Math Conjectures

Nakayama Conjecture

OPENMajorConjectureProposed 1958 · Standard version

Canonical statement

Let AA be a finite-dimensional algebra over a field and
0⟶AA⟶I0⟶I1⟶I2⟶⋯ 0\longrightarrow {}_AA\longrightarrow I^0\longrightarrow I^1 \longrightarrow I^2\longrightarrow\cdots
its minimal injective resolution as a left module. If every IiI^i is projective, then AA is self-injective.
View source LaTeX
Let \(A\) be a finite-dimensional algebra over a field and
\[
  0\longrightarrow {}_AA\longrightarrow I^0\longrightarrow I^1
    \longrightarrow I^2\longrightarrow\cdots
\] its minimal injective resolution as a left module. If every \(I^i\) is projective, then \(A\) is self-injective.
ALG-REP-010:[Nakayama1958][AuslanderReiten1975][Cummings2024Finitistic]
Algebra and representation theory

Notes

Nakayama's conjecture, from his 1958 paper on algebras with complete homology, concerns the minimal injective resolution 0→A→I0→I1→⋯0\to A\to I^0\to I^1\to\cdots of a finite-dimensional algebra AA as a left module over itself: if every term IiI^i is projective, then AA should already be self-injective [Nakayama1958]. In the language of dominant dimension, an algebra of infinite dominant dimension should be self-injective, which is why the statement is also called the dominant dimension conjecture.

The conjecture sits at the base of a hierarchy of homological conjectures. Auslander and Reiten proposed a generalized version in 1975 [AuslanderReiten1975], and the finitistic dimension conjecture is known to imply the Nakayama conjecture, so progress on finitistic dimension — including recent work on the left–right symmetry of its finiteness [Cummings2024Finitistic] — bears directly on it. The statement itself has been verified for many classes of algebras.

What is missing is the general implication from an infinite resolution by projective–injective modules to self-injectivity, and the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.