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, ${\displaystyle A,R}$, and ${\displaystyle S}$ refer to Noetherian commutative rings; ${\displaystyle R}$ will be a local ring with maximal ideal ${\displaystyle m_{R}}$, and ${\displaystyle M}$ and ${\displaystyle N}$ are finitely generated ${\displaystyle R}$-modules.

1. The Zero Divisor Theorem. If ${\displaystyle M\neq 0}$ has finite projective dimension and ${\displaystyle r\in R}$ is not a zero divisor on ${\displaystyle M}$, then ${\displaystyle r}$ is not a zero divisor on ${\displaystyle R}$.
2. Bass's Question. If ${\displaystyle M\neq 0}$ has a finite injective resolution then ${\displaystyle R}$ is a Cohen–Macaulay ring.
3. The Intersection Theorem. If ${\displaystyle M\otimes _{R}N\neq 0}$ 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 ${\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0}$ denote a finite complex of free R-modules such that ${\displaystyle \bigoplus \nolimits _{i}H_{i}(G_{\bullet })}$ has finite length but is not 0. Then the (Krull dimension) ${\displaystyle \dim R\leq n}$.
5. The Improved New Intersection Conjecture. Let ${\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0}$ denote a finite complex of free R-modules such that ${\displaystyle H_{i}(G_{\bullet })}$ has finite length for ${\displaystyle i>0}$ and ${\displaystyle H_{0}(G_{\bullet })}$ has a minimal generator that is killed by a power of the maximal ideal of R. Then ${\displaystyle \dim R\leq n}$.
6. The Direct Summand Conjecture. If ${\displaystyle R\subseteq S}$ 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]
7. The Canonical Element Conjecture. Let ${\displaystyle x_{1},\ldots ,x_{d}}$ be a system of parameters for R, let ${\displaystyle F_{\bullet }}$ be a free R-resolution of the residue field of R with ${\displaystyle F_{0}=R}$, and let ${\displaystyle K_{\bullet }}$ denote the Koszul complex of R with respect to ${\displaystyle x_{1},\ldots ,x_{d}}$. Lift the identity map ${\displaystyle R=K_{0}\to F_{0}=R}$ to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from ${\displaystyle R=K_{d}\to F_{d}}$ is not 0.
8. 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.
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 ${\displaystyle A\subseteq R\to S}$ 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 ${\displaystyle \operatorname {Tor} _{i}^{A}(W,R)\to \operatorname {Tor} _{i}^{A}(W,S)}$ is zero for all ${\displaystyle i\geq 1}$.
11. The Strong Direct Summand Conjecture. Let ${\displaystyle R\subseteq S}$ be a map of complete local domains, and let Q be a height one prime ideal of S lying over ${\displaystyle xR}$, where R and ${\displaystyle R/xR}$ are both regular. Then ${\displaystyle xR}$ is a direct summand of Q considered as R-modules.
12. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let ${\displaystyle R\to S}$ 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 ${\displaystyle B_{S}}$ 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.
13. Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that ${\displaystyle M\otimes _{R}N}$ has finite length. Then ${\displaystyle \chi (M,N)}$, defined as the alternating sum of the lengths of the modules ${\displaystyle \operatorname {Tor} _{i}^{R}(M,N)}$ is 0 if ${\displaystyle \dim M+\dim N<d}$, and is positive if the sum is equal to d. (N.B. Jean-Pierre Serre proved that the sum cannot exceed d.)
14. Small Cohen–Macaulay Modules Conjecture. If R is complete, then there exists a finitely-generated R-module ${\displaystyle M\neq 0}$ such that some (equivalently every) system of parameters for R is a regular sequence on M.

References

• Homological conjectures, old and new, Melvin Hochster, Illinois Journal of Mathematics Volume 51, Number 1 (2007), 151-169.
• On the direct summand conjecture and its derived variant by Bhargav Bhatt.