Mathematische Zeitschrift最新文献

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable ({infty })-categories. We study the relative Waldhausen S-construction (S_bullet (F)) of the spherical functor F and show that it has a natural paracyclic structure (“rotation symmetry”). This fulfills a part of the general program of perverse schobers which are conjectural categorical upgrades of perverse sheaves. If we view a spherical functor as defining a schober on a disk, then each component (S_n(F)) of the S-construction gives a categorification of the cohomology of a perverse sheaf on a disk with support in a union of ((n+1)) closed arcs in the boundary. In other words, (S_n(F)) can be interpreted as the Fukaya category of the disk with coefficients in the schober and with support (“stops”) at the boundary arcs. The importance of the paracyclic structure is that it allows us to naturally associate the above data to disks on oriented surfaces. The action of the paracyclic rotation is a categorical analog of the monodromy of a perverse sheaf.

We introduce a new class of almost periodic measures, and consider one-dimensional almost periodic Schrödinger operators with measure-valued potentials. For operators of this kind we introduce a rotation number in the spirit of Johnson and Moser.

We study the 2k-th moment of central values of the family of Dirichlet L-functions to a fixed prime modulus. We establish sharp lower bounds for all real (k ge 0) and sharp upper bounds for k in the range (0 le k le 1).

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand–Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko–Smirnov–Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.

Consider a complex normal surface singularity and its three plurigenera, the m-th (L^2)–plurigenus of Watanabe, the m-th plurigenus of Knöller and the m-th log-plurigenus of Morales. For any of these invariants we construct a double graded (mathbb {Z}[U])–module, whose Euler characteristic is the chosen plurigenus. The three outputs are compared with the analytic lattice cohomology of the germ, whose Euler characteristic is the classical geometric genus.

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to (mathbb {C}^2) and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on (x^2+y^2=2).

For a perfectoid ring R of characteristic 0 with tilt (R^{flat }), we introduce and study a tilting map ((-)^{flat }) from the set of p-adically closed ideals of R to the set of ideals of (R^{flat }) and an untilting map ((-)^{sharp }) from the set of radical ideals of (R^{flat }) to the set of ideals of R. The untilting map ((-)^{sharp }) is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic p introduced in the first author’s previous work. We prove that the two maps

define an inclusion-preserving bijection between the set of ideals J of R such that the quotient R/J is perfectoid and the set of (p^{flat })-adically closed radical ideals of (R^{flat }), where (p^{flat }in R^{flat }) corresponds to a compatible system of p-power roots of a unit multiple of p in R. Finally, we prove that the maps ((-)^{flat }) and ((-)^{sharp }) send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of ({{,textrm{Spec},}}(R)) consisting of prime ideals (mathfrak {p}) of R such that (R/mathfrak {p}) is perfectoid and the subspace of ({{,textrm{Spec},}}(R^{flat })) consisting of (p^{flat })-adically closed prime ideals of (R^{flat }). In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings.

Let R be a commutative noetherian ring and denote by ({{,mathrm{textsf{mod}},}}R) the category of finitely generated R-modules. In this paper, we study KE-closed subcategories of ({{,mathrm{textsf{mod}},}}R), that is, additive subcategories closed under kernels and extensions. We first give a characterization of KE-closed subcategories: a KE-closed subcategory is a torsion-free class in a torsion-free class. As an immediate application of the dual statement, we give a conceptual proof of Stanley-Wang’s result about narrow subcategories. Next, we classify the KE-closed subcategories of ({{,mathrm{textsf{mod}},}}R) when (dim R le 1) and when R is a two-dimensional normal domain. More precisely, in the former case, we prove that KE-closed subcategories coincide with torsion-free classes in ({{,mathrm{textsf{mod}},}}R). Moreover, this condition implies (dim R le 1) when R is a homomorphic image of a Cohen-Macaulay ring (e.g. a finitely generated algebra over a regular ring). Thus, we give a complete answer for the title.

Lai (Geom Topol 25:3629–3690, 2021) used singular Ricci flows, introduced by Kleiner and Lott (Acta Math 219(1):65–134, 2017), to construct a nonnegative Ricci curvature Ricci flow g(t) emerging from an arbitrary 3D complete noncompact Riemannian manifold ((M^3, g_0)) with nonnegative Ricci curvature. We show g(t) is complete for positive times provided (g_0) satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, 2016. arXiv:1603.08726) and Simon and Topping (J Differ Geom 122(3):467–518, 2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (J Eur Math Soc (JEMS) 17(12):3153–3194, 2015) can be adapted here to show g(t) is complete for positive times provided (g_0) is a compactly supported perturbation of a nonnegative sectional curvature metric.

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to N-complexes, one must find an appropriate candidate for the N-analogue of the stable category. We identify this “N-stable category” via the monomorphism category and prove Buchweitz’s theorem for N-complexes over a Grothendieck abelian category. We also compute the Serre functor on the N-stable category over a self-injective algebra and study the resultant fractional Calabi–Yau properties.