Base change conductors through intersection theory and quotient singularities

Intersectionality on the Horizon: Exploring Autism in Adulthood from a Unique Vantage Point.

Contents Introduction Chapter I. Affine toric varieties § 1. Cones, lattices, and semigroups § 2. The definition of an affine toric variety § 3. Properties of toric varieties § 4. Differential forms on toric varieties Chapter II. General toric varieties § 5. Fans and their associated toric varieties § 6. Linear systems § 7. The cohomology of invertible sheaves § 8. Resolution of singularities § 9. The fundamental group Chapter III. Intersection theory § 10. The Chow ring § 11. The Riemann-Roch theorem § 12. Complex cohomology Chapter IV. The analytic theory § 13. Toroidal varieties § 14. Quasi-smooth varieties § 15. Differential forms with logarithmic poles Appendix 1. Depth and local cohomology Appendix 2. The exterior algebra Appendix 3. Differentials References

Consider a non-commutative algebraic surface, X, and an effective divisor Y on X, as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting. We also apply our theory to some special cases, including the blow up of X in a point, and show that the self intersection of the exceptional divisor is -1. This is used to give an example of a non-commutative surface with a commutative P1 which cannot be blown down, because its self intersection is +1 rather than -1. We also get some results on Hilbert polynomials of modules on X.

* Preface * Conventions and Notation * Part I: Plane Algebraic Curves * Affine Algebraic Curves * Projective Algebraic Curves * The Coordinate Ring of an Algebraic Curve and the Intersections of Two Curves * Rational Functions on Algebraic Curves * Intersection Multiplicity and Intersection Cycle of Two Curves * Regular and Singular Points of Algebraic Curves. Tangents * More on Intersection Theory. Applications * Rational Maps. Parametric Representations of Curves * Polars and Hessians of Algebraic Curves * Elliptic Curves * Residue Calculus * Applications of Residue Theory to Curves * The Riemann-Roch Theorem * The Genus of an Algebraic Curve and of its Function Field * The Canonical Divisor Class * The Branches of a Curve Singularity * Conductor and Value Semigroup of a Curve Singularity * Part II: Algebraic Foundations * Algebraic Foundations * Graded Algebras and Modules * Filtered Algebras * Rings of Quotients. Localization * The Chinese Remainder Theorem * Noetherian Local Rings and Discrete Valuation Rings * Integral Ring Extensions * Tensor Products of Algebras * Traces * Ideal Quotients * Complete Rings. Completion * Tools for a Proof of the Riemann-Roch Theorem * References * Index * List of Symbols

In this chapter, we give some explicit formulas for Neron functions on Riemann surfaces, and an explicit construction due to Coleman. These are with respect to a canonical volume form. We shall also explain how the functions change when we change the metrics. We work complex analytically, in which case we can characterize the Neron functions in a complex analytic fashion, thus finding classical objects called Green’s functions. We shall give the definition of Green’s function and prove its basic properties ab ovo. Actually, we give several proofs for some of the basic theorems, depending on different explicit constructions. One of them depends on the Hodge decomposition and harmonic forms, for which a complete treatment is given in Griffiths-Harris. Another depends on the smoothness of the Green’s function, which may be constructed by theta functions. Different people at different times will use the different techniques for different purposes. Since the construction of the Green’s function by theta functions is given in detail in [La 1], Chapter 13, I do not reproduce this construction here. For the most part, in the application to the intersection theory and Riemann-Roch theorem, we use only the basic formal properties, and the construction of a Green’s function is irrelevant. In the proof of the existence of Faltings volumes, given in Chapter VI, we need to relate the Green’s function on the curve with the Green’s function on the Jacobian, associated with the theta divisor. At this point, we shall make a reference to [La 1], Chapter 13 for one particular property that is needed.

In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G -theory of algebraic stacks and topological G -theory for stacks: we define the latter as the localization of G -theory by topological K -homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G -theory (as well as rational G -theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G -theory on schemes as proved by Thomason.

Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann–Roch theorems of [21, 14]. In this paper, we prove a broad generalization of these Riemann–Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to symmetric monoidal bicategories and from the Euler class to traces of non-identity maps. Our generalization also implies spectral Riemann–Roch theorems. We regard this result as an instantiation of a 2-dimensional generalized cobordism hypothesis. This perspective draws the result close to many others that generalize results about Euler characteristics and classes to bicategorical traces.

In this note, we construct a sequence of l.t. surfaces (Xn)n ∈ ℕ such that KXn is ample for all n and such that (K2Xn)n ∈ ℕ is a strictly increasing series with limit equal to 1. This answers (in the affirmative) a question by Alexeev, cf. [Al], 11·1. Here, an l.t. surface is a normal complex projective surface with at most quotient singularities (which is the same as ‘at most log terminal singularities’). A main result of [Al] implies that it is impossible to find a sequence (Xn)n ∈ ℕ of l.t. surfaces with KXn ample for all n such that K2Xn is strictly decreasing. Although our construction is not too difficult, the example is new and has several interesting implications, see Section 4.Without further explanation, we use some fundamental tools concerning l.t. surfaces like Mumford's intersection theory or the notion of minimality; the reader should consult [Blb] and the references quoted there.

On Grothendieck’s Riemann–Roch theorem

The Riemann–Roch theorem and zero-energy solutions of the Dirac equation on the Riemann sphere

In this paper, we show that the combination of the constructions done in SGA 6 and the A1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann–Roch theorems.

The Riemann–Roch theorem without denominators in motivic homotopy theory

Usually, the Weierstras gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstras gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann– Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve X0(N).

The aim of this note is to give a friendly introduction to Arakelov geometry, starting with a modern reformulation of Minkowski’s geometry of numbers and arriving to the formulation of the arithmetic Grothendieck– Riemann–Roch theorem of Gillet–Soule. In between, we motivate and explain the bases of arithmetic intersection theory.

Gorenstein log del Pezzo surfaces of rank one