Segal’s Gamma rings and universal arithmetic

Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use G. Almkvist’s global Witt construction to explore the relation with J. Borger ${\mathbb F}_1$-geometry and compute the Witt functor-ring ${\mathbb W}_0({\mathbb S})$ of the simplest Γ-ring ${\mathbb S}$. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification ${\overline{{\rm Spec\,}{\mathbb Z}}}$ which acquires a structure sheaf of ${\mathbb S}$-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on ${\overline{{\rm Spec\,}{\mathbb Z}}}$, we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann–Roch problem for D.

HOLONOMIC FILTERED MODULES IN THE CATEGORY OF MICRO-STRUCTURE SHEAVES

Since the late sixties, Various Auslander regularity conditions have been widely investigated in both commutative and non-commutative cases, [6]. J. E. Bjork studied the Auslander regularity on graded rings and positively filtered Noetherian Noetherian rings, [7]. In [7] the notion of a holonomic module over positively filtered rings has been introduced. Recently, Huishi, in his Ph. D. Thesis [12], investigate Auslander regularity condition and holonomity of graded and filtered modules over Zariski filtered rings. In this work, using the micro-structure sheaf techniques we characterize a generalized Holonomic sheaf theory. We introduce a general study of Auslander regularity on the micro-structure sheaves. We calculate the global dimension of modules over the micro- structure sheaves O . The main results are contained in Theorem (2.4), Theorem (3.6) and Theorem (3.7).

Formality of -objects

We show that a$\mathbb{P}$-object and simple configurations of$\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov’s cup product lemma in which one factor is the (1, 0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a [Formula: see text]-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

Separable monoids in Dqc (X)

Suppose{({\mathscr{T}},\otimes,\mathds{1})}is a tensor triangulated category. In a number of recent articles Balmer defines and explores the notion of “separable tt-rings” in{{\mathscr{T}}}(in this paper we will call them “separable monoids”). The main result of this article is that, if{{\mathscr{T}}}is the derived quasicoherent category of a noetherian schemeX, then the only separable monoids are the pushforwards by étale maps of smashing Bousfield localizations of the structure sheaf.

Scheme representation for first-order logic

My research concerns a construction of "logical schemes," geometric entitieswhich represent logical theories in much the same way that algebraic schemesrepresent rings. These involve two components: a semantic spectral spaceand a syntactic structure sheaf. As in the algebraic case, we can recover atheory from its scheme representation (up to a conservative completion) andthe structure sheaf is local in a certain logical sense. From these ane pieceswe can build up a 2-category of logical schemes which share some of the niceproperties of algebraic schemes.

A Characterization of Ordinary Abelian Varieties by the Frobenius Push-Forward of the Structure Sheaf II

In this paper, we prove that a smooth projective variety X of characteristic p > 0 is an ordinary abelian variety if and only if KX is pseudo-effective and $F_{*}^{e}{\mathcal {O}}_{X}$ splits into a direct sum of line bundles for an integer e with pe > 2.

BERNSTEIN–SATO POLYNOMIALS AND TEST MODULES IN POSITIVE CHARACTERISTIC

In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein–Sato polynomials for the structure sheaf${\mathcal{O}}_{X}$and a hypersurface$(f=0)$in$X$, where$X$is a regular variety over an$F$-finite field of positive characteristic (see Mustaţă,Bernstein–Sato polynomials in positive characteristic, J. Algebra321(1) (2009), 128–151). He shows that the suitably interpreted zeros of his Bernstein–Sato polynomials correspond to the$F$-jumping numbers of the test ideal filtration${\it\tau}(X,f^{t})$. In the present paper we generalize Mustaţă’s construction replacing${\mathcal{O}}_{X}$by an arbitrary$F$-regular Cartier module$M$on$X$and show an analogous correspondence of the zeros of our Bernstein–Sato polynomials with the jumping numbers of the associated filtration of test modules${\it\tau}(M,f^{t})$provided that$f$is a nonzero divisor on$M$.