1 Introduction↩︎

Motivic homotopy theory, as developed by Morel and Voevodsky [1], [2], is centered on \(\mathbb{A}^1\)-homotopy invariance. This principle serves as a primary source of the theory’s strength; it yields robust invariants in algebraic geometry and leads to the algebraic cobordism spectrum \(\mathrm{MGL}\) as well as other oriented cohomology theories. However, \(\mathbb{A}^1\)-homotopy invariance necessitates the dévissage property, which indicates that any such cohomology theory is equivalent to that of the underlying reduced subscheme. For instance, Weibel’s homotopy \(K\)-theory \(\mathrm{KH}\) [3] famously exhibits the dévissage property and completely annihilates the \(K\)-theory of nilpotent algebras. This implies that any \(\mathbb{A}^1\)-invariant cohomology theory remains completely insensitive to nilpotent ideals, resulting in identical evaluations on a scheme and its underlying reduced subscheme. Therefore, the consideration of \(\mathbb{A}^1\)-local objects inevitably involves the destruction of the cohomology theory of nilpotent non-unital algebras, including all nilpotent thickenings and infinitesimal deformation data.

While this insensitivity often simplifies smooth geometry over a field, it significantly obstructs the study of mixed characteristic and perfectoid geometry. In these settings, phenomena are governed by adic topologies and towers of infinitesimal thickenings, making infinitesimal information an essential part of the geometric structure. A central problem is constructing cobordism-type invariants that retain essential deformation data by relinquishing strict \(\mathbb{A}^1\)-invariance. Derived algebraic cobordism \(\mathrm{dMGL}\), introduced by Lowrey–Schürg [4] and developed in a bivariant form by Annala [5], addresses this issue by utilizing quasi-smooth derived geometry instead of imposing \(\mathbb{A}^1\)-localization. The resulting theory effectively preserves the infinitesimal deformation data that classical motivic cobordism fails to capture. However, the unlocalized derived theory is significantly larger and more challenging to control directly. This paper’s guiding idea is to extract a finite-syntomic, non-unital fragment from this theory and study it using Goodwillie calculus.

Goodwillie calculus, introduced in [6]–[8], may be viewed as a Taylor expansion of functors. Its linear part is captured by excisive functors, and higher approximations form a tower \[\cdots \to P^n(F) \to P^{n-1}(F) \to \cdots \to P^1(F) \to P^0(F).\] We apply this idea not only to functors, but also to pointed \(\infty\)-categories. The category of non-unital algebras is naturally pointed, with the zero algebra serving as both the initial and terminal object, while square-zero extensions emerge as linear data in this pointed context. Thus, the \(1\)-excisive approximation provides a method to isolate and control infinitesimal deformation data without enforcing \(\mathbb{A}^1\)-invariance. Our formulation is tailored for the non-presentable categories that arise here; in the locally finitely presentable case, it aligns with the Goodwillie calculus of \(\infty\)-categories developed by Heuts [9], following Lurie’s \(\infty\)-categorical version of Goodwillie calculus [10, Ch. 6].

The first main application is perfectoid geometry with almost mathematics. We define finite non-unital derived algebraic cobordism \(\mathrm{dMGL}^{\rm nu,f}\) and its almost version \(\mathrm{dMGL}^{a,\rm fin}\). For a perfectoid valuation ring \(V\) and an integral perfectoid \(V\)-algebra \(A\), we prove that tilting preserves the finite non-unital theory: \[\mathrm{dMGL}^{\rm nu,f}(A) \simeq \mathrm{dMGL}^{\rm nu,f}(A^\flat)\] (Theorem 16). Transitioning to almost algebras reveals the primary invariant emphasized in this paper. Specifically, if \(V\) is an integral perfectoid valuation ring, tilting induces a weak equivalence \[\mathrm{dMGL}^{a,\rm fin}(A) \simeq \mathrm{dMGL}^{a,\rm fin}(A^\flat)\] for any integral perfectoid \(V\)-algebra \(A\) (Theorem 17).

The same calculus also gives a second family of applications. Given a natural transformation \(\alpha:F\to G\), we define the approximation functors \(P^n_G(F)\) as the homotopy fiber of \[G \to \mathrm{cok}\,\alpha \to P^n(\mathrm{cok}\,\alpha).\] We call \(P^n_G(F)\) the Goodwillie approximation of \(F\) to \(G\). Applying this construction to the canonical morphism from algebraic cobordism to homotopy algebraic \(K\)-theory produces approximations of cobordism that inherit key features of \(K\)-theory. In particular, we obtain Bott periodicity and rigidity statements for these approximations, including the lift of \(K\)-theoretic equivalences to periodic algebraic cobordism (Theorem 20) and an analogue of Gabber rigidity (Theorem 21).

We briefly indicate the organization of the paper. In Section 2, we recall Goodwillie calculus for functors and its higher excisive approximations. In Section 3, we develop the form of excisive approximation for pointed \(\infty\)-categories necessary for non-unital algebras and compare it to Heuts’s theory in the finitely presentable case. In Section 4, we construct the finite-syntomic non-unital and almost versions of derived algebraic cobordism and prove the tilting invariance theorems. Finally, in Section 5, we introduce approximation functors along natural transformations and apply them to the morphism \(\mathrm{MGL}\to \mathbb{K}\), obtaining Bott periodicity and Gabber rigidity for the resulting Goodwillie approximations.

Acknowledgments↩︎

I would like to thank Masaki Hanamura for the helpful discussion about Sections 2 and 3. Our discussion generalizes the universal property of \(n\)-excisive approximation for \(\infty\)-categories related to functors that admit a right adjoint.

1.0.0.1 Use of AI tools.

The author used Google’s Gemini-pro 3.1 as a conversational research aid for brainstorming and refining mathematical formulations, and OpenAI’s Prism (GPT-5.2) for editorial and expository assistance, including structural refinement. These tools improved the manuscript’s clarity and indicated where abbreviated arguments warranted fuller exposition. All results, proofs, and final formulations were reviewed, verified, and approved by the author, who bears sole responsibility for the content of this work.

2 A short review of Goodwillie calculus↩︎

In this section, following Lurie’s textbook [10], we recall the definition of Goodwillie calculus for functors between \(\infty\)-categories. We then define higher excisive objects of \(\infty\)-categories.

The category \(\mathrm{Set}_{\Delta}\) of simplicial sets has a standard simplicial model structure called the Kan–Quillen model structure. The simplicial nerve of \(\mathrm{Set}_{\Delta}\) is denoted by \(\mathcal{S}\), which is called the \(\infty\)-category of spaces or \(\infty\)-groupoids.

