In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, homological conjectures have been a focus of research activity in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promine ...
since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various
homological properties of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
to its internal ring structure, particularly its
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
and
depth.
The following list given by
Melvin Hochster is considered definitive for this area. In the sequel,
, and
refer to
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s;
will be a
local ring In abstract algebra, more specifically 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 nu ...
with maximal ideal
, and
and
are
finitely generated -modules.
# The Zero Divisor Theorem. If
has finite
projective dimension
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characteri ...
and
is not a
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
on
, then
is not a zero divisor on
.
# Bass's Question. If
has a finite
injective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to de ...
then
is a
Cohen–Macaulay ring
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finit ...
.
# The Intersection Theorem. If
has finite length, then the
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
of ''N'' (i.e., the dimension of ''R'' modulo the
annihilator of ''N'') is at most the
projective dimension
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characteri ...
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
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.
# The Canonical Element Conjecture. Let
be a
system of parameters for ''R'', let
be a free ''R''-resolution of the
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. Frequently, ''R'' is ...
of ''R'' with
, and let
denote the
Koszul complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its h ...
of ''R'' with respect to
. Lift the identity map
to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from
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
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
is zero for all
.
# The Strong Direct Summand Conjecture. Let
be a map of complete local domains, and let ''Q'' be a height one prime ideal of ''S'' lying over
, where ''R'' and
are both regular. Then
is a
direct summand
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of ''Q'' considered as ''R''-modules.
# Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let
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
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
has finite length. Then
, defined as the alternating sum of the lengths of the modules
is 0 if
, and is positive if the sum is equal to ''d''. (N.B.
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the i ...
proved that the sum cannot exceed ''d''.)
# Small Cohen–Macaulay Modules Conjecture. If ''R'' is complete, then there exists a finitely-generated ''R''-module
such that some (equivalently every) system of parameters for ''R'' is a
regular sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection.
Definitions
F ...
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 variantby Bhargav Bhatt.
{{DEFAULTSORT:Homological Conjectures In Commutative Algebra
Commutative algebra
Homological algebra
Conjectures
Unsolved problems in mathematics