Math Conjectures

Finitistic Dimension Conjecture

OPENMajorConjectureProposed 1960 · Full conjecture

Canonical statement

Let AA be a finite-dimensional algebra over a field, and let pd⁡AM\operatorname{pd}_A M denote the projective dimension of a finitely generated left AA-module MM. Then
findim⁡(A)=sup⁡{pd⁡AM:M∈A-mod, pd⁡AM<∞}<∞. \operatorname{findim}(A)= \sup\{\operatorname{pd}_A M: M\in A\text{-}\mathrm{mod},\ \operatorname{pd}_A M<\infty\}<\infty .
View source LaTeX
Let \(A\) be a finite-dimensional algebra over a field, and let \(\operatorname{pd}_A M\) denote the projective dimension of a finitely generated left \(A\)-module \(M\). Then
\[
  \operatorname{findim}(A)=
    \sup\{\operatorname{pd}_A M:
      M\in A\text{-}\mathrm{mod},\
      \operatorname{pd}_A M<\infty\}<\infty .
\]
ALG-REP-008:[Bass1960Finitistic][ZimmermannHuisgen1992][Cummings2024Finitistic]
Algebra and representation theory

Notes

Let AA be a finite-dimensional algebra over a field. The finitistic dimension conjecture asserts that findim⁡(A)\operatorname{findim}(A), the supremum of the projective dimensions pd⁡AM\operatorname{pd}_A M over all finitely generated left AA-modules MM of finite projective dimension, is itself finite. The question entered the literature through Bass's 1960 paper on finitistic dimension and homological generalizations of semi-primary rings [Bass1960Finitistic].

Although the supremum runs only over modules that individually have finite projective dimension, no general principle is known that bounds these dimensions uniformly, and this is where all known approaches stop. The conjecture has been verified for many structural classes of algebras, and the first decades of work on it, together with the web of related homological conjectures, are surveyed by Zimmermann-Huisgen [ZimmermannHuisgen1992]. Recent work continues to probe its basic behavior, for instance the question of left-right symmetry of having finite finitistic dimension [Cummings2024Finitistic].

The conjecture remains open in general: what is missing is a uniform mechanism controlling finite projective dimensions over an arbitrary finite-dimensional algebra, rather than class-by-class verifications.

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.