Vanishing self-extensions need not make a module projective

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Family 199 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

A projective module is a particularly flexible algebraic object: maps from it lift through surjections, and its higher extension groups vanish. Could a finite module that looks projective to the right collection of extension tests nevertheless fail to be projective?

The Auslander–Reiten conjecture proposes that vanishing extensions both to itself and to the algebra should suffice. Tachikawa’s second conjecture asks the corresponding question over self-injective algebras, where one of those vanishing conditions is automatic.

Family 199 claims counterexamples to both. One manuscript constructs an explicit triangular algebra and a nonprojective Gorenstein-projective module. The other obtains a symmetric algebra and develops consequences for several related homological conjectures. The setting is noncommutative and characteristic two, over a field containing transcendental parameters.

2. Background and essential definitions

What an extension group measures

For modules $M,N$ over an algebra $A$, the group $\operatorname{Ext}^1_A(M,N)$ classifies short exact sequences

$$0\longrightarrow N\longrightarrow E\longrightarrow M\longrightarrow0$$

up to the usual equivalence. The zero class represents a split sequence, where $E$ is a direct sum. Higher groups $\operatorname{Ext}^i$ record longer homological obstructions and can be computed from a projective resolution.

A projective module is a direct summand of a free module. If $M$ is projective, then $\operatorname{Ext}^i_A(M,N)=0$ for every target $N$ and every $i>0$. The Auslander–Reiten question asks whether testing only the targets $M$ and $A$ is enough:

$$\operatorname{Ext}^i_A(M,M)= \operatorname{Ext}^i_A(M,A)=0\quad(i>0).$$

A module with no positive self-extensions is called self-orthogonal. Vanishing in every positive degree is much stronger than vanishing only eventually or in a finite initial range.

Special kinds of algebra and module

An Artin algebra is finite over a commutative Artinian central base. Every finite-dimensional algebra over a field is an example. All modules in the main constructions are finite-dimensional left modules.

An algebra is self-injective when its regular module is injective. Injectivity is the dual lifting property to projectivity. For such an algebra, $\operatorname{Ext}^i_A(M,A)=0$ automatically, so Tachikawa’s second conjecture asks whether self-orthogonal finite modules must be projective.

An algebra is symmetric if it is isomorphic to its vector-space dual as a bimodule. Equivalently, it has a nondegenerate symmetric associative bilinear form. Symmetric algebras are self-injective, but the converse need not hold.

A module is Gorenstein-projective if it occurs inside a doubly infinite exact complex of finite projective modules that remains exact after applying $\operatorname{Hom}_A(-,A)$. Such a complex is totally acyclic. This property ensures vanishing of positive Ext into $A$, but need not make the module projective.

Dominant dimension

An injective resolution begins by embedding the algebra into injectives:

$$0\longrightarrow A\longrightarrow I^0\longrightarrow I^1\longrightarrow\cdots.$$

The dominant dimension counts how long its initial terms are also projective. Infinite dominant dimension means every term in the minimal injective resolution is projective. It does not, by definition alone, say that the algebra itself is injective or that the resolution terminates.

3. Development history and previous partial results

Auslander and Reiten’s work on the generalized Nakayama conjecture led to the projectivity question. Tachikawa’s 1973 monograph placed its self-injective version among the conjectures surrounding quasi-Frobenius rings and dominant dimension.

Many positive cases are known. Auslander, Ding, and Solberg treated complete local complete intersections in the commutative setting. Huneke and Leuschke proved the conjecture for specified classes including excellent Cohen–Macaulay normal domains containing the rationals. Christensen and Holm showed that Auslander’s condition, a uniform bound on eventual Ext vanishing for each fixed first module, implies the conjecture.

Radical-cube-zero algebras form another important positive class. Work of Huneke, Şega, and Vraciu, Ringel and Zhang, and Zhang and Zhou treats progressively broader versions. Recent results also cover Igusa–Todorov algebras and quantum complete intersections. The new triangular algebra has nonzero fourth radical power, so it does not fall into the radical-cube-zero class.

Earlier examples show why the exact formulation matters. Schulz constructed a nonprojective module without self-extensions using a skew-field setting outside the Artin-algebra framework considered here. His quantum-exterior examples have eventually vanishing self-extensions but retain a nonzero first one. Böhmler and Marczinzik gave a commutative self-injective example with the first two self-extension groups zero and the third nonzero. None meets the all-positive-degree vanishing condition.