Given a functor \(F: \mathcal{C} \to \mathcal{D}\) of \(\infty\)-categories under suitable conditions, one can regard excisive functors as linear approximations. In this setting, the natural morphism \(R^n(F) \to \Omega_{\mathcal{D}}\Sigma_{\mathcal{D}}R^n(F)\) is a weak equivalence, where \(\Omega_{\mathcal{D}}\) and \(\Sigma_{\mathcal{D}}\) denote the loop and suspension functors, respectively.

For any finite set \(S\), \(\mathbf{P}(S)\) denotes the set of subsets of \(S\), called the power set of \(S\), and \(\mathbf{P}_{ \le m}(S)\) denotes the set of subsets of \(S\) whose cardinality is at most \(m\). The sets \(\mathbf{P}_{ < m}(S)\), \(\mathbf{P}_{ \ge m}(S)\), and \(\mathbf{P}_{ > m}(S)\) are defined similarly.

Let \(\mathcal{C}\) be an \(\infty\)-category. A functor \(X: N(\mathbf{P}(S)) \to \mathcal{C}\) is called an \(S\)-cube. In the case \(S=[n]=\{0,\,1,\,\ldots,\,n\}\), \(S\)-cubes are called \((n+1)\)-cubes for any \(n \ge -1\). Here, \([-1]\) is the empty set. We let \(\mathrm{Cube}^m(\mathcal{C})\) denote the \(\infty\)-category of \(m\)-cubes of \(\mathcal{C}\) for any \(m \ge 0\).

The definition of higher excisive functors is as follows:

We recall the explicit construction of the \(n\)-excisive approximation \(P^n(F)\) for a functor \(F: \mathcal{C} \to \mathcal{D}\). For any finite subset \(S\) of \([n]\), the \(S\)-pointed cone functor \(C_S\) is the composition \[\mathcal{C} \simeq \mathrm{Cube}^S_{\le 0}(\mathcal{C}) \overset{i_{0!}}{\to} \mathrm{Cube}^S_{\le 1}(\mathcal{C}) \overset{i_{\le 1 *}}{\to} \mathrm{Cube}^S(\mathcal{C}) \overset{\mathrm{Ev}(S)}{\to} \mathcal{C},\] where \(i_{0!}\) denotes the right Kan extension along the inclusion \(i_0: \mathbf{P}_{\le 0}(S) \to \mathbf{P}(S)\), \(i_{\le 1 *}\) denotes the left Kan extension of \(\mathbf{P}_{\le 0}(S) \to \mathbf{P}(S)\), and \(\mathrm{Ev}(S)\) is the evaluation at the terminal vertex of cubes. Note that \(C_\emptyset\) is the identity and \(C_S\) is the final object of \(\mathcal{C}\) whenever \(|S|=1\). The functor \(T_n(F): \mathcal{C} \to \mathcal{D}\) is defined as follows: \[T_n(F)(X) = \varprojlim_{\emptyset \neq S \subset [n]} F(C_S(X))\] for any \(X \in \mathcal{C}\). The homotopy colimit of this sequence defines the \(n\)-excisive approximation \(P^n\): \[T_n(F) \to T_n(T_n(F)) \to T^3_n(F) \to \cdots \to T^k_n(F) \to \cdots\] By the definition of \(T_n\), if \(F\) is \(n\)-excisive, the canonical map \(\theta_F : F \to T_n(F)\) is a weak equivalence. Therefore, the colimit map \(F \to P^n(F)\) is also a weak equivalence. The universal property in Theorem 1 follows from the following lemmas.

For any set \(S' \subset S\), let \(\mathbf{P}_{S'}(S)\) denote the set of subsets of \(S\) that contain \(S'\). Then the projection \(\pi_{S'} : \mathbf{P}(S) \to \mathbf{P}_{S'}(S)\) is defined by \(\pi_{S'}(I)= I \cup S'\). The cube \(F(X_\blacksquare(S'))\) is the inverse image of \(\pi_{S'}\) of the restriction \(F(X)|_{\mathbf{P}_{S'}(S)}: \mathbf{P}_{S'}(S) \to \mathcal{D}\). Therefore, the homotopy limit \(\varprojlim_{I \neq \emptyset }F(X_I(S'))\) is weakly equivalent to the homotopy limit of \(F(X)|_{\mathbf{P}_{S'}(S)}: \mathbf{P}_{S'}(S) \to \mathcal{D}\), which is \(F(X_{S'}(S'))=F(X(S'))\). From \(F(X_\emptyset(S'))=F(X(S'))\), we obtain that \(F(X_\blacksquare(S'))\) is Cartesian, whenever \(S'\) is nonempty, and the limit \(Y_\blacksquare\) of Cartesian cubes is also Cartesian.

The identity \(F(X)|_{\mathbf{P}_{S'}(S)} \to F(X)|_{\mathbf{P}_{S'}(S)}\) induces the unit morphism \(F(X) \to F(X_\blacksquare(S'))\), and \(F(X) \to F(X_\blacksquare(\emptyset))\) is the identity. Therefore, there is a canonical map \(F(X) \to Y_\blacksquare\) of cubes. We assume that \(X\) is strongly coCartesian. Note that \(X_I|_{\mathbf{P}_{\le 1}(S)}\) is the left Kan extension of the restriction \(X|_{\mathbf{P}_{\le 1}(S \setminus I) }\) along \[i_I: \mathbf{P}_{\le 1}(S \setminus I) \to \mathbf{P}_{\le 1}(S ), \quad i_I (S') = S' \cup I.\] Hence, \(X_I\) is strongly coCartesian, and for any \(I \subset S\), the identity morphism \(X_I(\emptyset) \to X(I)\) induces a map \(X_I \to C_\blacksquare(X(I))\) of cubes, which is functorial for \(I\). The induced map of limits \(\varprojlim_{S' \neq \emptyset} F(X_I(S') ) \to \varprojlim_{S' \neq \emptyset}F(C_{S'}(X(I)))\) is functorial for \(I\), and we obtain the homotopically commutative square \[\xymatrix@1{ F(X_\blacksquare(\emptyset)) \ar[d] \ar[r]^\simeq & F(C_\emptyset(X)) \ar[d] \\ Y_\blacksquare \ar[r] & T_n(F)(X) }\] of \(S\)-cubes, which implies

the desired property. \(\Box\)

