This book introduces a theory of higher matrix factorizations for regular sequences and uses it
to describe the minimal free resolutions of high syzygy modules over complete intersections.
Such resolutions have attracted attention ever since the elegant construction of the minimal
free resolution of the residue field by Tate in 1957. The theory extends the theory of matrix
factorizations of a non-zero divisor initiated by Eisenbud in 1980 which yields a description
of the eventual structure of minimal free resolutions over a hypersurface ring. Matrix
factorizations have had many other uses in a wide range of mathematical fields from
singularity theory to mathematical physics.
