[Paper Review] Framed motives of smooth affine pairs

Andrei Druzhinin|arXiv (Cornell University)|Mar 30, 2018

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper extends the theory of framed motives to smooth affine pairs $(X, U)$, where $X$ is a smooth affine scheme over an infinite perfect field $k$ and $U \subset X$ is an open subscheme. It constructs an explicit motivically fibrant $\Omega$-resolution for the motivic suspension spectrum $\Sigma_{\mathbb{P}^1}^\infty(X_+/U_+)$ using framed correspondences, proving that the spectrum $\mathrm{M}_{\mathrm{fr}}(X_+/U_+)_{f}$ is stably equivalent to a fibrant $\Omega$-spectrum in positive degrees.

ABSTRACT

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular the motivically fibrant $Ω$-resolution in positive degrees of the motivic suspension spectrum $Σ_{\mathbb P^1}^\infty X_+$, where $X_+=X\amalg *$, for a smooth scheme $X\in \mathrm{Sm}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $(\mathbb A^l imes X,(\mathbb A^l-0) imes X)$, $X\in \mathrm{Sm}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety $X$ over $k$ and an open subscheme $U\subset X$. The result gives the explicit motivically fibrant $Ω$-resolution in positive degrees for the motivic suspension spectrum $Σ_{\mathbb P^1}^\infty (X_+/U_+)$ of the factor-sheaf $X_+/U_+$.

Motivation & Objective

• To generalize the theory of framed motives from smooth schemes to smooth affine pairs $(X, U)$ with $U \subset X$ open.
• To compute the motivically fibrant $\Omega$-resolution of the motivic suspension spectrum $\Sigma_{\mathbb{P}^1}^\infty(X_+/U_+)$ for smooth affine $X$.
• To establish that the framed motive construction $\mathrm{M}_{\mathrm{fr}}(X_+/U_+)_{f}$ yields a motivically fibrant $\Omega$-spectrum in positive degrees.
• To extend the Cone Theorem of Garkusha, Neshitov, and Panin to the case of affine pairs, providing a geometric model for $X/U$ in $\mathbf{SH}(k)$.

Proposed method

• Uses Voevodsky’s framed correspondences to define the pointed sheaf $\mathrm{Fr}(-, X/U)$ as a colimit over $n$ of $\mathrm{Fr}_n(-, X/U)$, capturing geometric data of framed maps.
• Constructs the $S^1$-spectrum $\mathrm{M}_{\mathrm{fr}}(X_+/U_+)$ via the $C^*$-construction on $\mathrm{Fr}(-, X/U \wedge S^i)$, applying simplicial co-approximation via $\Delta^\bullet \times -$.
• Applies levelwise injective local fibrant replacement $(-)_f$ to ensure the resulting spectrum is motivically fibrant in positive degrees.
• Defines the $\mathbb{P}^1$-spectrum $\mathrm{M}_{\mathbb{P}^1}(X,U)$ as the image under a canonical functor $\nu$ from $\mathbb{A}^1/\mathbb{G}_m$-spectra to $\mathbb{P}^1$-spectra, preserving stable equivalences.
• Establishes stable motivic weak equivalences between $\Sigma_{\mathbb{P}^1}^\infty(X_+/U_+)$ and $\mathrm{M}_{\mathbb{P}^1}(X,U)_{f}$, and between $\Sigma_{\mathbb{G}_m}^\infty \Sigma_{S^1}^\infty(X_+/U_+)$ and the $\mathbb{G}_m$-bi-spectrum $\mathrm{M}^{\mathbb{G}_m}_{\mathrm{fr}}(X,U)_{f}$.
• Relies on the key technical tool: the $\mathbb{A}^1$-local equivalence between $\mathbb{A}^1//\mathbb{G}_m$ and $\mathrm{pt}//\mathbb{G}_m$, which induces an equivalence between $\mathrm{Fr}((X/U) \wedge (\mathbb{A}^1//\mathbb{G}_m)^{\wedge i})$ and $\mathrm{Fr}((X/U) \wedge (\mathbb{G}_m^\wedge i \wedge S^i))$.

Experimental results

Research questions

• RQ1 How can the framed motive construction be extended from smooth schemes to smooth affine pairs $(X, U)$ with $U \subset X$ open?
• RQ2 Is the $S^1$-spectrum $\mathrm{M}_{\mathrm{fr}}(X_+/U_+)_{f}$ motivically fibrant in positive degrees for smooth affine $X$?
• RQ3 Does the canonical map $\Sigma_{\mathbb{P}^1}^\infty(X_+/U_+) \to \mathrm{M}_{\mathbb{P}^1}(X,U)_{f}$ induce a stable motivic weak equivalence?
• RQ4 Can the $\mathbb{P}^1$-spectrum $\mathrm{M}_{\mathbb{P}^1}(X,U)$ be realized as the image of an $\mathbb{A}^1/\mathbb{A}^1\setminus 0$-spectrum under a canonical functor $\nu$?
• RQ5 What is the relationship between the framed motive spectra $\mathrm{M}^{\mathbb{G}_m}_{\mathrm{fr}}(X,U)_{f}$ and the iterated suspension spectra $\Sigma_{\mathbb{G}_m}^\infty \Sigma_{S^1}^\infty(X_+/U_+)$?

Key findings

• The spectrum $\mathrm{M}_{\mathbb{P}^1}(X,U)_{f}$, defined as $C^*\mathrm{Fr}(-, X/U \wedge T^i)$ with $T = (\mathbb{A}^1, \mathbb{A}^1 \setminus 0)$, is a motivically fibrant $\Omega_{\mathbb{P}^1}$-spectrum in positive degrees.
• The canonical map $\Sigma_{\mathbb{P}^1}^\infty(X_+/U_+) \to \mathrm{M}_{\mathbb{P}^1}(X,U)_{f}$ is a stable motivic weak equivalence in $\mathbf{SH}(k)$.
• The $S^1$-spectrum $\mathrm{M}_{\mathrm{fr}}(X_+/U_+)_{f}$ is a motivically fibrant $\Omega$-spectrum in positive degrees and has the correct homotopy type in $\mathbf{SH}_{S^1}(k)$.
• The $\mathbb{G}_m$-bi-spectrum $\mathrm{M}^{\mathbb{G}_m}_{\mathrm{fr}}(X,U)_{f}$ is a motivically fibrant $\Omega$-bi-spectrum in $S^1$-positive degrees and is stably equivalent to $\Sigma_{\mathbb{G}_m}^\infty \Sigma_{S^1}^\infty(X_+/U_+)$.
• The $\mathbb{A}^1$-local equivalence $\mathbb{A}^1//\mathbb{G}_m \simeq \mathrm{pt}//\mathbb{G}_m$ induces a strong $\mathbb{A}^1$-local equivalence between the framed correspondence sheaves $\mathrm{Fr}((X/U) \wedge (\mathbb{A}^1//\mathbb{G}_m)^{\wedge i})$ and $\mathrm{Fr}((X/U) \wedge (\mathbb{G}_m^\wedge i \wedge S^i))$, enabling the comparison of spectra.
• The construction $\mathrm{M}_{\mathbb{P}^1}(X,U)$ is obtained as $\nu(\mathrm{M}^T_{\mathrm{fr}}(X,U))$, where $\nu$ is the functor induced by $\mathbb{P}^1 \to \mathbb{A}^1/\mathbb{A}^1\setminus 0$, and this preserves stable equivalences.