Let \(X\) be a strongly coCartesian \(n\)-cube. By Lemma 1, the canonical map \[\theta_{T_n^{k-1}(F)}(X): T_n^{k-1}(F)(X) \to T_n^k(F)(X)\] factors through a Cartesian \(n\)-cube \(Y_\blacksquare^k\). Therefore, \[P^n(F)(X) \simeq \varinjlim_k Y_\blacksquare^k.\] Since finite limits commute with filtered colimits, \(\varinjlim_k Y_\blacksquare^k\) is Cartesian. Hence \(P^n(F)(X)\) is Cartesian. \(\Box\)

Let \(\mathrm{Fun}(\mathcal{C},\,\mathcal{D})[T_n^{-1}]\) denote the localization of \(\mathrm{Fun}(\mathcal{C},\,\mathcal{D})\) by the family of morphisms \(\theta_F :F \to T_n(F)\). Lemma 3 implies that the functor \(P^n : \mathrm{Fun}(\mathcal{C},\,\mathcal{D}) \to \mathrm{Exc}^n(\mathcal{C},\,\mathcal{D})\) factors through \(\mathrm{Fun}(\mathcal{C},\,\mathcal{D})[T_n^{-1}]\). Any object \(F\) of \(\mathrm{Exc}^n(\mathcal{C},\,\mathcal{D})\) satisfies \(F \simeq T_n(F) \simeq P^n(F)\). Therefore, the \(n\)-excisive approximation induces a categorical equivalence \[P^n : \mathrm{Fun}(\mathcal{C},\, \mathcal{D})[T_n^{-1}] \to \mathrm{Exc}^n(\mathcal{C},\,\mathcal{D})\] of \(\infty\)-categories. Thus, we can identify the \(n\)-excisive approximation \(P^n\) with the localization functor \(\mathrm{Fun}(\mathcal{C}, \, \mathcal{D}) \to \mathrm{Fun}(\mathcal{C},\,\mathcal{D})[T_n^{-1}]\).

3 Higher excisive approximations of \(\infty\)-categories↩︎

In this section, using the \(\infty\)-categorical Yoneda lemma, we define \(n\)-excisive objects of \(\infty\)-categories. First, in Section 3.1, we recall the \(\infty\)-categorical Yoneda lemma. Next, in Section 3.2, we define \(n\)-excisive objects and prove their universal property. Finally, in Section 3.3, we note that locally presentable excisive \(\infty\)-categories are stable.

3.1 The \(\infty\)-categorical Yoneda lemma and \(n\)-excisive \(\infty\)-categories↩︎

We write \(\mathrm{Pre}(\mathcal{C}) =\mathrm{Fun}(\mathcal{C}^{\rm op},\, \mathcal{S})\) for the \(\infty\)-category of presheaves on \(\mathcal{C}\), and the fully faithful functor \(\mathbb{Y}\) is called the Yoneda functor (or Yoneda embedding). Dually, the functor \[\begin{align} \mathbb{Y}^\vee_\mathcal{C}: \mathcal{C} &\to \mathrm{Fun}(\mathcal{C},\, \mathcal{S})\\ X &\mapsto \mathrm{Hom}_\mathcal{C}(X,\,-) \end{align}\] is also fully faithful, which is called the coYoneda functor.

We say that such \(S\) in Definition 4 is a generator of \(\mathcal{C}\). For any \(\infty\)-category \(\mathcal{C}\), \(\mathcal{C}_0\) denotes the full subcategory spanned by compact objects. By the definition, if \(\mathcal{C}\) is compactly generated, the \(\mathcal{C}_0 \to \mathcal{C}\) induces a categorical equivalence \(\mathrm{Ind}(\mathcal{C}_0) \to \mathcal{C}\), where \(\mathrm{Ind}(\mathcal{C}_0)\) denotes the ind-category whose objects are filtered colimits of objects of \(\mathcal{C}_0\). Along the Yoneda embedding \(\mathbb{Y}_{\mathcal{C}_0} : \mathcal{C}_0 \to \mathrm{Pre}(\mathcal{C}_0)\), the ind-category \(\mathrm{Ind}(\mathcal{C}_0)\) is identified with the full subcategory of \(\mathrm{Pre}(\mathcal{C})\) spanned by left exact presheaves.

An \(\infty\)-category that is both locally presentable and compactly generated is called locally finitely presentable.

3.2 The definition of higher excisive objects↩︎

Let \(F: \mathcal{C} \to \mathcal{D}\) be a functor between pointed \(\infty\)-categories. The induced functor \(F_*:\mathrm{Fun}(\mathcal{C}^{\rm op},\, \mathcal{S} ) \to \mathrm{Fun}(\mathcal{D}^{\rm op},\, \mathcal{S} )\) is the left Kan extension of \(\mathbb{Y}_\mathcal{D} \circ F: \mathcal{C} \to \mathrm{Fun}(\mathcal{D}^{\rm op},\, \mathcal{S} )\) along \(\mathbb{Y}_\mathcal{C}:\mathcal{C} \to \mathrm{Fun}(\mathcal{C}^{\rm op},\, \mathcal{S} )\). This correspondence gives an equivalence of \(\infty\)-categories \[\mathrm{Fun}^{\rm L}\left( \mathrm{Pre}(\mathcal{C}),\, \mathrm{Pre}(\mathcal{D}) \right) \to \mathrm{Fun}\left( \mathcal{C},\, \mathrm{Pre}(\mathcal{D}) \right)\] where \(\mathrm{Fun}^{\rm L}\left( \mathrm{Pre}(\mathcal{C}),\, \mathrm{Pre}(\mathcal{D}) \right)\) denotes the \(\infty\)-category of functors admitting a right adjoint. (See [11, p. 324], Theorem 5.1.5.6). Furthermore, if \(F: \mathcal{C} \to \mathcal{D}\) is a functor between compactly generated \(\infty\)-categories sending compact objects to compact objects, \(F\) is the left Kan extension of its restriction to the full subcategory \(\mathcal{C}_0\) spanned by compact objects, and admits a right adjoint \(F^*: \mathcal{D} \to \mathcal{C}\), which is the restriction of the inverse image functor \(F^{-1}:\mathrm{Pre}(\mathcal{D}_0) \to \mathrm{Pre}(\mathcal{C}_0)\).

Since any corepresentable functor is left exact, the coYoneda embedding \(\mathbb{Y}^\vee_\mathcal{C} : \mathcal{C} \to \mathrm{Fun}(\mathcal{C},\,\mathcal{S})\) factors through the full subcategory \(\mathrm{Fun}^{\mathrm{lex}}(\mathcal{C},\,\mathcal{S})\) spanned by left exact functors. Hence, \(P^n (\mathcal{C})\) coincides with \[\mathrm{Exc}^{n \mathrm{lex}} ( \mathcal{C},\, \mathcal{S}) \times_{ \mathrm{Fun}^\mathrm{lex} ( \mathcal{C},\, \mathcal{S}) } \mathcal{C},\] where \(\mathrm{Exc}^{n \mathrm{lex}} ( \mathcal{C},\, \mathcal{S})\) is the full subcategory spanned by left exact \(n\)-excisive functors.

