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

1. The Zero Divisor Theorem. If [math]\displaystyle{ M \ne 0 }[/math] has finite projective dimension and [math]\displaystyle{ r \in R }[/math] is not a zero divisor on [math]\displaystyle{ M }[/math], then [math]\displaystyle{ r }[/math] is not a zero divisor on [math]\displaystyle{ R }[/math].
2. Bass's Question. If [math]\displaystyle{ M \ne 0 }[/math] has a finite injective resolution then [math]\displaystyle{ R }[/math] is a Cohen–Macaulay ring.
3. The Intersection Theorem. If [math]\displaystyle{ M \otimes_R N \ne 0 }[/math] 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 [math]\displaystyle{ 0 \to G_n\to\cdots \to G_0\to 0 }[/math] denote a finite complex of free R-modules such that [math]\displaystyle{ \bigoplus\nolimits_i H_i(G_{\bullet}) }[/math] has finite length but is not 0. Then the (Krull dimension) [math]\displaystyle{ \dim R \le n }[/math].
5. The Improved New Intersection Conjecture. Let [math]\displaystyle{ 0 \to G_n\to\cdots \to G_0\to 0 }[/math] denote a finite complex of free R-modules such that [math]\displaystyle{ H_i(G_{\bullet}) }[/math] has finite length for [math]\displaystyle{ i \gt 0 }[/math] and [math]\displaystyle{ H_0(G_{\bullet}) }[/math] has a minimal generator that is killed by a power of the maximal ideal of R. Then [math]\displaystyle{ \dim R \le n }[/math].
6. The Direct Summand Conjecture. If [math]\displaystyle{ R \subseteq S }[/math] 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 [math]\displaystyle{ x_1, \ldots, x_d }[/math] be a system of parameters for R, let [math]\displaystyle{ F_\bullet }[/math] be a free R-resolution of the residue field of R with [math]\displaystyle{ F_0 = R }[/math], and let [math]\displaystyle{ K_\bullet }[/math] denote the Koszul complex of R with respect to [math]\displaystyle{ x_1, \ldots, x_d }[/math]. Lift the identity map [math]\displaystyle{ R = K_0 \to F_0 = R }[/math] to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from [math]\displaystyle{ R = K_d \to F_d }[/math] 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 [math]\displaystyle{ A \subseteq R \to S }[/math] 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 [math]\displaystyle{ \operatorname{Tor}_i^A(W,R) \to \operatorname{Tor}_i^A(W,S) }[/math] is zero for all [math]\displaystyle{ i \ge 1 }[/math].
11. The Strong Direct Summand Conjecture. Let [math]\displaystyle{ R \subseteq S }[/math] be a map of complete local domains, and let Q be a height one prime ideal of S lying over [math]\displaystyle{ xR }[/math], where R and [math]\displaystyle{ R/xR }[/math] are both regular. Then [math]\displaystyle{ xR }[/math] is a direct summand of Q considered as R-modules.
12. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let [math]\displaystyle{ R \to S }[/math] 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 [math]\displaystyle{ B_S }[/math] 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 [math]\displaystyle{ M \otimes_R N }[/math] has finite length. Then [math]\displaystyle{ \chi(M, N) }[/math], defined as the alternating sum of the lengths of the modules [math]\displaystyle{ \operatorname{Tor}_i^R(M, N) }[/math] is 0 if [math]\displaystyle{ \dim M + \dim N \lt d }[/math], 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 [math]\displaystyle{ M \ne 0 }[/math] such that some (equivalently every) system of parameters for R is a regular sequence on M.

## References

1. ↑ André, Yves (2018). "La conjecture du facteur direct". Publications Mathématiques de l'IHÉS 127: 71–93. doi:10.1007/s10240-017-0097-9.

* 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.

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