Macaulay is a system for computing in the fields of algebraic geometry and commutative algebra; it is intended to provide a computational research tool for working mathematicians. The system performs algebraic manipulation on polynomials, matrices, ideals, polynomial rings, modules, maps between rings, and complexes of modules. The possible manipulations include the computation of standard (Gröbner) bases, modules of syzygies, finite free resolutions, Hilbert polynomials and functions. Using these basic operations, a variety of derived operations are possible, such as projections, ideal intersections, and the computation of coherent sheaf cohomology groups.
Macaulay differs in a number of significant ways from other computer algebra systems. The computation of standard bases is its fundamental operation, rather than simplification and factoring. Unlike other systems which provide for the computation of standard bases, submodules of free modules can uniformly be used wherever ideals can be used. Macaulay is written in the language C and its design has been optimized for execution on small systems; Macaulay is available, and reasonably powerful, on a Macintosh microcomputer. Computations in Macaulay are interruptable, with the full power of the system available to study and use partial as well as complete results of computations. Macaulay is command-driven, and is quickly learned by mathematicians having no experience with computers.
Macaulay has been continuously evolving since 1977. Macaulay\ is currently implemented on the Macintosh and Sun, and has been used by mathematicians in a number of countries, including Canada, France, Germany, Italy, Norway, and the USA.