# Homological conjectures in commutative algebra

In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. 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.

The following list given by Melvin Hochster is considered definitive for this area. In the sequel, $A,R$ , and $S$ refer to Noetherian commutative rings; $R$ will be a local ring with maximal ideal $m_{R}$ , and $M$ and $N$ are finitely generated $R$ -modules.

1. The Zero Divisor Theorem. If $M\neq 0$ has finite projective dimension (i.e., $M$ has a finite projective (=free when $R$ is local) resolution: the projective dimension is the length of the shortest such) and $r\in R$ is not a zero divisor on $M$ , then $r$ is not a zero divisor on $R$ .
2. Bass's Question. If $M\neq 0$ has a finite injective resolution then $R$ is a Cohen–Macaulay ring.
3. The Intersection Theorem. If $M\otimes _{R}N\neq 0$ 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 → Gn → … → G0 → 0 denote a finite complex of free R-modules such that iHi(G) has finite length but is not 0. Then the (Krull dimension) dim R ≤ n.
5. The Improved New Intersection Conjecture. Let 0 → Gn → … → G0 → 0 denote a finite complex of free R-modules such that Hi(G) has finite length for i > 0 and H0(G) has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n.
6. The Direct Summand Conjecture. If R ⊆ S is a module-finite ring extension with R regular (here, R need not be local but the problem reduces at once to the local case), then R is a direct summand of S as an R-module. The conjecture was proven by Yves André using a theory of perfectoid spaces.
7. The Canonical Element Conjecture. Let x1, …, xd be a system of parameters for R, let F be a free R-resolution of the residue field of R with F0 = R, and let K denote the Koszul complex of R with respect to x1, …, xd. Lift the identity map R = K0 → F0 = R to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from R = Kd → Fd is not 0.
8. Existence of Balanced Big Cohen–Macaulay Modules Conjecture. There exists a (not necessarily finitely generated) R-module W such that mRW ≠ W and every system of parameters for R is a regular sequence on W.
9. Cohen-Macaulayness of Direct Summands Conjecture. If R is a direct summand of a regular ring S as an R-module, then R is Cohen–Macaulay (R need not be local, but the result reduces at once to the case where R is local).
10. The Vanishing Conjecture for Maps of Tor. Let A ⊆ R → S be homomorphisms where R is not necessarily local (one can reduce to that case however), with A, S regular and R finitely generated as an A-module. Let W be any A-module. Then the map ToriA(W,R) → ToriA(W,S) is zero for all i ≥ 1.
11. The Strong Direct Summand Conjecture. Let R ⊆ S be a map of 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.
12. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let R → S be a local homomorphism of complete local domains. Then there exists an R-algebra BR that is a balanced big Cohen–Macaulay algebra for R, an S-algebra BS that is a balanced big Cohen-Macaulay algebra for S, and a homomorphism BR → BS such that the natural square given by these maps commutes.
13. Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that M ⊗R N has finite length. Then χ(M, N), defined as the alternating sum of the lengths of the modules ToriR(M, N) is 0 if dim M + dim N < d, and positive if the sum is equal to d. (N.B. Jean-Pierre Serre proved that the sum cannot exceed d.)
14. Small Cohen–Macaulay Modules Conjecture. If R is complete, then there exists a finitely-generated R-module M ≠ 0 such that some (equivalently every) system of parameters for R is a regular sequence on M.