A note on higher topological Hochschild homology

This paper demonstrates that the homotopy fixed points spectrum of higher topological Hochschild homology applied to a commutative ring spectrum detecting $v_n$-elements yields a spectrum detecting $v_{n+k}$-elements with $k > 1$, thereby establishing a mechanism for higher chromatic redshift beyond the single-step increase typically observed in algebraic K-theory.

The Gist

Here is an explanation of the paper "A Note on Higher Topological Hochschild Homology" by Rixin Fang, translated into simple language with creative analogies.

The Big Picture: The "Mathematical Elevator"

Imagine you are in a building where each floor represents a different level of mathematical complexity. In the world of algebraic topology (a branch of math dealing with shapes and spaces), there is a famous rule called Chromatic Redshift.

Think of Algebraic K-theory as a magical elevator. If you take a mathematical object (like a ring spectrum, which is a fancy type of number system) and put it in this elevator, the elevator takes you up one floor.

• If you start on Floor $n$, the elevator takes you to Floor $n+1$.
• This is a "redshift" because, in physics, shifting to a higher energy level is often associated with shifting toward the red end of the spectrum.

The Question:
Mathematicians have known about this "one-floor elevator" for a long time. But Rixin Fang asks: Can we build an elevator that takes us up multiple floors at once?

Specifically, if we start on Floor $n$, can we find a machine that takes us straight to Floor $n+k$ (where $k$ is more than 1)?

The Machine: "Higher Topological Hochschild Homology"

To build this multi-floor elevator, the author uses a tool called Higher Topological Hochschild Homology (THH).

• The Analogy: Imagine you have a piece of clay (your mathematical object).
  • Standard THH: You wrap the clay in a single loop of string. This creates a new shape that is slightly more complex.
  • Higher THH: Instead of one loop, you wrap the clay in a whole net of strings, or a multi-dimensional grid (like a cube made of string).
  • The "Fixed Point" Step: After wrapping the clay, you look at the shape from a specific angle (mathematically called "homotopy fixed points"). This filters out the noise and reveals the core structure.

The paper investigates what happens when you take a mathematical object, wrap it in this complex multi-dimensional string net, and then look at the result.

The Discovery: Jumping Floors

The author proves that this "multi-string wrapping" machine is incredibly powerful.

1. The Setup: You start with a specific type of mathematical object (a "ring spectrum") that lives on a certain floor (let's say Floor $n$).
2. The Process: You apply the "Higher THH" machine to it.
3. The Result: The output isn't just on the next floor; it's on a much higher floor!

The Main Finding:
If you start with an object that detects a specific "signal" at height $n$, the output of this machine detects a signal at height $n + k$ (where $k$ is greater than 1).

In the paper's specific examples, the author shows that by using this method, you can jump two floors at a time.

• Start at Floor $n$.
• Apply the machine.
• End up at Floor $n+2$.

This is a "super-redshift." It's like finding a secret stairwell that lets you skip the intermediate floors entirely.

How Did They Prove It? (The Detective Work)

To prove this works, the author acts like a detective connecting clues:

• The Clue: They knew that for simple cases (like the number system $\mathbb{F}_p$), this machine worked and jumped floors.
• The Connection: They built a bridge (a "ring map") connecting their new, complex objects to the simple ones they already understood.
• The Logic:
  • "If Object A connects to Object B..."
  • "And Object B is known to jump floors..."
  • "Then Object A must also jump floors."

By chaining these connections together, they showed that the "multi-floor jump" isn't just a fluke; it's a general rule that works for a wide variety of mathematical objects.

Why Does This Matter?

In the world of mathematics, understanding these "floors" (heights) helps us classify the fundamental building blocks of the universe of shapes and numbers.

• Before: We had a slow elevator that took us up one step at a time.
• Now: We have discovered a "turbo-charger" that lets us leap ahead.

This helps mathematicians solve problems that were previously too hard because they were stuck on a lower floor. It suggests that there are deeper, more powerful structures hidden inside these mathematical objects that we can now reach.