The Yoneda functor \(\mathbb{Y}_\mathcal{C}\) preserves all small limits. By the definition of \(T_n\), one has a weak equivalence: \[T_n(\mathbb{Y}_\mathcal{C})(X) = \varprojlim_{\emptyset \neq S} \mathbb{Y}_\mathcal{C}(C_S(X)) \simeq \mathbb{Y}_\mathcal{C}( \varprojlim_{\emptyset \neq S}C_S (X) )= \mathbb{Y}_\mathcal{C}(T_n (\mathrm{id}_\mathcal{C})(X)),\] implying that the natural map \(T_n \circ \mathbb{Y}_\mathcal{C} \to \mathbb{Y}_\mathcal{C} \circ T_n\) is a weak equivalence. Furthermore, if \(\mathcal{C}\) is compactly generated, the homotopy colimit map \(P^n \circ \mathbb{Y}_\mathcal{C} \to \mathbb{Y}_\mathcal{C} \circ P^n\) is also a weak equivalence.

Let \(Y\) be a strongly coCartesian \(n\)-cube. Since \(\mathrm{Hom}_\mathcal{C}(X,\,-)\) commutes with all small limits, the \(\infty\)-categorical Yoneda lemma implies that conditions (1) and (2) are equivalent to the statement that \(Y(\emptyset) \to \varprojlim_{\emptyset \neq S} Y(S)\) is a weak equivalence. Hence \(Y\) is also Cartesian. Since the Yoneda functor \(\mathbb{Y}_\mathcal{C}\) preserves all small limits, (2) and (3) are equivalent.

We assume that \(\mathcal{C}\) is compactly generated. To prove implication (4)\(\Rightarrow\)(2), it suffices to verify that \(\mathrm{id}_\mathcal{C} \to P^n(\mathrm{id}_\mathcal{C}): \mathcal{C} \to \mathcal{C}\) is a weak equivalence. Any object \(X\) of \(\mathcal{C}\) is homotopically equivalent to a filtered colimit of compact objects. Since filtered colimits commute with finite limits and small colimits, Condition (4) implies Condition (2). \(\Box\)

For any object \(X\) of the essential image of \(P^n(\mathrm{id}_\mathcal{C}): \mathcal{C} \to \mathcal{C}\), the induced map \(X \to T_n(X)\) is also a weak equivalence. Therefore, the essential image of \(P^n(\mathrm{id}_\mathcal{C})\) can be identified with the localization \(\mathcal{C}[T_n^{-1}]\).

By Proposition 2, the embedding \(P^n(\mathcal{C}[T_n^{-1}]) \to \mathcal{C}[T_n^{-1}]\) is a categorical equivalence. Hence \(\mathcal{C}[T_n^{-1}]\) is contained in the full subcategory \(P^n(\mathcal{C})\). \(\Box\)

Let \(X_\bullet\) be a strongly coCartesian \(n\)-cube of \(\mathcal{C}\). Since \(F\) preserves all small colimits, \(F(X_\bullet)\) is also strongly coCartesian. Therefore, by Lemma 2, \(F(X_\bullet)\) is Cartesian.

Let \(Y_\bullet\) be a strongly coCartesian \(n\)-cube. Then \(Y_\bullet\) is also Cartesian. Since the right adjoint preserves all small limits, \(F^*(Y_\bullet)\) is Cartesian. \(\Box\) A restriction of an \(n\)-excisive functor is also \(n\)-excisive.

For any pointed \(\infty\)-category \(\mathcal{C}\) admitting finite colimits, the \(n\)-excisive approximation \(P^n(\mathcal{C})\) has the following universal property:

It is sufficient to prove that the natural transformation \(\mathrm{id} \to P^n\) induces a weak equivalence \[(F \circ \mathrm{id}_{\mathcal{C}} \to F \circ P^n( \mathrm{id}_{\mathcal{C}} ) ) : \mathcal{C} \to \mathcal{D}.\] Since the restriction \(F: \mathcal{C} \to \mathcal{D}\) preserves all small colimits and final objects, one has \[P^n(F^{*} \circ F ) \simeq P^n(F^*) \circ F \simeq F^{*} \circ F\] by Proposition 4, which implies that the endofunctor \(F^{*} \circ F: \mathcal{C} \to \mathcal{C}\) is \(n\)-excisive. Therefore, the unit \[u:\mathrm{id}_{\mathcal{C}} \to F^{*} \circ F\] of the induced adjunction \(F : \mathcal{C} \rightleftarrows \mathcal{D} : F^{*}\) factors through the \(n\)-excisive approximation \(P^n(\mathrm{id}_{\mathcal{C}})\), yielding the induced natural transformation \(F(\mathrm{id}_{\mathcal{C}}) \to F(P^n(\mathrm{id}_{\mathcal{C}}))\) which is homotopically split. Since the \(n\)-excisive approximation \(P^n\) is homotopically idempotent, the natural transformation \(F(P^n(\mathrm{id}_{\mathcal{C}})) \to F(P^n(P^n(\mathrm{id}_{\mathcal{C}})))\) is a weak equivalence, implying that its retract \(F(\mathrm{id}_{\mathcal{C}}) \to F(P^n(\mathrm{id}_{\mathcal{C}}))\) is also a weak equivalence. \(\Box\) By Lurie [11], we have the following:

Note that any object of \(\mathcal{C}\) is identified with a filtered colimit of corepresentable functors on compact objects, each of which is left exact. Let \(X\) be an \(n\)-excisive object. Then one has a chain of weak equivalences \[P^n(X) \simeq P^n(\varinjlim \mathrm{Hom}_{\mathcal{C}_0}(C_\alpha,\, - )) \simeq \varinjlim \mathrm{Hom}_{\mathcal{C}_0}(C_\alpha,\, P^n(-) ).\] Therefore, \(X \to P^n(X)\) is a weak equivalence if and only if \(X\) is a filtered colimit of objects of \(P^n(\mathcal{C}_0)\). \(\Box\)