The parameter-shifting syzygies in Schulz’s quantum-exterior work are a direct antecedent of the construction. The new manuscripts add two independently scaled actions to remove all positive self-extensions, rather than merely moving them to finitely many degrees.

Recent reductions of Enomoto and Marczinzik connect the universal Auslander–Reiten, Nakayama, and Tachikawa questions. Other work of Chen, Fang, and Xi studies symmetric transfer through endomorphism algebras and related constructions. These implications explain why one sufficiently strong example has several consequences. The present symmetric construction additionally controls negative degrees in a complete resolution; positive Ext vanishing alone would not justify its final transfer.

4. What the new manuscripts claim

Both manuscripts are dated 23 September 2026 and work over

$$k=\mathbb F_2(q,H_1,H_2),$$

where the three displayed variables are algebraically independent. This is a rational-function field, not a finite field.

The triangular and symmetric examples

M1 claims a finite-dimensional algebra $\Lambda$ and a nonprojective finite left module $Z$ with

$$\operatorname{Ext}^i_\Lambda(Z,Z)= \operatorname{Ext}^i_\Lambda(Z,\Lambda)=0 \quad\text{for every }i>0.$$

The module is Gorenstein-projective. The algebra is split with eight simple-module isomorphism classes, meaning its semisimple quotient is $k^8$, and its fourth radical power is nonzero. The example therefore claims to refute both Auslander–Reiten and the Gorenstein-projective restriction.

M2 claims a finite-dimensional symmetric algebra $A$ and a nonprojective finite module $M$ with $\operatorname{Ext}^i_A(M,M)=0$ for every positive $i$. This refutes Tachikawa’s second conjecture and also gives a symmetric Auslander–Reiten counterexample.

Both constructions and the stated conclusions persist after every field extension of $k$. Taking an algebraic closure gives examples over an algebraically closed field of characteristic two containing the transcendental parameters. This is not the algebraic closure of a finite field.

Consequences of the symmetric example

For any extension $K/k$, write $A_K=K\otimes_kA$, $M_K=K\otimes_kM$, and form

$$\Gamma_K=\operatorname{End}_{A_K}(A_K\oplus M_K)^{\mathrm{op}}.$$

The opposite algebra fixes the left-module convention. M2 claims the following properties:

• Every term of the minimal injective resolution of $\Gamma_K$ is projective, but its self-injective dimension is infinite. This contradicts the classical Nakayama assertion and the stated Auslander–Gorenstein implication.
• A nonzero simple module $S$ satisfies $\operatorname{Ext}^i_{\Gamma_K}(S,\Gamma_K)=0$ for all $i\ge0$, including Hom in degree zero. This contradicts the generalized and strong Nakayama assertions.
• There is a projective-injective Wakamatsu tilting module that is not tilting. In this terminology, Wakamatsu tilting permits an infinite exact coresolution with a prescribed Hom-exactness condition; tilting requires a finite one, together with finite projective dimension and self-orthogonality.

The same injective resolution supplies finite modules $C_n$ with

$$\operatorname{pd}_{\Gamma_K}C_n=n\quad(n\ge1).$$

Thus its little and big left finitistic dimensions are infinite, independently of the complex example in family 198.

Further stated applications use existing results to produce infinitely many distinct one-summand tilting complements for a projective module over $\Gamma_K^{\mathrm{op}}$, and auxiliary symmetric algebras with arbitrarily long idempotent stratifications and associated derived-category decompositions. These are consequences of the main example and established transfer theorems, not additional independent constructions of self-orthogonal modules.

5. Why the advance matters

Provisional importance: 5/5. If correct, the family refutes a central projectivity criterion and several longstanding conjectures linked to it. The symmetric example is particularly restrictive: even a strong duality between the algebra and its vector-space dual would not force the proposed conclusion.

The common theme is the gap between homological tests and actual projectivity. Vanishing every positive self-extension, and even vanishing every positive extension to the algebra, would still leave a nonprojective module.

The consequences show how this gap propagates. Endomorphism algebras convert a self-orthogonal module into an indefinitely projective injective resolution, failures of detection by Ext, and unbounded finite resolution lengths. The significance comes from this network of precise implications, rather than treating the list of conjectures as unrelated headlines.

6. The main idea

Build a controllable extension algebra

The common starting point is a ten-dimensional algebra $C$ with a simple module $s$. Its symmetric trivial extension $T=C\ltimes DC$, where $D$ is vector-space duality, has a claimed self-extension algebra

$$\operatorname{Ext}^*_T(s,s)=k[\tau],\qquad \deg\tau=3.$$

Taking two tensor copies gives a module $X=s\otimes s$ over the symmetric algebra $E=T\otimes T$, with self-extensions $k[\tau_1,\tau_2]$, both generators in degree three. This polynomial structure makes the nonzero degrees and the effects of scaling explicit.

