Introduction

Local cohomology was invented by Grothendieck to prove Lefschetz-type

theorems in algebraic geometry. This book seeks to provide an introduction

to the subject which takes cognizance of the breadth of its interactions with

other areas of mathematics. Connections are drawn to topological, geo-

metric, combinatorial, and computational themes. The lectures start with

basic notions in commutative algebra, leading up to local cohomology and

its applications. They cover topics such as the number of defining equations

of algebraic sets, connectedness properties of algebraic sets, connections to

sheaf cohomology and to de Rham cohomology, Grobner bases in the com-

mutative setting as well as for D-modules, the Probenius morphism and

characteristic p methods, finiteness properties of local cohomology modules,

semigroup rings and polyhedral geometry, and hypergeometric systems aris-

ing from semigroups.

The subject can be introduced from various perspectives. We start from

an algebraic one, where the definition is elementary: given an ideal a in a

Noetherian commutative ring, for each module consider the submodule of

elements annihilated by some power of a. This operation is not exact, in the

sense of homological algebra, and local cohomology measures the failure of

exactness. This is a simple-minded algebraic construction, yet it results in

a theory rich with striking applications and unexpected interactions.

On the surface, the methods and results of local cohomology concern

the algebra of ideals and modules. Viewing rings as functions on spaces,

however, local cohomology lends itself to geometric and topological interpre-

tations. From this perspective, local cohomology is sheaf cohomology with

support on a closed set. The interplay between invariants of closed sets and

the topology of their complements is realized as an interplay between local

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