In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the homological conjectures have been a focus of research activity in commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

 since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

 properties of a commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

 to its internal ring structure, particularly its Krull dimension

Krull dimension

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....

 and depth.

The following list given by Melvin Hochster

Melvin Hochster

Melvin Hochster is an eminent American mathematician, regarded as one of the leading commutative algebraists active today. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.Hochster attended Stuyvesant High School, where he was...

 is considered definitive for this area. A, R, and S refer to Noetherian

Noetherian

In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...

 commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

s. R will be a local ring

Local ring

In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...

 with maximal ideal m_R, and M and N are finitely-generated R-modules.

1. The Zerodivisor Theorem. If M ≠ 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 ∈ R is not a zerodivisor on M, then r is not a zerodivisor on R.
2. Bass's Question. If M ≠ 0 has a finite injective resolution then R is a Cohen-Macaulay ring
   Cohen-Macaulay ring
   In mathematics, a Cohen–Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality....
   
   .
3. The Intersection Theorem. If M ⊗_R N ≠ 0 has finite length, then the Krull dimension of N (i.e., the dimension of R modulo the annihilator
   Annihilator (ring theory)
   In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
   
    of N) is at most the projective dimension of M.
4. The New Intersection Theorem. Let 0 → G_n → … → G₀ → 0 denote a finite complex of free R-modules such that ⊕_iH_i(G_•) has finite length but is not 0. Then the (Krull dimension) dim R ≤ n.
5. The Improved New Intersection Conjecture. Let 0 → G_n → … → G₀ → 0 denote a finite complex of free R-modules such that H_i(G_•) has finite length for i > 0 and H₀(G_•) has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n.
6. 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.
7. The Canonical Element Conjecture. Let x₁, …, x_d be a system of parameters
   System of parameters
   In commutative algebra, a system of parameters for a local ring of Krull dimension d with maximal ideal m is a set of elements x1, ..., xd that satisfies any of the following equivalent conditions:...
   
    for R, let F_• be a free R-resolution of the residue field
   Residue field
   In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
   
    of R with F₀ = R, and let K_• denote the Koszul complex
   Koszul complex
   In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
   
    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.
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 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.
11. 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.
12. 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.
13. Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures
    Serre's multiplicity conjectures
    In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry...
    
    .) 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. 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 M ≠ 0 such that some (equivalently every) system of parameters for R is a regular sequence
    Regular sequence
    In mathematics, a regular sequence may be:* Regular sequence , in commutative algebra, a sequence of elements defining the depth of a module* Regular Cauchy sequence, in real analysis, a quickly converging Cauchy sequence...
    
    on M.