CGP-TR: Unified Homotopy Theory Framework

• CGP-TR is a unified framework that structures topological restriction homology via cyclotomic spectra, Cartier module enhancements, and pro-curve formulations in K-theory.
• It establishes explicit vanishing theorems under chromatic and telescopic localizations and introduces new adjunctions, improving computational approaches in algebraic K-theory.
• The framework’s practical applications extend to p-power torsion algebras, connective Morava K-theories, and Thom spectra, impacting arithmetic geometry and derived algebraic studies.

CGP-TR (Cyclotomic–Cartier–Pro/Curve–Topological Restriction Homology) is a foundational structure in the study of cyclotomic spectra, algebraic K-theory, and chromatic homotopy theory. The CGP-TR framework recognizes that topological restriction homology (TR) exhibits a threefold structure: corepresentability by cyclotomic spectra, enhancement to Cartier modules, and a pro-theoretic realization as the spectrum of curves in K-theory. This synthesis yields both deep conceptual insights and technical results, such as explicit vanishing theorems for localized TR in chromatic settings and new adjunctions at the level of spectral algebra.

1. Overview of CGP-TR

CGP-TR structures topological restriction homology (TR) as follows:

• Cyclotomic: TR is corepresentable by reduced topological Hochschild homology (THH) of the affine line and functions as the right adjoint in the adjunction between the ∞-categories of spectra with Frobenius lifts ($\mathrm{Sp}^{Fr}$) and cyclotomic spectra ($\mathrm{CycSp}$).
• Cartier: TR naturally acquires the structure of a topological Cartier module (TCart), with explicit Frobenius and Verschiebung maps obeying Cartier relations. TR becomes a right adjoint to the free Cartier module construction.
• Pro/Curve: For a connective $E_1$-ring $R$, TR evaluates to the pro-spectrum of curves in algebraic K-theory, realized concretely as $\varprojlim_n\,\Omega\,K(R[t]/t^n,(t))$.

This synthesis, articulated in McCandless (McCandless, 2021), underpins both basic definitions and the formulation of new theorems concerning the behavior of TR under Bousfield localization and chromatic vanishing phenomena (Keenan et al., 2023).

2. Cyclotomic Spectra, Corepresentability, and Adjunctions

A cyclotomic spectrum $X$ is a spectrum with an $S^1$-action, together with compatible $S^1$-equivariant Frobenius maps $\varphi_p: X \to X^{tC_p}$ for each prime $p$, where $X^{tC_p}$ denotes Tate fixed points. The construction is formalized in $\infty$-categories: $X$ is an object of $\mathrm{Fun}(S^1,\,\mathrm{Sp})$.

For any cyclotomic spectrum $X$, topological restriction homology is corepresentable:

$$\mathrm{TR}(X) \simeq \mathrm{Map}_{\mathrm{CycSp}}\bigl(\widetilde{\mathrm{THH}}(S[t]),\, X \bigr)$$

where $\widetilde{\mathrm{THH}}(S[t])$ is the fiber of the canonical map $\mathrm{THH}(S[t]) \to S$. This realizes TR as the right adjoint in the adjunction

$$\mathrm{Sp}^{Fr} \;\rightleftarrows\; \mathrm{CycSp}$$

with the left adjoint forgetting Frobenius lifts, and the right adjoint given by TR itself (McCandless, 2021).

3. Cartier Module Structure and Mackey Functor Formalism

The “Witt monoid” $W = S^1 \rtimes \mathbb{N}^\times$ acts via Frobenius lifts, creating the ∞-category $$\mathrm{Sp}^{Fr} = \mathrm{Fun}(BW^{op},\, \mathrm{Sp})$$ of spectra with Frobenius lifts. Topological Cartier modules (TCart) are spectral Mackey functors on $BW$:

$$\mathrm{TCart} = \mathrm{Mack}_{\mathrm{Sp}}(BW) = \mathrm{Fun}^\times ( \mathrm{Span}(\mathrm{Fin}_{BW}),\, \mathrm{Sp} )$$

Objects $M \in \mathrm{TCart}$ are spectra with $S^1$-action, equipped for each $k$ with:

• Verschiebung $V_k: M_{hC_k}\to M$
• Frobenius $F_k: M \to M^{hC_k}$

satisfying the Cartier relation:

$F_m V_n = g\, V_{n/g} F_{m/g},\quad g = \gcd(m, n)$

A Segal–tom Dieck splitting identifies the free Cartier module on $X$ as $\bigoplus_{n\ge1}X_{hC_n}$. The functor $$\mathrm{TR}: \mathrm{CycSp} \rightarrow \mathrm{TCart}$$ serves as the right adjoint to the “free” functor from Cartier modules to cyclotomic spectra, aligning with divided fixed-point constructions (McCandless, 2021).