The argument works in the stable category, where maps factoring through projective modules are identified with zero. Cones of maps representing $\tau_1$ and $\tau_2$ act like a Koszul construction that removes most of the polynomial extension structure. The manuscripts track the remaining small collection of stable Hom groups, including negative degrees where needed.

Use two independent twists to remove every positive degree

An automorphism fixes $C$ and scales its dual summand by a nonzero parameter $H$. Its action fixes the underlying simple module but rescales a degree-$3m$ self-extension by $H^{-m}$.

The construction combines twists with parameters $H_1,H_2$ through genuine finite bimodule maps. The fact that the maps exist at the bimodule level is important: a map after evaluating on one module would not automatically define a tensor construction for the algebra.

In M1, the comparison on a positive degree reduces to the matrix

$$\begin{pmatrix} H_1^{-m}&1\\ H_2^{-m}&1 \end{pmatrix}.$$

Because $H_1/H_2$ has infinite multiplicative order, this matrix is invertible for every $m>0$. The comparison therefore removes positive self-extension classes uniformly over all degrees. In degree zero, a nonzero stable endomorphism survives, certifying nonprojectivity.

A triangular algebra built from $E$ and a finite bimodule turns that stable construction into an ordinary finite module. A totally acyclic resolution separately provides the vanishing of Ext into the algebra. The two vanishings have different roles and are both retained.

Add the negative-degree control needed for symmetry

M2 uses a related but different cone and bimodule construction. It obtains a totally acyclic resolution $P_Z$ such that its complete self-Hom cohomology vanishes for degrees

$$a>0\quad\text{or}\quad a\le-2.$$

The middle degrees are not simply declared zero. The negative tail is needed because, after passing to the symmetric trivial extension, positive extensions into a dual-tensor term correspond to negative complete-resolution degrees with an index shift.

The transfer forms $A=\Lambda\ltimes D\Lambda$ and induces $M=A\otimes_\Lambda Z$. The additional negative vanishing makes the induced module self-orthogonal while preserving nonprojectivity. Infinite complexes prove the all-degree statements, but the final algebras, kernels, cokernels, and modules are finite-dimensional.

Finally, the classical endomorphism-algebra correspondence applied to $A\oplus M$ supplies infinite dominant dimension. Successive cokernels in its projective-injective resolution yield modules with each prescribed finite projective dimension, producing the finitistic-dimension consequence.

7. Scope, limitations, and open questions

Both examples are noncommutative. They do not settle the commutative restriction of the Auslander–Reiten question. Their characteristic-two field and transcendental parameters are also substantive: specialization to arbitrary finite fields or to characteristic zero is not asserted.

All-positive-degree vanishing is essential. Earlier eventual-vanishing or finite-range examples are not equivalent to these claims. Conversely, the degree-zero vanishing in the strong Nakayama consequence applies to its particular simple module over $\Gamma_K$, not to the self-endomorphisms of the original nonzero module.

M1’s eight-simple structural description belongs to its triangular algebra. It should not be transferred to M2’s final symmetric algebra. Similarly, the consequences for $\Gamma_K$ belong to the associated endomorphism algebra, not automatically to $A$ itself.

The two manuscripts reproduce the ingredients needed for their respective constructions. They share a starting mechanism, but the symmetric transfer requires extra negative-degree control and is not merely an automatic consequence of M1’s headline.

8. Sources and further reading

Tachikawa’s Quasi-Frobenius Rings and Generalizations develops the classical setting. Schulz’s nonprojective self-extension example and his quantum-exterior work explain important predecessors and their different hypotheses.

Chen and Xi’s theory of self-orthogonal modules, Chen–Fang–Xi’s symmetric and endomorphism-algebra constructions, and Enomoto–Marczinzik’s homological-conjecture reductions describe the implication network.

For positive restricted cases, see Zhang and Zhou’s radical-cube-zero theorem and Xia’s quantum-complete-intersection theorem. The source manuscripts below state the new examples and the precise consequences summarized here.

Source manuscripts

• M1. OpenAI, An explicit counterexample to the Auslander-Reiten conjecture (September 23, 2026).
• M2. OpenAI, A counterexample to Tachikawa’s second conjecture (September 23, 2026).

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