3.3 Stable and excisive \(\infty\)-categories↩︎

Let \(\mathcal{C}\) be an \(\infty\)-category admitting finite colimits and a final object, \(\mathcal{D}\) a Goodwillie-differentiable \(\infty\)-category, and \(F: \mathcal{C} \to \mathcal{D}\) an excisive functor. For any object \(X\) of \(\mathcal{C}\), the canonical morphism \(F(X) \to \Omega_{\mathcal{D}} F (\Sigma_\mathcal{C} X)\) is a weak equivalence in \(\mathcal{D}\). Therefore, \[\Omega_{\mathcal{D}} \circ (-): \mathrm{Exc}^1(\mathcal{C},\, \mathcal{D} ) \to \mathrm{Exc}^1(\mathcal{C},\, \mathcal{D} )\] is a categorical equivalence, whose inverse is \((-)\circ \Sigma_{\mathcal{C}}\). In addition, the stabilization functor \(\Sigma_{\mathcal{D}}^\infty: \mathcal{D} \to \mathrm{Sp}(\mathcal{D})\) induces a categorical equivalence \[\Sigma_{\mathcal{D}}^\infty \circ (-): \mathrm{Exc}^1(\mathcal{C},\, \mathcal{D} ) \to \mathrm{Exc}^1(\mathcal{C},\, \mathrm{Sp}(\mathcal{D}) ),\] whose inverse is induced by the forgetful functor \(\Omega^\infty_{\mathcal{D}}: \mathrm{Sp}(\mathcal{D}) \to \mathcal{D}\).

Applying this to the case \(\mathcal{D} = \mathcal{S}_{*}\), one has the following:

First, we assume that \(\mathcal{C}\) is excisive. Then, for any integer \(n \ge 0\), the unit \[\mathrm{Id} \to \Omega_\mathcal{C}^n \circ \Sigma_\mathcal{C}^n : \mathcal{C} \to \mathcal{C}\] is a weak equivalence by Lemma 4.

In the case of condition (2), the stabilization functor \[\Sigma_+^\infty \mathcal{C} \to \mathrm{Sp}(\mathcal{C})\] is fully faithful. Indeed, for any \(X\) and \(Y\), \[\mathrm{Hom}_{\mathrm{Sp}(\mathcal{C})}( \Sigma_+^\infty X ,\, \Sigma^\infty_+ Y) \simeq \varinjlim_{n } \mathrm{Hom}_\mathcal{C} (\Sigma^n X ,\, \Sigma^n Y )\] is weakly equivalent to the constant space \(\mathrm{Hom}_\mathcal{C} ( X ,\, Y)\). Again by Lemma 4, the unit \(Y \to \Omega_+^\infty( \Sigma_+^\infty(Y))\) is a weak equivalence.

Finally, we assume that the condition \((3)\) holds. Then, for any object \(X\) of \(\mathcal{C}\), the coYoneda functor \[\mathbb{Y}^\vee(X) \simeq (\Sigma^\infty_+)_*( \mathbb{Y}^\vee( \Sigma_+^\infty X) )= \mathrm{Hom}_{\mathrm{Sp}(\mathcal{C})}( \Sigma_+^\infty X ,\, \Sigma^\infty_+ (-)): \mathcal{C} \to \mathrm{Sp}(\mathcal{C}) \to \mathcal{S}\] is excisive. \(\Box\)

4 Application to the finite derived algebraic cobordism of perfectoid algebras↩︎

For a locally presentable category \(\mathcal{C}\) with a final object, the functor category \(\mathrm{Set}_{\Delta}^{\mathcal{C}^\omega}\) has the covariant model structure representing the \(\infty\)-category \(\mathrm{Fun}^{\rm L}(N(\mathcal{C}),\,\mathcal{S})\). By [11], the \(n\)-excisive approximation \(P^n :\mathrm{Fun}^{\rm L}(\mathcal{C}, \, \mathcal{S}) \to \mathrm{Exc}^n ( N(\mathcal{C}),\, \mathcal{S})\) induces a Bousfield localization \(P^{n}:\mathrm{Set}_{\Delta}^{\mathcal{C}^\omega} \to P^n(\mathrm{Set}_{\Delta}^{\mathcal{C}^\omega})\). Since \(P^n\) has a right adjoint, it is accessible, and \(P^n\) sends compact objects to compact objects. Hence, the \(n\)-excisive category \(P^n(\mathcal{C}^\omega)\) is homotopically equivalent to the full subcategory of \(P^n(\mathrm{Set}_{\Delta}^{\mathcal{C}^\omega})\) spanned by compact objects.

In this section, we consider \(\mathbb{E}_{\infty}\)-rings. Ordinary rings, whose homotopy groups are concentrated in degree zero, are called discrete \(\mathbb{E}_{\infty}\)-rings, or simply discrete rings.

First, in Section 4.1, to formulate a finite-syntomic analogue of derived algebraic cobordism for non-unital \(\mathbb{E}_{\infty}\)-rings, we define finite syntomic (i.e., finite quasi-smooth) morphisms of \(\mathbb{E}_{\infty}\)-rings. In Section 4.2, we review the excisive approximation of the pointed category of augmented commutative algebras. In Section 4.3, we verify that this finite derived algebraic cobordism preserves tilting equivalences between integral perfectoid algebras (Theorem 16).

4.1 Finite syntomic morphisms of \(\mathbb{E}_{\infty}\)-rings↩︎

We fix the base commutative unital ring \(V\), which is a discrete \(\mathbb{E}_{\infty}\)-ring. The category \(\mathrm{CAlg}_{V//V}\) of augmented commutative \(V\)-algebras is a pointed category, whose initial and terminal object is \(V\). The operadic Smith-ideal theory [12] induces an adjoint categorical equivalence \[\mathrm{cok}: \mathrm{CAlg}^{\rm nu}_{V}(\mathcal{C}) \rightleftarrows \mathrm{CAlg}_{V//V}(\mathcal{C}): \ker\] of \(\infty\)-categories (see [13]). We define cotangent complexes by the language of Goodwillie calculus and Smith ideal theory.

In derived algebraic geometry, intersection-theoretic data is unconditionally preserved by replacing flat transversality with quasi-smoothness.

The definition of flatness of \(\mathbb{E}_{\infty}\)-algebras is described as follows:

By definition, if \(A\) is connected (resp. discrete), any flat \(A\)-algebra is connected (resp. discrete).

We define finite syntomic morphisms for \(\mathbb{E}_{\infty}\)-rings, which correspond to finite quasi-smooth morphisms. Given an \(\mathbb{E}_{\infty}\)-ring \(A\), let \(\mathrm{CAlg}_{A}^{\flat}\) denote the full subcategory of \(\mathrm{CAlg}_{A}\) spanned by flat \(A\)-algebras.

