One finite algebra with arbitrarily long terminating resolutions

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Family 198 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

Solving algebraic relations often proceeds in stages: describe generators, then relations among them, then relations among those relations, and so on. A projective resolution records this process. Some modules require infinitely many stages; others eventually terminate.

The little finitistic-dimension conjecture asks a subtle uniformity question. For one fixed finite-dimensional algebra, is there a common upper bound on the lengths of all the resolutions that do terminate?

Family 198 claims a negative answer over the complex numbers. It constructs a single finite-dimensional algebra with finite-dimensional modules whose projective dimensions are finite but arbitrarily large. Changing the module does not change the algebra. That fixed-algebra requirement is the central distinction between the claimed counterexample and familiar families of increasingly complicated algebras.

2. Background and essential definitions

Resolutions and projective dimension

A finite-dimensional complex algebra $A$ is a finite-dimensional vector space with associative multiplication and an identity. It need not be commutative. A left $A$-module is a vector space on which algebra elements act linearly and compatibly with multiplication.

A free module is a direct sum of copies of $A$. A projective module is a direct summand of a free module. Projective modules are the building blocks used in resolutions because maps from them lift through surjections.

A finite projective resolution of a module $N$ has the form

$$0\longrightarrow P_r\longrightarrow\cdots \longrightarrow P_1\longrightarrow P_0 \longrightarrow N\longrightarrow0,$$

where each $P_i$ is projective and the sequence is exact. Exactness says that each stage records precisely the relations left by the previous one. The least possible $r$ is the projective dimension $\operatorname{pd}_A N$. If no finite resolution exists, the dimension is infinite.

For a finite-dimensional algebra, a finitely generated module is also finite-dimensional over the coefficient field. Its successive kernels in a projective resolution are called syzygies.

Three suprema with different meanings

The global dimension takes a supremum over projective dimensions without discarding infinite values. The little finitistic dimension instead keeps only finite projective dimensions of finitely generated modules:

$$\operatorname{fin.dim}A= \sup\{\operatorname{pd}_A N: N\text{ finite-dimensional},\ \operatorname{pd}_A N<\infty\}.$$

The big finitistic dimension, written $\operatorname{Fin.dim}A$, permits all modules, including infinitely generated ones, while still retaining only those of finite projective dimension. Therefore the little dimension is at most the big one.

An algebra can have modules of infinite projective dimension and nevertheless have finite finitistic dimension. Such modules are excluded from the defining supremum. The counterexample must supply increasingly long finite resolutions, not simply one infinite resolution.

For a noncommutative algebra, left and right modules need not behave alike. Right $A$-modules are left modules over the opposite algebra $A^{\mathrm{op}}$, whose multiplication reverses the order of factors.

3. Development history and previous partial results

Bass publicized finitistic-dimension questions in 1960. Two questions must be distinguished: whether little and big finitistic dimensions are equal, and whether they are finite.

Huisgen-Zimmermann disproved equality by constructing, for each $n\ge2$, a finite-dimensional monomial algebra with little dimension $n$ and big dimension $n+1$. Those examples vary with $n$, and both dimensions remain finite for each algebra. They therefore do not refute the universal finiteness conjecture.

Several large classes satisfy little finitistic finiteness. Green, Kirkman, and Kuzmanovich proved it for finite-dimensional monomial algebras, whose defining relations are paths rather than linear combinations of paths. Green and Huisgen-Zimmermann proved it for Artin algebras with radical cube zero. The radical is the ideal carrying the algebra’s nonsemisimple behavior; requiring its cube to vanish restricts how long products of such elements can persist.

Igusa and Todorov introduced syzygy invariants giving another proof of the radical-cube-zero case and finiteness for representation dimension at most three. These results show that homological complexity is bounded under important structural restrictions, without settling arbitrary finite-dimensional algebras.