Summary in a Nutshell

• The Problem: Can we jump up multiple levels of mathematical complexity at once?
• The Tool: A complex wrapping technique called "Higher Topological Hochschild Homology."
• The Result: Yes! By wrapping a mathematical object in a multi-dimensional grid and analyzing it, you can boost its complexity by two or more levels.
• The Analogy: It's like taking a simple toy car, wrapping it in a complex web of springs, and discovering that when you let it go, it doesn't just roll forward—it suddenly flies to a higher dimension.

This paper is a blueprint for building those "super-elevators," opening up new territories in the landscape of modern mathematics.

Technical Summary

Here is a detailed technical summary of the paper "A Note on Higher Topological Hochschild Homology" by Rixin Fang.

1. Problem Statement and Motivation

Context: Chromatic Redshift
The paper is grounded in the phenomenon of chromatic redshift, which posits that algebraic $K$-theory increases the "height" of a commutative ring spectrum by one.

• Definition: A commutative ring spectrum $X$ has height $n$ if $K(n)_* X \neq 0$ and $K(n+1)_* X = 0$.
• The Question: While algebraic $K$-theory ($K$) is known to shift height by 1 (i.e., $ht(K(X)) = ht(X) + 1$), the author investigates whether there exists a functor $F^n: CAlg \to CAlg$ that shifts the height by an integer $n > 1$. Specifically, the paper asks if the homotopy fixed points of higher Topological Hochschild Homology (THH) can detect higher chromatic heights.

The Specific Question (Question 1.6):
Let $X$ be a commutative ring spectrum of height $n$. Is the spectrum $(LT^m X)^{hT^m}$ non-zero when detected by $k(n+m)_*$?

• Here, $LT^m X$ denotes the Loday construction (higher THH) over the torus $T^m = (S^1)^m$.
• $(-)^{hT^m}$ denotes the homotopy fixed points under the $T^m$-action.
• The goal is to determine if $k(n+m)_* (LT^m X)^{hT^m} \neq 0$, implying the detection of $v_{n+m}$-elements.

2. Methodology

The paper employs a combination of trace methods, inductive arguments, and ring map constructions within the framework of stable homotopy theory and $\infty$-categories.

Key Tools:

1. Loday Construction ($LT^n$): The author formalizes the Loday construction $X \otimes A$ for a simplicial set $X$ and a commutative algebra $A$. It is established that $T^n \otimes A \simeq THH^{(n)}(A)$, where $THH^{(n)}$ is the iterated THH ($n$ times).
2. Trace Maps: Utilizing the trace map $K \to THH$ and its higher analogues to construct ring maps between different spectra.
3. Inductive Proofs: The proofs often rely on reducing the problem to known cases (e.g., $F_p$) and extending them via ring maps to more complex spectra like $\mathbb{Z}$, $BP\langle n \rangle$, and $j_p$.
4. Homotopy Fixed Point Spectral Sequences: Analyzing the convergence and differentials in spectral sequences computing $k(n)_* (LT^n X)^{hT^n}$.
5. Comparison with Known Results: Leveraging previous work by Veen (on $F_p$), Ausoni–Rognes, and Hahn to build a chain of implications.

3. Key Contributions and Results

The paper provides affirmative answers to the redshift question for specific classes of ring spectra, demonstrating that higher THH homotopy fixed points can detect elements of higher height.

A. Theoretical Framework (Section 2)

• Defined the Loday construction rigorously in the context of presentable stable symmetric monoidal $\infty$-categories.
• Proved the equivalence $LT^n A \simeq THH^{(n)}(A)$, establishing the $T^n$-action on higher THH.
• Analyzed the homotopy groups of $LT^n k$ for a perfect field $k$ of characteristic $p$, relating them to Hopf algebras and tensor products involving $B_n$.

B. Height Bounds (Section 3)

• Upper Bound Theorem (Theorem 3.3 & 3.5): Proved that the height of $(LT^n F_p)^{hT^n}$ is at most $n-1$, and for a general height $n$ spectrum $R$, the height of $(LT^n R)^{hT^n}$ is at most $2n$. This sets the theoretical ceiling for redshift via this method.