In other words, the \(\infty\)-category \(\mathrm{CAlg}_{A}^{\rm FSyn}\) of finite syntomic \(A\)-algebras is defined by the homotopy Cartesian product \[\mathrm{CAlg}_{A}^{\rm FSyn}= \mathrm{CAlg}_{A}^{\flat} \times_{\mathrm{CAlg}_{\pi_0(A)}^{\flat} } \mathrm{CAlg}_{\pi_0(A)}^{\rm FSyn}.\]

With this definition, finite syntomic morphisms of \(\mathbb{E}_{\infty}\)-rings are locally represented as pullbacks of universal finite syntomic morphisms \(p_d: V_d \to U_d\) \((d \ge 1)\) (see the Stacks Project [14]):

The corresponding statement for \(\mathbb{E}_{\infty}\)-rings is as follows.

By definition, \(f\) is finite syntomic if and only if \(B\) is flat over \(A\) and \(\pi_0(A)\to \pi_0(B)\) is finite syntomic. By Proposition 10, the latter condition is equivalent to the existence of the stated local charts on \(\mathrm{Spec}{\,\pi_0(A)}\). Since flatness is preserved by base change and can be checked on the corresponding homotopy Cartesian squares, these charts lift to the derived morphism \(\mathrm{Spec}{B} \to \mathrm{Spec}{A}\). Conversely, any such local pullback of \(p_d\) is flat and has finite syntomic \(\pi_0\), so \(f\) is finite syntomic. \(\Box\)

4.2 Excisive approximation of augmented algebras↩︎

The excisive approximation of the category of augmented \(V\)-algebras is described as follows:

Since \(\Sigma^{\infty}_{A}A\) is contractible, the induced morphism \(M \to \Sigma^\infty_{A}\tilde{A}\) is a weak equivalence. The unit \(P^1_{A}(\tilde{A}) \to \Omega^\infty_{A} \Sigma_{A}^\infty \tilde{A} \simeq A \oplus L_{\tilde{A} / A}\) is a weak equivalence. Therefore, \(M \to L_{\tilde{A} / A}\) is a weak equivalence. \(\Box\)

A commutative non-unital \(V\)-algebra \(I\) is said to be homotopically square-zero if the product \(I \otimes_{V} I \to I\) is null-homotopic.

This follows directly from the properties of exact functors and Ken Brown’s lemma applied to \(A \otimes_V A \simeq A \times_V A\). \(\Box\)

4.3 Finite derived algebraic cobordism of non-unital algebras↩︎

The classical motivic cobordism \(\mathrm{MGL}\) enforces strict \(\mathbb{A}^1\)-invariance. Over mixed-characteristic bases, this destroys nilpotent thickenings and makes linear approximations trivial. To retain infinitesimal deformation data, we work with unlocalized derived algebraic cobordism \(\mathrm{dMGL}\) [4], [5].

By utilizing spans of quasi-smooth morphisms, derived algebraic cobordism unconditionally preserves higher intersection data. In this section, we construct a finite-syntomic analogue of non-unital derived cobordism by directly using the unlocalized \(\infty\)-groupoid of finite syntomic \(\mathbb{E}_{\infty}\)-algebras, bypassing the \(\mathbb{A}^1\)-localization entirely.

Crucially, because we do not impose \(\mathbb{A}^1\)-localization, the functor \(\mathrm{FSyn}_*^{\rm aug}\) faithfully detects the infinitesimal thickenings governed by the derived cotangent complex. Applying Corollary 4, \(\mathrm{dMGL}^{\rm nu,f}\) factors through the excisive approximation \(P^1(\mathrm{CAlg}_{V//V})\).

Property (1) follows from \(B \otimes_A (A \oplus M[1]) \otimes_{A \oplus M[1]} A \simeq B\). Property (2) follows from Corollary 5, as the base change \(A \oplus M[1] \to \tilde{B}\) along the finite syntomic algebra homomorphism \(A \to B\) is also finite syntomic. \(\Box\)

We recall the following \(\infty\)-categorical deformation theory:

Put \[R=\Sigma^1 P^1_A(\tilde{A})\simeq A\oplus M[1].\] By Lemma 5, the functor \[s:\mathrm{FSyn}^{\rm aug}(A)\to \mathrm{FSyn}^{\rm aug}(R),\qquad B\mapsto B\otimes_A R\] is well-defined, and the composite \((-)\otimes_R A\circ s:\mathrm{FSyn}^{\rm aug}(A) \to \mathrm{FSyn}^{\rm aug}(A)\) is equivalent to the identity functor on \(\mathrm{FSyn}^{\rm aug}(A)\).

Conversely, let \(C\) be a finite syntomic \(R\)-algebra and put \(B=C\otimes_R A\). By Corollary 5, \(B\) is finite syntomic over \(A\), hence flat over \(A\). Since \(C\) is flat over \(R\) and \(\pi_0(R)=A\), the canonical morphism \(B\otimes_A R \to C\) induces an isomorphism on \(\pi_0\). For each \(i\ge 0\), one has \[\pi_i(B\otimes_A R)\cong \pi_0(B)\otimes_A \pi_i(R) \cong \pi_0(C)\otimes_A \pi_i(R) \cong \pi_i(C),\] where the first isomorphism uses that \(B\) is flat over \(A\), and the last one uses that \(C\) is flat over \(R\). Therefore \(B\otimes_A R \to C\) is a weak equivalence of connective \(R\)-algebras. Thus, the composition \(s\circ((-)\otimes_R A)\) is also equivalent to the identity. Hence \((-)\otimes_R A\) is an equivalence of unlocalized spaces. \(\Box\)

We also recall the Milnor exact sequence for towers of spectra: for any inverse system \(\{X_n\}_{n\ge 1}\), one has a short exact sequence \[0\to \varprojlim\nolimits^1_n \pi_{*-1}(X_n)\to \pi_*(\varprojlim_n X_n)\to \varprojlim_n \pi_*(X_n)\to 0.\]

Put \(F:=\mathrm{FSyn}^{\rm aug}_*:\mathrm{CAlg}_{V//V}\to\mathcal{S}\). Because \(F\) is unlocalized, it accurately captures infinitesimal deformations and factors through \(P^1(\mathrm{CAlg}_{V//V})\); hence \(F\) is reduced and \(1\)-excisive.

Choose a topologically nilpotent element \(\omega\in V\), and let \(\omega^\flat\in V^\flat\) denote its tilt.

