constructible sheaves in rigid geometry

We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.

On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups

Let (R,𝔪,𝕜) be a three-dimensional Cohen–Macaulay analytically unramified local ring and I an 𝔪-primary R-ideal. Write X=Proj(⊕n∈ℕIn¯tn). We prove some consequences of the vanishing of H2(X,𝒪X), whose length equals the constant term e¯3(I) of the normal Hilbert polynomial of I. Firstly, X is Cohen–Macaulay. Secondly, if the extended Rees ring A := ⊕n∈ℤIn¯tn is not Cohen–Macaulay, and either R is equicharacteristic or I¯=m, then e¯2(I)−lengthRI2¯/II¯≥3; this estimate is proved using Boij–Söderberg theory of coherent sheaves on ℙ𝕜2. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, HE2(X,Im𝒪X)=0 for all integers m, where E is the exceptional divisor in X. Finally, if additionally R is regular and X is pseudorational, then the adjoint ideals In ˜, n≥1 satisfy In ˜=IIn−1 ˜ for every n≥3. The last two results are related to conjectures of J. Lipman (1994).

Neutral-fermionic presentation of the K-theoretic Q-function

We show a new neutral-fermionic presentation of Ikeda-Naruse's $K$-theoretic $Q$-functions, which represent a Schubert class in the $K$-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields straightforward proof of two types of Pfaffian formulas for them. We present a dual space of $G\Gamma$, the vector space generated by all $K$-theoretic $Q$-functions, by constructing a non-degenerate bilinear form that is compatible with the neutral fermionic presentation. We give a new family of dual $K$-theoretic $Q$-functions, their neutral-fermionic presentations, and Pfaffian formulas.

Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20].

Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs

We develop a theory of sheaves and cohomology on the category of proper modulus pairs. This complements [KMSY21], where a theory of sheaves and cohomology on the category of non-proper modulus pairs has been developed.

$n$-quasi-abelian categories vs $n$-tilting torsion pairs

It is a well established fact that the notions of quasi-abelian categories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t -structures. Firstly, we extend this picture into a hierarchy of n -quasi-abelian categories and n -tilting torsion classes. We prove that any n -quasi-abelian category \mathcal{E} admits a "derived" category D(\mathcal{E}) endowed with a n -tilting pair of t -structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t -structures as quotient categories of coherent functors, generalizing Auslander's Formula. Thirdly, we apply our results to Bridgeland's theory of perverse coherent sheaves for flop contractions. In Bridgeland's work, the relative dimension 1 assumption guaranteed that f_\ast -acyclic coherent sheaves form a 1 -tilting torsion class, whose associated heart is derived equivalent to D(Y) . We generalize this theorem to relative dimension 2 .

On the formal degree conjecture for simple supercuspidal representations

We prove the formal degree conjecture for simple supercuspidal representations of symplectic groups and quasi-split even special orthogonal groups over a p-adic field, under the assumption that p is odd. The essential part is to compute the Swan conductor of the exterior square of an irreducible local Galois representation with Swan conductor 1. It is carried out by passing to the equal characteristic local field and using the theory of Kloosterman sheaves.

TOPOLOGIES ON SCHEMES AND MODULUS PAIRS

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

It has been proven by Serre, Larsen–Pink and Chin, that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have “the same” pi _0 and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent F-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent F-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent F-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects.