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.  

“The text provides a wonderful introduction describing the background which led to the development of higher matrix factorizations and includes (with proofs and examples) all the theory required to understand the new material and put it in context.” (Benjamin P. Richert, Mathematical Reviews, May, 2017)