Cummings later proved an equivalence between universal little finitistic finiteness and a universal left–right symmetry assertion for finiteness. This provides a way to convert a counterexample into pronounced asymmetry, but is not itself a construction of infinite finitistic dimension.

Rickard established another connection: if injective modules generate the unbounded derived category in the appropriate sense, then big finitistic dimension is finite. Derived categories organize complexes while identifying maps that preserve homology; allowing arbitrary coproducts permits infinite direct sums. The new counterexample therefore has a consequence for injective generation as well.

The construction draws on universal localization and on formulas for trivial extension algebras developed by Minamoto and Yamaura. A separate characteristic-two construction in family 199 also claims infinite finitistic dimension. The present manuscript emphasizes that its complex example is distinct and constructed independently.

4. What the new manuscripts claim

The manuscript, dated 23 September 2026, claims a finite-dimensional unital complex algebra $A$ such that for every integer $m\ge1$, there is a finite-dimensional left module $N_m$ with

$$2m-2\le\operatorname{pd}_A N_m<\infty.$$

The same $A$ works for all $m$. Hence

$$\operatorname{fin.dim}A=\operatorname{Fin.dim}A=\infty.$$

The second equality follows because the big supremum includes all the modules already witnessing the first. This also refutes the corresponding universal finiteness statement for Artin algebras, since finite-dimensional complex algebras belong to that class.

By Rickard’s theorem, injective left $A$-modules do not generate the unbounded derived category of all left modules.

An auxiliary algebra with opposite extremes

Combining the example with Cummings’ triangular-algebra construction, the manuscript claims another finite-dimensional complex algebra $\Lambda$ satisfying

$$\operatorname{fin.dim}\Lambda= \operatorname{Fin.dim}\Lambda=\infty,$$

while

$$\operatorname{fin.dim}(\Lambda^{\mathrm{op}})= \operatorname{Fin.dim}(\Lambda^{\mathrm{op}})=0.$$

All displayed dimensions use left modules. Thus $\Lambda$ has infinite finitistic dimensions on the left and zero on the right. Its injective left modules do not generate, whereas its injective right modules do.

This is a statement about the auxiliary $\Lambda$. The manuscript makes no corresponding right-side assertion about the original algebra $A$. Keeping the two algebras separate prevents an unjustified strengthening of the main construction.

5. Why the advance matters

Provisional importance: 5/5. If correct, the example disproves a central conjecture in the representation theory and homological algebra of finite-dimensional algebras.

Its force comes from the difference between finite description and uniformly bounded homological behavior. A fixed algebra has finitely many basis elements and a finite multiplication table, yet the claimed modules can encode terminating processes of arbitrarily large depth.

The consequences also challenge expectations of symmetry. The auxiliary algebra would support arbitrarily long finite resolutions on one side while every module of finite projective dimension on the other side is already projective. This does not make left and right global behavior identical to those finite suprema; it identifies an extreme asymmetry in the specific finitistic invariants.

6. The main idea

Start with a process that eventually dies, but at no fixed time

The first stage constructs a finitely presented auxiliary complex algebra $R$, a central idempotent $e$, and an automorphism $\alpha$. A central idempotent satisfies $e^2=e$ and commutes with the algebra, so $eY$ is a direct summand of every module $Y$.

Define a selection operation $H$ by taking $eY$ and changing its action through $\alpha$. The manuscript constructs finite-dimensional modules $Y_m$ satisfying

$$H^j(Y_m)\ne0\quad(0\le j<m), \qquad H^m(Y_m)=0.$$

The auxiliary algebra is a group algebra whose group contains an indexed family of central involutions. The automorphism shifts that family. Finite matrix quotients detect arbitrarily long independent segments, and prescribed characters make selection survive for exactly the desired number of rounds.

At this stage, $R$ is finitely presented but need not be finite-dimensional. The remaining work is to encode its selection behavior inside one finite-dimensional algebra without changing the encoding when $m$ changes.

