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Math-2540 · mathematics · Algebraic geometry

The cohomological Hikita isomorphism for quivers

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any finite oriented quiver, allowing loops and multiple arrows, put $G=\prod GL(V_i)$ and $N=\oplus _{i\to j}Hom(V_i,V_j)\oplus _i$ $Hom(V_i,W_i)$. Let F be any linear flavor torus commuting with G, and choose a character $\nu $ for which stable equals semistable in the zero cotangent moment fiber and G acts freely. Set $X=\mu ^-1(0)^{\nu -stable}/G$. For the commutative flavor-deformed BFN Coulomb algebra $A_F$ without loop rotation, let $A_F^\nu $ denote the quotient by its nonzero $\nu$-weights, without reduction. Must the natural coefficient maps from $R=H*_{G×F}(point;C)$ onto $H*_F(X;C)$ and $A_F^\nu $ be surjective with identical kernels, giving a graded R-algebra isomorphism? Empty X with zero ring is included.

Short English statement

For any finite oriented quiver, allowing loops and multiple arrows, put $G=\prod GL(V_i)$ and $N=\oplus _{i\to j}Hom(V_i,V_j)\oplus _i$ $Hom(V_i,W_i)$. Let F be any linear flavor torus commuting with G, and choose a character $\nu $ for which stable equals semistable in the zero cotangent moment fiber and G acts freely. Set $X=\mu ^-1(0)^{\nu -stable}/G$. For the commutative flavor-deformed BFN Coulomb algebra $A_F$ without loop rotation, let $A_F^\nu $ denote the quotient by its nonzero $\nu$-weights, without reduction. Must the natural coefficient maps from $R=H*_{G×F}(point;C)$ onto $H*_F(X;C)$ and $A_F^\nu $ be surjective with identical kernels, giving a graded R-algebra isomorphism? Empty X with zero ring is included.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2540-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2540

Manuscript 1 (family 044, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

The cohomological Hikita isomorphism for quivers

OpenAI's claim

The equivariant cohomological Hikita correspondence is claimed for arbitrary finite quivers under the stated free stable-action hypothesis, as an algebra isomorphism preserving nilpotent structure. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Take any finite oriented quiver, allowing loops and parallel arrows, vector spaces $V_i,W_i$, and \[ G=\prod_i\operatorname{GL}(V_i),\qquad N=\bigoplus_{a:i\to j}\operatorname{Hom}(V_i,V_j)\oplus\bigoplus_i\operatorname{Hom}(V_i,W_i). \] Let a flavor torus $F$ act on $N$ commuting with $G$. Choose a character $\nu$ for which the cotangent moment-map semistable locus at zero equals the stable locus and $G$ acts freely, and set $X=\mu^{-1}(0)^{\nu\text{-stable}}/G$. Put $\mathcal O=\mathbb C[[t]]$, $\mathcal K=\mathbb C((t))$ and $\mathcal R=\{[g,s]\in G(\mathcal K)\times_{G(\mathcal O)}N(\mathcal O):gs\in N(\mathcal O)\}$. Let $A_F=H_*^{G(\mathcal O)\times F}(\mathcal R;\mathbb C)$ be its commutative BFN convolution algebra, without loop-rotation parameter. The character $\nu$ gives a topological-torus grading $A_F=\bigoplus_m(A_F)_m$. OpenAI claims that the Kirwan and Coulomb coefficient maps from $R=H^*_{G\times F}(\mathrm{pt};\mathbb C)$ are surjective with identical kernels, giving \[ H_F^*(X;\mathbb C)\simeq A_F/\langle(A_F)_m:m\ne0\rangle \] as graded $R$-algebras, with cohomological versus BFN homological degree and linear coefficient parameters of degree two. The quotient is scheme-theoretic: any nilpotents are retained, with no reduction applied; empty stable locus gives zero rings.

Outline of the argument

Manuscript 1 (family 044, 2026-09-24): Correspondences first identify cohomology with an intermediate algebra using the full topological torus and its coefficient maps. A separate stable-pair and Laurent comparison identifies the stability-cocharacter fixed scheme with that same algebra; keeping scheme-theoretic quotients throughout retains nilpotents and empty cases.

Formal verification

Family 044 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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