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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