Simulate selection with a fixed derived tensor operation

A quadratic presentation of $R$ is encoded in a directed algebra $B$ with three vertices and global dimension at most two. A module $M(Y)$ places the vector space $Y$ at each vertex, with arrows recording the generator actions.

The manuscript constructs one bounded complex $P$ of $B$-bimodules such that

$$P\otimes_B^{\mathbf L}M(Y) \simeq M(H(Y))\oplus M(H(Y))[3].$$

A bimodule has compatible left and right actions. The derived tensor product uses resolutions so that it retains higher homological information. The symbol $[3]$ shifts a complex’s degree by three. Thus the formula simulates selection twice in different degrees, rather than demanding that one selected summand lift by itself.

Uniformity is the key issue. Denominators are cleared for a single finite diagram before any test module is chosen. Relations initially hold only up to chain homotopy; extra terms in a three-column differential turn those homotopies into an actual bimodule complex. The separation between the two surviving degrees exceeds $B$‘s global dimension, giving the displayed splitting. The argument does not need a splitting chosen naturally for every module simultaneously.

Turn survival time into resolution length

A further finite-dimensional construction replaces the bounded complex by an ordinary bimodule $X$ over an algebra $D$ of finite global dimension. Let

$$F=X\otimes_D^{\mathbf L}-.$$

Two iterations of $F$ simulate one selection step, up to fixed shifts and direct sums. The final algebra is the trivial extension

$$A=D\ltimes X,$$

whose underlying vector space is $D\oplus X$, with multiplication

$$(d,x)(d',x')=(dd',dx'+xd').$$

In particular, the ideal $X$ has square zero. This does not say that the radical of the whole algebra has cube zero; the earlier positive theorem for that class is not being contradicted.

For an inflated $D$-module $N$, the bar construction decomposes the relevant derived tensor expression according to the number of occurrences of $X$:

$$D\otimes_A^{\mathbf L}N \simeq\bigoplus_{r\ge0}(F^rN)[r].$$

The manuscript treats derived powers explicitly, including when $X$ is not flat. Eventual disappearance of the iterates implies finite projective dimension, while a nonzero $r$-th iterate forces projective dimension at least $r$.

For $N_m=M(Y_m)$, it obtains $F^{2m}N_m\simeq0$ but $F^{2m-2}N_m\not\simeq0$. These give the finite upper status and the lower bound $2m-2$. All algebras, complexes, and bimodules were fixed beforehand; only the test module varies.

7. Scope, limitations, and open questions

The lower bound is not an assertion that $\operatorname{pd}_A N_m$ equals exactly $2m-2$. The theorem needs finiteness and unboundedness, which are established by different parts of the transfer.

The coefficient field is $\mathbb C$. The manuscript does not claim the identical construction over every field; its characteristic-two comparison comes from a separate companion.

The result does not imply that all modules have finite projective dimension. Nor does it alter the positive theorems for monomial algebras, radical-cube-zero algebras, or the other restricted classes described earlier.

The extreme right-side conclusions belong to $\Lambda$, not automatically to $A$. More explicit small examples, quantitative upper bounds for the particular test modules, and classifications of the remaining positive classes are separate questions from the universal counterexample.

8. Sources and further reading

Bass’s Finitistic dimension and a homological generalization of semi-primary rings (1960) is the historical starting point. Huisgen-Zimmermann’s survey distinguishes the equality and finiteness questions and describes earlier positive results.

Igusa and Todorov’s On the finitistic global dimension conjecture for Artin algebras (2005) develops syzygy invariants. Cummings’ Left–right symmetry of finite finitistic dimension (2024) supplies the auxiliary asymmetry construction.

Neeman, Ranicki, and Schofield’s universal-localization work and Minamoto–Yamaura’s trivial-extension formulas explain the encoding and transfer background. Rickard’s injective-generation theorem supplies the derived-category consequence.

Source manuscripts

• M1. OpenAI, An algebra of infinite little finitistic dimension (September 23, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.