Holger Brenner, University of Sheffield
Grothendieck topologies and ideal closure operations
In this talk I would like to report on a project (in progress, joint work with
Manuel Blickle, Essen), intended to relate Grothendieck topologies and closure operations for ideals and submodules. Grothendieck topologies are ageneralization of topological spaces forgetting everything except the notionof a covering. There exist in general non-trivial coverings consisting of only
one map. This allows still to define sheaves and cohomology, and has led tothe construction of the flat and of the etale topology (or site), etale cohomology and the solution to the Weil conjectures in algebraic geometry.
On the other hand, closure operations play an important role in commutative
algebra, like the radical, integral closure, tight closure, Frobenius closure, plusclosure, solid closure. The basic idea is to understand such closure operations
as sheafification in a suitable, non-flat Grothendieck topology, in order to use
this highly developed machinery.
In the classical flat Grothendieck topologies, every coherent ideal can beconsidered as an ideal sheaf in the topology, and the global evaluation givesback the ideal (equivalently, coherent sheaves on affine schemes have trivial
cohomology in the flat or in the etale site). In this sense the classical topologies yield the trivial (identical) closure operation. For non-flat Grothendiecktopologies however an ideal gives at once only an ideal presheaf, and its
sheafification cuts back to an ideal which might be bigger than the original one. So Grothendieck topologies yield closure operations with certain
structural properties, e.g. they are persistent.
If, on the other hand, a given closure operation has these structural properties, then it is possible to construct a (minimal) Grothendieck topology
yielding back this operation: The affine coverings Spec B → Spec A, whichconstitutes the topology, are essentially given by the forcing algebras
B = A[T1 , · · · , Tn ]/(f1 T1 + · · · + fn Tn + f ) ,where f ∈ (f1 , · · · , fn )c , c denoting the given closure operation. Forcing
algebras appeared first in the work of M. Hochster on solid closure and are
fundamental for the geometric understanding of tight closure.
Under the viewpoint of Grothendieck topologies the closure operation itself
is believed to be only a tip of an iceberg, while the topology gives at once anew structure sheaf, a cohomology theory (hard to compute), a concept ofexactness and of resolution, a concept of stalks. For example, in the tight
topology (which is not yet constructed), the stalks are hoped to give big
Cohen-Macaulay algebras in a natural way, like the stalks in the etale site
give the Henselisation of a local ring, and exactness in the tight topology isrelated to phantom acyclity.
Special emphasis will be given on the integral closure, where the correspondence is so far best understood. The “natural” Grothendieck topology for theintegral closure is given by taking the universally Zariski-submersive mappings as coverings. Zariski-submersive means that a subset with an open
preimage must itself be open. This realtionship relies on the fact that bothproperties can be tested with base change to discrete valuation domains (workof Picavet).