For each \(n\ge 1\), write \(A_n=A/\omega^nA\) and \(A_n^\flat=A^\flat/(\omega^\flat)^nA^\flat\). For each \(n\ge 2\), \(0\to \omega^{n-1}A/\omega^nA\to A_n\to A_{n-1}\to 0\) is a square-zero extension. Applying Proposition 15 to \(A_n\to A_{n-1}\), we obtain a weak equivalence \(F(\Sigma^1_{A_{n-1}}A_n)\to F(A_{n-1})\).

Since \(F\) is reduced and \(1\)-excisive, \(F(\Sigma X)\simeq \Omega F(X)\). Therefore, \(\Omega F(A_n)\simeq F(A_{n-1})\), yielding \(F(A_n)\simeq \Sigma^{n-1}F(A_1)\). Using \(A_1\cong A_1^\flat\), we get weak equivalences \(F(A_n)\to F(A_n^\flat)\) for all \(n\ge 1\).

By Proposition 11, finite syntomic algebras are Zariski locally obtained from the finitely presented universal families of [14]. For the derived descent and completion formalism that allows these finite-presentation charts to be used for connective \(\mathbb{E}_{\infty}\)-rings and their nilpotent thickenings, see Lurie’s derived algebraic geometry [15], [16]. Hence [14] implies that compatible systems of finite syntomic algebras over the towers \(\{A_n\}_n\) and \(\{A_n^\flat\}_n\) algebraize uniquely, so the canonical maps \[F(A)\to \varprojlim_n F(A_n),\qquad F(A^\flat)\to \varprojlim_n F(A_n^\flat)\] are weak equivalences.

Applying the preceding Milnor exact sequence to the towers \(X_n=F(A_n)\) and \(X_n=F(A_n^\flat)\), we obtain short exact sequences \[0\to \varprojlim\nolimits^1_n\pi_{*-1}(F(A_n)) \to \pi_*(\varprojlim_n F(A_n))\to \varprojlim_n\pi_*(F(A_n))\to 0\] and \[0\to \varprojlim\nolimits^1_n\pi_{*-1}(F(A_n^\flat)) \to \pi_*(\varprojlim_n F(A_n^\flat))\to \varprojlim_n\pi_*(F(A_n^\flat))\to 0.\] Since \(F(A_n)\simeq F(A_n^\flat)\) for all \(n\ge 1\), the left and right terms are isomorphic, hence the middle terms are also isomorphic. Thus \[\varprojlim_n F(A_n)\to \varprojlim_n F(A_n^\flat)\] is a weak equivalence. Combining this with the canonical identifications above, we obtain that \[(-)^\flat:F(A)\to F(A^\flat),\] is a weak equivalence, implying that \[\mathrm{dMGL}^{\rm nu,f}(A)\simeq \Sigma^{\infty}F(A) \simeq \Sigma^{\infty}F(A^\flat) \simeq \mathrm{dMGL}^{\rm nu,f}(A^\flat).\] \(\Box\)

4.4 Finite derived algebraic cobordism of almost algebras↩︎

Following the Smith-ideal formulation of almost mathematics in [13], almost mathematics is described by base rings and their idempotent ideals. We consider the case where the base ring \(V\) has an idempotent ideal \(\mathfrak{m}\). We strictly assume that \(\mathfrak{m} \subsetneq V\) is a proper idempotent ideal, and \(\tilde{\mathfrak{m}}= \mathfrak{m} \otimes_V \mathfrak{m}\) is a flat \(V\)-module.

To relate this almost construction to the non-unital one, note that the almost unitalization satisfies \[\tilde{\mathfrak{m}}\otimes_V(-)_{!!}\simeq \tilde{\mathfrak{m}}\otimes_V(-)\] as functors on almost \(V\)-algebras. Thus tensoring back with \(\tilde{\mathfrak{m}}\) recovers the original almost algebra from its almost unitalization. By the Smith-ideal approach to almost mathematics [13], this exhibits \(\mathrm{dMGL}^{a,\rm fin}(A)\) as a homotopy direct factor of \(\mathrm{dMGL}^{\rm nu,f}(V\oplus (\tilde{\mathfrak{m}}\otimes_V A))\). Hence Theorem 16 immediately gives the almost version of tilting equivalences:

5 Approximation of algebraic cobordism to \(K\)-theory↩︎

In this section, we apply Goodwillie calculus to algebraic cobordism and \(K\)-theory. In Section 5.1, we define approximation functors. In Section 5.2, we apply them to algebraic cobordism and \(K\)-theory, and prove Bott periodicity and Gabber rigidity.

5.1 Approximation functors↩︎

Let \(\alpha:F \to G\) be a natural transformation of functors \(F,G:\mathcal{C} \to \mathcal{D}\), and let \(\mathrm{cok}\,\alpha\) be its homotopy cokernel. Under suitable conditions, let \(P^n_{G}(F):\mathcal{C} \to \mathcal{D}\) be the fiber of \[G \to \mathrm{cok}\,\alpha \to P^n(\mathrm{cok}\,\alpha).\] Then \(P^0_G(F) \to G\) is an equivalence. Thus we get a tower \[\cdots \to P^n_G(F) \to P^{n-1}_G(F) \to \cdots \to P^1_G(F) \to P^0_G(F) \simeq G : \mathcal{C} \to \mathcal{D}.\] We call \(P^n_{G}(F)\) the \(n\)-Goodwillie approximation of \(F\) to \(G\).

The full subcategory \(\mathrm{Exc}^n(\mathcal{C},\,\mathcal{D})\) is closed under small limits in \(\mathrm{Fun}(\mathcal{C},\,\mathcal{D})\). In particular, the fiber of \(G \to P^n(\mathrm{cok}(\alpha))\) is \(n\)-excisive. \(\Box\)

5.2 Approximation to \(\mathbb{K}\)-theory↩︎

Let \(\mathrm{CAlg}(\mathrm{MSp}_\infty)\) be the \(\infty\)-category of motivic \(\mathbb{E}_{\infty}\)-rings. If we forget the unit, a motivic \(\mathbb{E}_{\infty}\)-ring becomes a motivic non-unital \(\mathbb{E}_{\infty}\)-ring. Since algebraic cobordism is universal among oriented motivic spectra, there is a canonical morphism \(\alpha:\mathrm{MGL} \to \mathbb{K}\).

For the universal morphism \(\alpha: \mathrm{MGL} \to \mathbb{K}\), the \(n\)-Goodwillie approximation \(P^n_\mathbb{K}(\mathrm{MGL})\) inherits properties from \(\mathbb{K}\).

