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 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 $R\subseteq S$ 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 $x_{1},\ldots ,x_{d}$ be a system of parameters for R, let $F_{\bullet }$ be a free R-resolution of the residue field of R with $F_{0}=R$ , and let $K_{\bullet }$ denote the Koszul complex of R with respect to $x_{1},\ldots ,x_{d}$ . Lift the identity map $R=K_{0}\to F_{0}=R$ to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from $R=K_{d}\to F_{d}$ 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 $A\subseteq R\to S$ 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 $\operatorname {Tor} _{i}^{A}(W,R)\to \operatorname {Tor} _{i}^{A}(W,S)$ is zero for all $i\geq 1$ .
The Strong Direct Summand Conjecture. Let $R\subseteq S$ 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 $R\to S$ 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 $M\otimes _{R}N$ has finite length. Then $\chi (M,N)$ , defined as the alternating sum of the lengths of the modules $\operatorname {Tor} _{i}^{R}(M,N)$ is 0 if $\dim M+\dim N<d$ , and is 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 $M\neq 0$ such that some (equivalently every) system of parameters for R is a regular sequence on M.