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.

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 $R$ .
Bass's Question. If $M\neq 0$ has a finite injective resolution then $R$ is a Cohen–Macaulay ring.
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.
The New Intersection Theorem. Let $0\to G_{n}\to \cdots \to G_{0}\to 0$ denote a finite complex of free R-modules such that $\bigoplus \nolimits _{i}H_{i}(G_{\bullet })$ has finite length but is not 0. Then the (Krull dimension) $\dim R\leq n$ .
The Improved New Intersection Conjecture. Let $0\to G_{n}\to \cdots \to G_{0}\to 0$ denote a finite complex of free R-modules such that $H_{i}(G_{\bullet })$ has finite length for $i>0$ and $H_{0}(G_{\bullet })$ has a minimal generator that is killed by a power of the maximal ideal of R. Then $\dim R\leq n$ .
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.^[1]
The Canonical Element Conjecture. Let x₁, …, x_d 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, …, x_d. 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 = K_d → F_d is not 0.
Existence of Balanced Big Cohen–Macaulay Modules Conjecture. There exists a (not necessarily finitely generated) R-module W such that m_RW ≠ W and every system of parameters for R is a regular sequence on W.
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).
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 Tor_i^A(W,R) → Tor_i^A(W,S) is zero for all i ≥ 1.
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.
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 B_R that is a balanced big Cohen–Macaulay algebra for R, an S-algebra B_S that is a balanced big Cohen-Macaulay algebra for S, and a homomorphism B_R → B_S such that the natural square given by these maps commutes.
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 Tor_i^R(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.)
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.