We discuss a number of the local homological conjectures, many of which are now theorems in equal characteristic and conjectures in mixed characteristic. One focus is the syzygy theorem of Evans and Griffith, and its connection with the direct summand conjecture, the existence of big Cohen-Macaulay modules and algebras, and tight closure theory.
"Homological conjectures, old and new." Illinois J. Math. 51 (1) 151 - 169, Spring 2007. https://doi.org/10.1215/ijm/1258735330