Homological conjectures in commutative algebra

In commutative algebra, the "Homological Conjectures" have been the focus of intense research activity for the last 30 years. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth. Here is a definitive list given by Mel Hochster. A, R, and S will always refer to Noetherian commutative rings. R will be a local ring, and M and N are finitely-generated R-modules.

1. The Zerodivisor Theorem. Let R &sube S be complete local domains, and let Q be a height one prime ideal of S lying over xR, where R and R/xR are both regular. Then xR is a direct summand of Q considered as R-modules.
2. Bass's Question. If M &ne 0 has a finite injective resolution then R is a Cohen-Macaulay ring.
3. The Intersection Theorem. If M &otimes_R N &ne 0 has finite length, then the Krull dimension of N (i.e., the dimension of R modulo the annihilator of N) is at most the projective dimension of M.
4. The New Intersection Theorem. Let 0 &rarr G_n &rarr &hellip &rarr G₀ &rarr 0 denote a finite complex of free R-modules such that &oplus_iH_i(G_&bull) has finite length but is not 0. Then the (Krull dimension) dim R &le n.
5. The Improved New Intersection Conjecture.
6. The Direct Summand Conjecture.
7. The Canonical Element Conjecture.
8. Existence of Balanced Big Cohen-Macaulay Modules Conjecture.
9. Cohen-Macaulayness of Direct Summands Conjecture.
10. The Vanishing Conjecture for Maps of Tor.
11. The Strong Direct Summand Conjecture.
12. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture.
13. Serre's Multiplicity Conjectures.
14. Small Cohen-Macaulay Modules Conjecture.