Since \(\mathbb{K}\) has Bott periodicity, \(P^0_\mathbb{K}(\mathrm{MGL})\) is periodic and is the trivial \(\mathbb{K}\)-augmented \(\mathbb{K}\)-algebra. We prove that \(P^n_\mathbb{K}(\mathrm{MGL})\) is periodic and a \(\mathbb{K}\)-augmented algebra for each \(n\) by induction on \(n \ge 0\). The homotopy Cartesian square \[\xymatrix@1{ P^{n}(\mathrm{cok} \alpha) \ar[r] \ar[d]& P^{n-1}(\mathrm{cok} \alpha ) \ar[d]\\ {*} \ar[r] & R^n (\mathrm{cok} \alpha ) }\] induces a homotopy Cartesian square \[\xymatrix@1{ P_\mathbb{K}^n(\mathrm{MGL}) \ar[r] \ar[d] & P_\mathbb{K}^{n-1}(\mathrm{MGL}) \ar[d]\\ \mathbb{K}\ar[r]& R^n_\mathbb{K}(\mathrm{MGL}) }\] of motivic \(\mathbb{E}_{\infty}\)-rings. Since \(P^{n-1} \circ R^n\) is contractible for \(n \ge 1\), \(R^n_\mathbb{K}(\mathrm{MGL}) \to P_\mathbb{K}^{n-1}( \mathrm{MGL}) \simeq \mathbb{K}\) is homotopic to the augmentation to \(\mathbb{K}\). Therefore, \(R_\mathbb{K}^n(\mathrm{MGL})\) has a \(\mathrm{PMGL}\)-algebra structure implying that it is periodic. Therefore, by the assumption of the induction, \(P^n_\mathbb{K}(\mathrm{MGL})\) is periodic and \(\mathbb{K}\)-augmented. \(\Box\)

Since \(P^n_\mathbb{K}(\mathrm{MGL})\) is periodic, \(\mathrm{MGL}\to P^n_\mathbb{K}(\mathrm{MGL})\) factors through \(\mathrm{PMGL}\). Therefore \(P^n(p): P^n_\mathbb{K}(\mathrm{MGL}) \to P^n_\mathbb{K}(\mathrm{PMGL})\) is homotopically split. \(\Box\)

The cofiber sequence \(I \to \tilde{A}\to A\) induces \(P^1_A(\tilde{A}) \simeq \Omega_{A}\Sigma_{A}(P^1_A(\tilde{A})) \simeq A \oplus L_{A/P^1_A(\tilde{A})}[-1] \simeq A \oplus P^1_{A}(I)\). The homotopy biCartesian square \[\xymatrix@1{ P^1_A(\tilde{A}) \ar[r] \ar[d] & A \ar[d] \\ A \ar[r]& A\otimes_{P^1_A(\tilde{A})} A }\] induces a weak equivalence \[L_{A/P^1_A(\tilde{A})}[-1] \to \mathrm{fib}( A \to A \otimes_{P^1_A(\tilde{A})}A).\] \(\Box\)

For each \(n\ge 1\), by construction of relative Goodwillie approximations, there is a homotopy Cartesian square \[\xymatrix@1{ P^n_\mathbb{K}(\mathrm{PMGL}) \ar[r] \ar[d] & P^{n-1}_\mathbb{K}(\mathrm{PMGL}) \ar[d] \\ \mathbb{K}\ar[r] & R^n_\mathbb{K}(\mathrm{PMGL}). }\] Hence it is enough to prove that, for each \(n\ge 1\), \[R^n_\mathbb{K}(\mathrm{PMGL})(Y)\to R^n_\mathbb{K}(\mathrm{PMGL})(X)\] is a weak equivalence.

For \(n=1\), Lemma 6 gives that \(L_{\mathbb{K}/P^1_\mathbb{K}(\mathrm{PMGL})}[-1]\) is a homotopy retract of \(\mathbb{K}\). Therefore \[L_{\mathbb{K}(Y)/P^1_\mathbb{K}(\mathrm{PMGL})(Y)}[-1]\to L_{\mathbb{K}(X)/P^1_\mathbb{K}(\mathrm{PMGL})(X)}[-1]\] is a weak equivalence, and so are \[P^1_\mathbb{K}(\mathrm{PMGL})(Y)\to P^1_\mathbb{K}(\mathrm{PMGL})(X),\qquad R^1_\mathbb{K}(\mathrm{PMGL})(Y)\to R^1_\mathbb{K}(\mathrm{PMGL})(X).\]

Now let \(n\ge 2\). Let \(\mathcal{H}^n_\mathbb{K}\) denote the full subcategory of \(n\)-homogeneous functors in the ambient functor category. By Goodwillie theory ([10]), \(\mathcal{H}^n_\mathbb{K}\) is a stable \(\infty\)-category. Hence the unit map \[R^n_\mathbb{K}(\mathrm{PMGL})\to \Omega^1_\mathbb{K}\Sigma^1_\mathbb{K}R^n_\mathbb{K}(\mathrm{PMGL})\] is a weak equivalence, and the augmentation \(\varepsilon_n:R^n_\mathbb{K}(\mathrm{PMGL})\to \mathbb{K}\) is a homotopically square-zero extension.

Applying Lemma 6 to \(\varepsilon_n\) objectwise at \(Y\) and \(X\), the relative cotangent complexes \[L_{\mathbb{K}(Y)/R^n_\mathbb{K}(\mathrm{PMGL})(Y)}\to L_{\mathbb{K}(X)/R^n_\mathbb{K}(\mathrm{PMGL})(X)}\] are identified with the corresponding fibers of the augmentation squares. Since \(f^*: \mathbb{K}(Y)\to\mathbb{K}(X)\) is a weak equivalence, base change along \(f^*\) induces an equivalence on module categories; therefore the above map of relative cotangent complexes is a weak equivalence. By the square-zero description of \(\varepsilon_n\), this implies \[R^n_\mathbb{K}(\mathrm{PMGL})(Y)\to R^n_\mathbb{K}(\mathrm{PMGL})(X)\] is a weak equivalence.

Finally, applying the first homotopy Cartesian square objectwise at \(Y\) and \(X\), and using induction on \(n\), we conclude that \[P^n_\mathbb{K}(\mathrm{PMGL})(Y)\to P^n_\mathbb{K}(\mathrm{PMGL})(X)\] is a weak equivalence for any \(n\ge 0\). \(\Box\)

This is a direct result of Theorem 20 and Corollary 6. \(\Box\)

We give an analogue of the Gabber rigidity theorem for nilpotent approximation in Kato2023nilpotent?:

[17], [18] [19] [13] [1] [20] [21] [22] [23] [24] [25] [26] [11] [10] [2] [27] preygel?

References↩︎