The Direct Summand Conjecture in Dimension Three

In [Ho1], Hochster proved the following results: Theorem (Hochster). If R is a regular Noetherian ring which contains a field and S ⊃ R is a module-finite R-algebra, then R is a direct summand of S as an R-module. Theorem (Hochster). If S is any local ring which contains a field and x1,...,xn is a system of parameters for S, then for every integer k ≥ 0, (x1 · · · xn) k ̸ ∈ ( x k+1 1,...,x k+1 n S. The mixed characteristic case of these results is easy for dimR ≤ 2 but relatively little is known for dimR> 2. The general statements, which are equivalent, became known as the direct summand and monomial conjectures. The principal advance in this subject occurred in Hochster’s 1983 article [Ho2] in which he introduced the canonical element conjecture. He demonstrated that this new conjecture was equivalent to the other two both overall and in the case of fixed dimension. He also showed that it was sufficiently strong to imply the validity of a number of other homological conjectures which had been shown to follow from the big Cohen-Macaulay modules conjecture.