Hans Schoutens

Projective Dimension and the Singular Locus

Abstract For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.

Bounds in cohomology

We introduce a measure of complexity for affine algebras and their finitely generated modules, in terms of the degrees of the polynomials used in their description. We then study how various cohomological operations and numerical invariants are uniformly bounded with respect to these complexities. We apply this to give first order characterisations of certain algebraic-geometric properties. This enables us to apply the Lefschetz Principle to transfer properties between various characteristics. As an application, we obtain the following version of the Zariski-Lipman Conjecture in positive characteristic: letR be the local ring of a pointP on a hypersurface over an algebraically closed fieldK such that the module ofK-invariant derivations onR is free, thenP is a non-singular point, provided the characteristic is larger than some bound only depending on the degree of the hypersurface.

Log-terminal singularities and vanishing theorems via non-standard tight closure

Generalizing work of Smith and Hara, we give a new characterization of logterminal singularities for finitely generated algebras over C, in terms of purity properties of ultraproducts of characteristic p Frobenii. As a first application we obtain a Boutot-type theorem for log-terminal singularities: given a pure morphism Y → X between affine Q-Gorenstein varieties of finite type over C, if Y has at most a log-terminal singularities, then so does X . The second application is the Vanishing for Maps of Tor for log-terminal singularities: if A ⊆ R is a Noether Normalization of a finitely generated C-algebra R and S is an R-algebra of finite type with log-terminal singularities, then the natural morphism Tori (M, R) → Tori (M, S) is zero, for every A-module M and every i ≥ 1. The final application is Kawamata-Viehweg Vanishing for a connected projective variety X of finite type over C whose affine cone has a log-terminal vertex (for some choice of polarization). As a corollary, we obtain a proof of the following conjecture of Smith: if G is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety X , then for any numerically effective line bundle L on any GIT quotient Y := X//G, each cohomology module Hi(Y,L) vanishes for i > 0, and, if L is moreover big, then Hi(Y,L−1) vanishes for i < dim Y .

Asymptotic Homological Conjectures in Mixed Characteristic

In this final chapter, we discuss some of the homological conjectures. Although now theorems in equal characteristic, many remain conjectures in mixed characteristic. Whereas there may be no consensus as to which conjectures count as ‘homological’, an extensive list of them together with their interconnections, can be found in Hochster’s authoritative treatise [43].

LEFSCHETZ PRINCIPLE APPLIED TO SYMBOLIC POWERS

In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein, Lazarsfeld and Smith [3] (for the affine case over ) and to Hochster and Huneke [4] (for the general case). Let A be a regular ring containing a field K. Let be a radical ideal of A and let h be the maximum of the heights of its minimal primes. Then for all n, we have an inclusion , where the first ideal denotes the hnth symbolic power of . In prime characteristic, this result admits an easy tight closure proof due to Hochster and Huneke. In this paper, the characteristic zero version is obtained from this by an application of the Lefschetz Principle.

Existentially Closed Models of the Theory of Artinian Local Rings

The class of all Artinian local rings of length at most l is ∀ 2 -elementary, axiomatised by a finite set of axioms τ t l . We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory o τ l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τ t l is companionable, with model-companion o τ l .

Computably Categorical Fields via Fermat’s Last Theorem

We construct a computable, computably categorical field of infinite transcendence degree over the rational numbers, using the Fermat polynomials and assorted results from algebraic geometry. We also show that this field has an intrinsically computable (infinite) transcendence basis.

On the vanishing of Tor of the absolute integral closure

Let R be an excellent local domain of positive characteristic with residue field k and let R + be its absolute integral closure. If Tor R (R + ,k) vanishes, then R is weakly F-regular. If R has at most an isolated singularity or has dimension at most two, then R is regular.

A non-standard proof of the Briançon-Skoda theorem

Using a tight closure argument in characteristic p and then lifting the argument to characteristic zero with the aid of ultraproducts, I present an elementary proof of the Briancon-Skoda Theorem: for an m-generated ideal a of C[[X 1 ,...,X n ]], the m-th power of its integral closure is contained in a. It is well-known that as a corollary, one gets a solution to the following classical problem. Let f be a convergent power series in n variables over C which vanishes at the origin. Then f lies in the ideal generated by the partial derivatives of f.

Pure subrings of regular rings are pseudo-rational

We prove a generalization conjectured by Aschenbrenner and Schoutens (2003) of the Hochster-Roberts-Boutot-Kawamata Theorem: let R → S be a pure homomorphism of equicharacteristic zero Noetherian local rings. If S is regular, then R is pseudo-rational, and if R is moreover Q-Gorenstein, then it is pseudo-log-terminal.