In this chapter we discuss several conjectures that arose from the theory of intersection multiplicities and that have come to be known as the homological conjectures. Some of these questions came directly from questions on Serre's intersection multiplicity, while others concern related problems on systems of parameters, modules of finite projective dimension, and properties of complexes.

Many of these conjectures can be verified directly in low dimension (usually dimension at most two). For proving them in higher dimension, three basic methods have been used. The first method, which goes back to the time before some of these questions were explicitly stated, is to use Cohen-Macaulay properties to show that certain homology groups must vanish. The second method is the application of the Frobenius map for rings of positive characteristic, which was introduced to this subject by Peskine and Szpiro in. In addition to proving some of the conjectures in positive characteristic, they introduced a method of reduction from characteristic zero to positive characteristic; this method was completed by Hochster to settle the characteristic zero case of several of these questions. The Frobenius methods will be discussed in the next chapter.

The third method is the most recent and is the use of local Chern characters, which can be used to prove a few of these conjectures in the most difficult case, that of mixed characteristic.