Computations of standard bases and syzygies are the heart of Macaulay. The command std is the workhorse of Macaulay\ which computes standard (Gröbner) bases. The commands res and nres are important commands which compute finite free resolutions. These three basic commands are described in section 7.1. The other computations are somewhat more specialized, but still quite useful. These include intersect, lift-std, modulo, quotient and syz. These commands are described in section 7.2.
Each matrix which is input to the commands of this chapter must be graded: see section 6.5 for details. In fact, the commands in this section are the only commands of Macaulay\ which require that their input parameters be graded. If the matrix is not graded, an error message is issued. In this case, either homogenize the matrix, or use the setdegs or setcoldegs command to make the matrix graded.
The commands in this chapter compute a standard basis and/or syzygy
module of a submodule of a free module corresponding to the columns of
a matrix. (see the introduction of Chapter 6). For each
computation command except res and nres, the result is a
matrix together with a standard basis for the corresponding submodule.
Both the standard basis and the matrix have the same name, the name
given to the 
result computation 
. The result of the commands
nres and res is the set of syzygy matrices and
corresponding standard bases. In these cases, the various matrices of
the resolution are indexed starting at 1: e.g. w.1, w.2, etc.
In every case except one the result matrix or matrices correspond to minimal generators of the result submodule(s). The exception is with the res command. The first matrix of the result has a standard basis computed, but the matrix is the original matrix. However, the rest of the matrices do correspond to minimal generators of their submodules.
Each of these commands is designed to work in the case where the base ring is a quotient ring.