Joseph Lipman

Rational singularities with applications to algebraic surfaces and unique factorization

§ o. Some terminology and notation . 196 198 I. Applications to the birational theory of surfaces · . . .. . .. .. . . . . .. . . . 199 § I. Birational behavior of rational singularities . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . 199 § 2. Resolution of singularities by quadratic transformations and normalization (method of Zariski) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~o I § 3. Conclusion of the proof: a key proposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 § 4. Resolution of rational singularities; factorization of proper birational maps into quadratic transformations · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~04

Local homology and cohomology on schemes

Abstract We present a sheafified derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendiecks local duality theorem): for a quasi-compact separated scheme X and a “proregular” subscheme Z-for example, any separated noetherian scheme and any closed subscheme-there is a sort of adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, left-derived completion can be identified with local homology, i.e., the homology of R H em•(RΛZ Q X−). Generalizations of a number of duality theorems scattered about the literature result: the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.

On Complete Ideals in Regular Local Rings

Publisher Summary This chapter presents an approach to unique factorization of complete ideals in two-dimensional regular local rings, based on a decomposition theorem that is valid in all dimensions. The theory of complete ideals in two-dimensional regular local rings was founded by Zariski. Zariskis work was motivated by the birational theory of linear systems on smooth surfaces. Zariski first raised the question of higher-dimensional generalizations, but not much has happened in this respect during the intervering fifty years. The chapter presents the assumption that R is a noetherian local domain, with maximal ideal m and fraction field K , then a prime divisor of R is a valuation ν of K whose valuation ring R ν dominates R such that the transcendence degree of the field R ν /m ν over R / m is as large as possible, that is, dim R – 1.

Foundations of grothendieck duality for diagrams of schemes

Joseph Lipman: Notes on Derived Functors and Grothendieck Duality.- Derived and Triangulated Categories.- Derived Functors.- Derived Direct and Inverse Image.- Abstract Grothendieck Duality for Schemes.- Mitsuyasu Hashimoto: Equivariant Twisted Inverses.- Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors.- Sheaves on Ringed Sites.- Derived Categories and Derived Functors of Sheaves on Ringed Sites.- Sheaves over a Diagram of S-Schemes.- The Left and Right Inductions and the Direct and Inverse Images.- Operations on Sheaves Via the Structure Data.- Quasi-Coherent Sheaves Over a Diagram of Schemes.- Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes.- Simplicial Objects.- Descent Theory.- Local Noetherian Property.- Groupoid of Schemes.- Bokstedt-Neeman Resolutions and HyperExt Sheaves.- The Right Adjoint of the Derived Direct Image Functor.- Comparison of Local Ext Sheaves.- The Composition of Two Almost-Pseudofunctors.- The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams.- Commutativity of Twisted Inverse with Restrictions.- Open Immersion Base Change.- The Existence of Compactification and Composition Data for Diagrams of Schemes Over an Ordered Finite Category.- Flat Base Change.- Preservation of Quasi-Coherent Cohomology.- Compatibility with Derived Direct Images.- Compatibility with Derived Right Inductions.- Equivariant Grothendiecks Duality.- Morphisms of Finite Flat Dimension.- Cartesian Finite Morphisms.- Cartesian Regular Embeddings and Cartesian Smooth Morphisms.- Group Schemes Flat of Finite Type.- Compatibility with Derived G-Invariance.- Equivariant Dualizing Complexes and Canonical Modules.- A Generalization of Watanabes Theorem.- Other Examples of Diagrams of Schemes.

Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes

Part 1: Duality and flat base change on formal schemes by L. Alonso, A. Jeremias, and J. Lipman Part 2: Greenlees-May duality on formal schemes by L. Alonso, A. Jeremias, and J. Lipman Part 3: Non-noetherian Grothendieck duality by J. Lipman Index.

A numerical criterion for simultaneous normalization

We investigate conditions for simultaneous normalizability of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective C-varieties (Xy)y∈Y with normal parameter space Y—algebraic or analytic—admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure OXy is locally independent of y. When the Xy are curves projectivity is not needed, and the statement reduces to the well known δ-constant criterion of Teissier. Proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta, etc.) to more abstract algebraic ones.

Equisingularity and Simultaneous Resolution of Singularities

Zariski defined equisingularity on an n-dimensional hypersurface V via stratification by “dimensionality type,” an integer associated to a point by means of a generic local projection to affine n-space. A possibly more intuitive concept of equisingularity can be based on stratification by simultaneous resolvability of singularities. The two approaches are known to be equivalent for families of plane curve singularities. In higher dimension we ask whether constancy of dimensionality type along a smooth subvariety W of V implies the existence of a simultaneous resolution of the singularities of V along W. (The converse is false.)

The multiple-point schemes of a finite curvilinear map of codimension one

AbstractLetX andY be smooth varieties of dimensionsn−1 andn over an arbitrary algebraically closed field,f: X→Y a finite map that is birational onto its image. Suppose thatf is curvilinear; that is, for allxεX, the Jacobian ϱf(x) has rank at leastn−2. Forr≥1, consider the subschemeNr ofY defined by the (r−1)th Fitting ideal of the

Bivariance, Grothendieck duality and Hochschild homology, II: The fundamental class of a flat scheme-map☆

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