C. Main Detection Theorems

The core contribution is proving that specific maps send $v$-elements to non-zero elements, confirming "higher redshift" in specific cases.

1. Base Case ($F_p$): Reaffirmed Veen's result (Theorem 3.1): For $p \ge 5$, the map $k(n-1)_* \to k(n-1)_* (LT^n F_p)^{hT^n}$ sends $v_{n-1}$ to a non-zero element.

2. Extension to $\mathbb{Z}$ and $BP\langle 0 \rangle$ (Theorem 3.6):

   • Showed that for $p \ge 5$, the maps $k(n)_* \to k(n)_* (LT^n \mathbb{Z})^{hT^n}$ and $k(n)_* \to k(n)_* (LT^n BP\langle 0 \rangle)^{hT^n}$ send $v_n$ to a non-zero element.
   • This is achieved by constructing ring maps from $K(F_p)$ and $K(\mathbb{Z})$ to the target spectra.

3. Main Result: Higher Redshift for $j_p$ (Theorem 3.13):

   • Hypothesis: Let $p \ge 5$ and $1 \le n+2 \le p$.
   • Result: The unit map $k(n+1)_* \to k(n+1)_* (LT^n j_p)^{hT^n}$ sends $v_{n+1}$ to a non-zero element.
   • Significance: This demonstrates a redshift of $n+1$ (from height 0 of $j_p$ to height $n+1$) via the $n$-th iteration of THH.
   • Mechanism: The proof constructs a chain of ring maps:
     $$j_p \to K(\mathbb{Z}_p)^\wedge_p \to (THH(\mathbb{Z}_p)^{hS^1})^\wedge_p \to (LT^2 F_p)^{hT^2}$$
     By iterating this, one maps $(LT^n j_p)^{hT^n}$ to $(LT^{n+2} F_p)^{hT^{n+2}}$. Since the target is known to detect $v_{n+1}$ (via Veen's theorem), the source must also detect it.

4. Generalization (Proposition 3.15):

   • If a ring spectrum $R$ admits a map to $THH(\mathbb{Z}_p)^{hS^1}$, then $(LT^n R)^{hT^n}$ detects $v_{n+1}$ (under the same prime constraints).

4. Significance and Future Directions

Significance:

• Higher Redshift: The paper provides concrete evidence that higher Topological Hochschild Homology (iterated THH) is a viable mechanism for achieving chromatic redshift greater than 1. Specifically, it shows that applying the $n$-th Loday construction and taking homotopy fixed points can shift the height from $0$(for$j_p$) to$n+1$.
• Methodological Advancement: It successfully extends the trace method and detection theorems from the classical $S^1$-case (standard THH) to the $T^n$-case (higher THH).
• Connection to $K$-Theory: It bridges the gap between the known redshift of $K$-theory and the potential redshift of higher THH, suggesting a unified framework for understanding height increases.

Limitations and Open Questions (Section 4):

• The current method relies on maps to $THH(\mathbb{Z}_p)^{hS^1}$, which does not currently exist for $kup$ (connective complex K-theory) or $BP\langle 1 \rangle$.
• Question 4.1: Can similar non-vanishing results be proven for $LT^n \mathbb{Z}_p[\zeta_{p^\infty}]$?
• Question 4.2: Can one find a height $n-1$ spectrum $R_{n-1}$ such that $E_n \to THH(R_{n-1})^{hS^1}$? This would generalize the redshift phenomenon to Lubin-Tate theories ($E_n$).

Conclusion

Rixin Fang's paper establishes that the homotopy fixed points of higher Topological Hochschild Homology serve as a powerful detector for higher chromatic heights. By constructing specific ring maps from $j_p$ to iterated THH constructions, the author proves that these spectra detect $v_{n+1}$-elements, providing a rigorous mathematical foundation for "higher chromatic redshift" phenomena beyond the standard $K$-theory case.