Pushforward of tensor products and restriction

Open

OAI.MaximalSeshadri.OpenBaseChange.tensor_square

by wurtle · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

maximal-seshadrioai-math

For $f:X\to Y$ , an open $U\subset Y$ , and modules $M,N$ on $X$ , the comparison PushforwardTensor.hom between $f_*M\otimes f_*N$ and $f_*(M\otimes N)$ is compatible with restriction to $U$ through iso f U and moduleTensorRestrict.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.

Preamble

import Mathlib
import Definitions.Def_MaximalSeshadriConstants
import Definitions.Def_SeshadriPrelim6

namespace OAI

section

namespace MaximalSeshadri.OpenBaseChange
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite
open MaximalSeshadri.Geometry MaximalSeshadri.TensorPure

variable {X Y : Scheme.{0}} (f : X ⟶ Y) (U : Y.Opens)

Formal statement

theorem tensor_square (M N : X.Modules) :
    (Scheme.Modules.restrictFunctor U.ι).map (PushforwardTensor.hom f M N) ≫
      (iso f U (moduleTensor X M N)).hom ≫
      (Scheme.Modules.pushforward (f ∣_ U)).map
        (moduleTensorRestrict (f ⁻¹ᵁ U) M N).hom =
    (moduleTensorRestrict U ((Scheme.Modules.pushforward f).obj M)
      ((Scheme.Modules.pushforward f).obj N)).hom ≫
      moduleTensorMap (iso f U M).hom (iso f U N).hom ≫
      PushforwardTensor.hom (f ∣_ U) (M.restrict (f ⁻¹ᵁ U).ι) (N.restrict (f ⁻¹ᵁ U).ι) := by
  sorry

end
end MaximalSeshadri.OpenBaseChange
end
end OAI

Source

OpenAI, Maximal Seshadri constants on arbitrary polarized surfaces, September 23, 2026, https://github.com/openai/math (paper: preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf; Lean: lean/OAI/AlgebraicGeometry/Seshadri, Apache License 2.0); Lean declaration `OAI.MaximalSeshadri.OpenBaseChange.tensor_square`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Sheaves/TensorOpenSquare.lean#L61

View graph