CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The internal and homological shifts are not interchangeable

Statement refuted

Let $m \geq 1$ and let $C_{m} = K^{b} ({proj}^{g r} A_{m})$ be the bounded homotopy category of finite graded projective left $A_{m}$ -modules of The bounded projective homotopy category C_m and the two shifts, carrying the internal shift ${r}$ of Associative graded algebras, bimodules, and internal shifts and the homological shift $[1]$ . The following over-generalisation is false:

Refuted claim. The functors ${1}$ and $[1]$ on $C_{m}$ are naturally isomorphic, so that shifting a complex internally by one degree and shifting it homologically by one degree give the same object up to a natural identification.

It is not so. For $1 \leq i \leq m$ take the vertex projective $P_{i} = A_{m} e_{i}$ of Finite graded A_m-modules, internal shifts and the vertex projectives, concentrated in homological degree $0$ . Then $P_{i} {1}$ has its only nonzero term in homological degree $0$ , namely $P_{i} {1}$ , while $P_{i} [1]$ has its only nonzero term in homological degree $- 1$ , namely $P_{i}$ ; since a morphism of complexes has components in each homological degree, there is not even a nonzero degree-zero chain map $P_{i} {1} \to P_{i} [1]$ , let alone an isomorphism, whereas a natural isomorphism ${1} ≅ [1]$ would give one for every object. The two shifts are therefore different functors, and the internal shift leaves homological placement fixed while the homological shift lowers it by one in the indexing of this page. The comparison in $K_{0} (C_{m})$ is kept quantitative rather than identifying the shifts: $[X [1]] = - [X]$ by Homological and internal shifts on K_0(C_m) and $[X {1}] = q [X]$ with $q$ invertible, so the two shift functors act on $K_{0} (C_{m})$ by $- 1$ and by $q$ respectively.

Facts & Assumptions

Given: An integer $m \geq 1$ , the algebra $A_{m}$ with its vertex projectives $P_{i} = A_{m} e_{i}$ for $0 \leq i \leq m$ , the category $C_{m} = K^{b} ({proj}^{g r} A_{m})$ with its homological shift $[1]$ , its internal shift ${r}$ and its group $K_{0} (C_{m})$ .

[L1]

Objects of $C_{m}$ are bounded complexes $X = (X^{n}, d_{X}^{n})$ of finitely generated graded projective left $A_{m}$ -modules with degree-zero differentials, and a morphism $f : X \to Y$ is a homotopy class of chain maps, each represented by a family of degree-zero $A_{m}$ -linear maps $f^{n} : X^{n} \to Y^{n}$ with $f^{n + 1} d_{X}^{n} = d_{Y}^{n} f^{n}$ ; the homological shift is $(X [1])^{n} = X^{n + 1}$ with $d_{X [1]} = - d_{X}$ , and the internal shift is $(X {r})^{n} = X^{n} {r}$ with $d_{X {r}} = d_{X}$ (The bounded projective homotopy category C_m and the two shifts, Associative graded algebras, bimodules, and internal shifts).

[L2]

$P_{i} = A_{m} e_{i}$ is a finitely generated graded projective left $A_{m}$ -module and $P_{i} \neq 0$ for every $0 \leq i \leq m$ , since $e_{i}$ is one of the $4 m + 1$ $Z$ -linearly independent basis classes of $A_{m}$ (Finite graded A_m-modules, internal shifts and the vertex projectives, The 4m+1 path basis).

[L3]

The internal shift of graded modules satisfies $(M {r})_{d} = M_{d - r}$ and has the same underlying ungraded abelian group as $M$ , so $M {r} \neq 0$ whenever $M \neq 0$ (Associative graded algebras, bimodules, and internal shifts).

[L4]

In $K_{0} (C_{m})$ one has $[X [1]] = - [X]$ for every object, and the internal shift induces an automorphism $q$ with $[X {r}] = q^{r} [X]$ , so $q$ is invertible in the endomorphism ring of $K_{0} (C_{m})$ (Homological and internal shifts on K_0(C_m), The triangulated K_0 of the Khovanov–Seidel projective category).