4. Curves on K-theory and the Pro-theoretic Perspective

For connective $E_1$-rings, TR is realized in terms of the spectrum of curves:

$$\mathrm{TR}(R) \simeq \varprojlim_{n} \Omega\, K(R[t]/t^n, (t))$$

where $K(R[t]/t^n, (t))$ is the relative algebraic K-theory spectrum. The key steps include:

1. Utilizing the Dundas–Goodwillie–McCarthy theorem to replace $K$ by $\mathrm{TC}$ in inverse limits.
2. Identifying $\mathrm{TR}(R)$ with limits over relative topological cyclic homology.
3. Applying excision to relate THH of the “truncated polynomial” extensions to THH(R).

This aligns TR with the inverse system of “curves” in K-theory, unifying the cyclotomic, Cartier, and pro-theory perspectives (McCandless, 2021).

5. Chromatic Vanishing and Localized TR

A central application of the CGP-TR framework is the chromatic vanishing theorem for telescopically localized TR. For a connective $E_1$-ring $R$ that is $L_n^{p,f}$-acyclic (i.e., $L_n^{p,f}R \simeq 0$), the localized TR vanishes:

$L_{T(k)} \mathrm{TR}(R) \simeq *$

for all $1 \leq k \leq n$, where $L_{T(k)}$ denotes Bousfield localization with respect to the telescope $T(k)$ and $T(k)$ is associated with a $v_k$-self-map on a finite $p$-local complex. This is established through:

• The preservation of products by algebraic K-theory and the behavior of the “curves” model for TR under products (Keenan et al., 2023).
• Analyses of the module category, weight structures, and application of vanishing criteria for $K$-theory given the acyclicity of $R$.
• Careful manipulation of fiber sequences arising from the definition of curves and the non-commutation of Bousfield localization with inverse limits.

This result yields immediate consequences for $p$-power torsion rings, connective Morava $K$-theories, and Thom spectra, where TR vanishes after telescopic localization (Keenan et al., 2023).

6. Corollaries and Applications

Three significant applications follow from the chromatic vanishing theorem:

• $p$-power torsion algebras: For any connective $E_1$-algebra over $\mathbb{Z}/p^j$, $L_{T(k)}\,\mathrm{TR}(R) \simeq 0$ for all relevant $k$.
• Connective Morava $K$-theories: For $k(n)$, the connective cover of Morava $K$-theory, $L_{T(k)}\,\mathrm{TR}(k(n)) \simeq 0$ for $k < n$.
• Thom spectra $y(n)$: For the Mahowald–Ravenel–Shick Thom spectrum, $L_{T(k)}\,\mathrm{TR}(y(n)) \simeq 0$ for $k < n$.

These results derive from the spectrum-of-curves model for TR and the vanishing criteria for $K$-theory under chromatic acyclicity (Keenan et al., 2023).

7. Significance and Future Directions

The CGP-TR structure presents a unified conceptual apparatus for studying TR, with implications for equivariant homotopy theory, arithmetic geometry, and derived algebraic geometry. The combination of cyclotomic, Cartier, and pro-theoretic structures invites further refinement of computational tools for THH/TC, the development of new spectral vanishing criteria, and the exploration of deeper connections with chromatic localization, telescopic functors, and the arithmetic of $K$-theory. The explicit product-vanishing arguments and the preservation of infinite products by $K$-theory provide new leverage for handling highly structured ring spectra and their TR invariants.

References:

• (McCandless, 2021) – “On curves in K-theory and TR” (McCandless)
• (Keenan et al., 2023) – “A chromatic vanishing result for TR”
• [Cor23] – $K$-theory preserves products of additive $\infty$-categories
• [LMT20] – Vanishing of $K$-theory for certain endomorphism rings
• [HS21], [MNN15] – Telescopic and chromatic localization results.