# 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, , and refer to Noetherian commutative rings; will be a local ring with maximal ideal , and and are finitely generated -modules.

**The Zero Divisor Theorem.**If has finite projective dimension (i.e., has a finite projective (=free when is local) resolution: the projective dimension is the length of the shortest such) and is not a zero divisor on , then is not a zero divisor on .**Bass's Question.**If has a finite injective resolution then is a Cohen–Macaulay ring.**The Intersection Theorem.**If 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 denote a finite complex of free*R*-modules such that has finite length but is not 0. Then the (Krull dimension) .**The Improved New Intersection Conjecture.**Let denote a finite complex of free*R*-modules such that has finite length for and has a minimal generator that is killed by a power of the maximal ideal of*R*. Then .**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*x*be a system of parameters for_{1}, …, x_{d}*R*, let*F*be a free_{•}*R*-resolution of the residue field of*R*with*F*, and let_{0}= R*K*denote the Koszul complex of_{•}*R*with respect to*x*. Lift the identity map_{1}, …, x_{d}*R = K*to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from_{0}→ F_{0}= R*R = K*is not_{d}→ F_{d}*0*.**Existence of Balanced Big Cohen–Macaulay Modules Conjecture.**There exists a (not necessarily finitely generated)*R*-module*W*such that*m*and every system of parameters for_{R}W ≠ W*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*is zero for all_{i}^{A}(W,R) → Tor_{i}^{A}(W,S)*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*that is a balanced big Cohen–Macaulay algebra for_{R}*R*, an*S*-algebra*B*that is a balanced big Cohen-Macaulay algebra for_{S}*S*, and a homomorphism*B*such that the natural square given by these maps commutes._{R}→ B_{S}**Serre's Conjecture on Multiplicities.**(cf.**Serre's multiplicity conjectures.**) Suppose that*R*is regular of dimension*d*and that*M ⊗*has finite length. Then_{R}N*χ(M, N)*, defined as the alternating sum of the lengths of the modules*Tor*is_{i}^{R}(M, N)*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*.

## References

- André, Yves (2018). "La conjecture du facteur direct".
*Publications Mathématiques de l'IHÉS*.**127**: 71–93. arXiv:1609.00345. doi:10.1007/s10240-017-0097-9. MR 3814651.

- 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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