Proof

technique · direct

1.1

The witness is nonzero. By [L2] the module $P_{i} = A_{m} e_{i}$ is a nonzero finitely generated graded projective left $A_{m}$ -module for each $0 \leq i \leq m$ , in particular for $1 \leq i \leq m$ ; let $\underline{P_{i}}$ denote the complex with ${\underline{P_{i}}}^{0} = P_{i}$ and ${\underline{P_{i}}}^{n} = 0$ for $n \neq 0$ , an object of $C_{m}$ by [L1].

L1L2

2.1

The two shifted complexes. By [L1] the internal shift acts termwise, so $(\underline{P_{i}} {1})^{n} = {\underline{P_{i}}}^{n} {1}$ is nonzero exactly for $n = 0$ , where it equals $P_{i} {1}$ , and its differentials are those of $\underline{P_{i}}$ , all zero; the homological shift reindexes, so $(\underline{P_{i}} [1])^{n} = {\underline{P_{i}}}^{n + 1}$ is nonzero exactly for $n = - 1$ , where it equals $P_{i}$ , and again all differentials vanish. In particular $(\underline{P_{i}} {1})^{0} = P_{i} {1} \neq 0$ by [L3] while $(\underline{P_{i}} [1])^{0} = 0$ .

step 1.1L1L3

3.1

No morphism, hence no isomorphism. Let $f : \underline{P_{i}} {1} \to \underline{P_{i}} [1]$ be a morphism in $C_{m}$ represented by a chain map. By [L1] its degree- $0$ component is a degree-zero $A_{m}$ -linear map $f^{0} : P_{i} {1} \to 0$ , which must be the zero map; every other component of $f$ has either zero source or zero target by step 2.1, so $f = 0$ and the only morphism between the two objects is the zero morphism. The identity of the one-term nonzero complex $\underline{P_{i}} {1}$ is nonzero in $C_{m}$ : with both differentials zero, a homotopy cannot make its degree- $0$ identity map null. As its Hom group to $\underline{P_{i}} [1]$ is zero, the two objects are not isomorphic in $C_{m}$ ; consequently there is no natural isomorphism between the functors ${1}$ and $[1]$ , because such a natural isomorphism would supply an isomorphism $\underline{P_{i}} {1} ≅ \underline{P_{i}} [1]$ for this particular object.

step 2.1L1

4.1

The $K_{0}$ comparison. By [L4] one has $[\underline{P_{i}} [1]] = - [\underline{P_{i}}]$ and $[\underline{P_{i}} {1}] = q [\underline{P_{i}}]$ with $q$ an invertible endomorphism of $K_{0} (C_{m})$ ; the two shift functors therefore act on the class of the witness by the operators $- 1$ and $q$ . The displayed classes are not asserted to be unequal: deciding that would require the additional input that $[\underline{P_{i}}]$ is not annihilated by $q + 1$ , that is, a basis computation in $K_{0} (C_{m})$ , which this counterexample does not use. The non-isomorphism of step 3.1 is a homological-support statement and needs no such computation.

step 3.1L4

5.1

Conclusion. For every $1 \leq i \leq m$ the objects $\underline{P_{i}} {1}$ and $\underline{P_{i}} [1]$ of $C_{m}$ have nonzero terms in the distinct homological degrees $0$ and $- 1$ by step 2.1, there is no nonzero morphism between them by step 3.1, and consequently the internal shift ${1}$ and the homological shift $[1]$ are not naturally isomorphic functors on $C_{m}$ ; the refuted claim fails already on a single vertex projective concentrated in degree $0$ . The internal shift moves internal degrees and leaves homological placement fixed, the homological shift reindexes without touching internal degrees, and the two are compared in $K_{0} (C_{m})$ by the operators $q$ and $- 1$ of step 4.1 rather than identified. No choice principle is used, and the witness is the single module $P_{i}$ in one homological degree.

step 2.1step 3.1step 4.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2c, printed pp. 10-11 (standard reference, not scraped)
The Stacks Project, Derived Categories, section 28, K-groups (tag 0FCM), Definition 13.28.1 (standard reference, not scraped)