--- title: Nygaard Filtration in p-adic Cohomology url: https://www.emergentmind.com/topics/nygaard-filtration type: topic --- # Nygaard Filtration in p-adic Cohomology Nygaard filtration is a filtration attached to \(p\)-adic cohomological objects in which Frobenius divisibility is encoded integrally. In the classical de Rham–Witt setting it is a filtered subcomplex on which a divided Frobenius \(\varphi/p^n\) is defined, while in modern prismatic theory it appears as a filtration on absolute or relative prismatic cohomology, often on a Frobenius-twisted object, and can be packaged by Rees constructions and stacks. The recent literature uses several conventions—decreasing versus increasing indexing, filtrations on \(\Delta\) or on \(F^*\Delta\), and, in some positive-characteristic treatments, only the first Nygaard piece—so the term denotes a closely related family of Frobenius-divisibility structures rather than a single universal formula [2505.03546] [2512.19348] [2507.08568]. ## 1. Classical formulations and Frobenius divisibility In the de Rham–Witt complex of a smooth \(X/F_p\), one classical form of the Nygaard filtration is the filtered subcomplex \[ \mathrm N^{\geq n}W\Omega_X = p^{n-1}VW\mathcal O_X \to p^{n-2}VW\Omega_X^1 \to \cdots \to VW\Omega_X^{n-1} \to W\Omega_X^n \to \cdots, \] where \(V\) is the Verschiebung. The defining feature is that Frobenius on this subcomplex is divisible by \(p^n\), so one obtains a divided Frobenius \[ \varphi_n:=\varphi/p^n:\mathrm N^{\geq n}W\Omega_X\to W\Omega_X. \] This is used to define the syntomic complex \[ Z_p(n)(X):=\mathrm{fib}\bigl(\varphi_n-1:\mathrm N^{\geq n}W\Omega_X\to W\Omega_X\bigr), \] and the associated graded description \[ \mathrm N^{\geq n}W\Omega_X/\mathrm N^{\geq n+1}W\Omega_X = \tau^{\leq n}\Omega^\bullet_{X/F_p} \] identifies the Nygaard filtration as the mechanism relating Frobenius-divisible Witt complexes to truncated de Rham data [2404.18367]. A different but closely related positive-characteristic formulation isolates only the first Nygaard piece of crystalline cohomology. For a smooth \(k\)-scheme \(X\) over a perfect field \(k\) of characteristic \(p\), one defines \[ \Nyg \Rcris(X/\mathbf Z):=\Rcris(X,\mathcal I), \] where \(\mathcal I\) is the kernel of the reduction map \(\mathcal O\to G_a\) on the big fppf crystalline site. This fits into the exact triangle \[ \Nyg\Rcris(X/\mathbf Z)\to \Rcris(X/\mathbf Z)\to \R\Gamma(X,\mathcal O_X), \] and, for elementary quasiregular semiperfect algebras \(C\), admits the explicit characterization \[ \Nyg \Acris(C)=\{a\in \Acris(C)\mid F(a)\in p\,\Acris(C)\}. \] Here the Nygaard condition is precisely the condition needed to define \(F/p\), and hence the operator \(F/p-1\), on an integral subobject of crystalline cohomology [2507.08568]. These formulations share the same structural content: a Nygaard piece is the part of an integral cohomological object on which Frobenius is divisible by the appropriate power of \(p\) or of the prism ideal. This suggests that Nygaard filtration is best viewed as an integral Frobenius-divisibility device before it is viewed as a comparison filtration. ## 2. Prismatic and stack-theoretic formulations In absolute prismatic cohomology, the Nygaard filtration is a filtered object \[ \Fil^\bullet_N R\Gamma_\Delta(X) \] attached to a bounded \(p\)-adic formal scheme \(X\) that is \(p\)-quasisyntomic and qcqs. A stack-theoretic model is the Nygaard-filtered prismatisation \[ \pi_{X^N}:X^N\to Z_p^N, \] together with the Rees map \[ t:Z_p^N\to A^1_-/G_m. \] The pushforward along \(t_X:=t\circ \pi_{X^N}\) computes the Nygaard filtration: \[ t_{X,*}\mathcal O_{X^N}\simeq \Fil^\bullet_N R\Gamma_\Delta(X). \] In this language, \(X^\prism\) computes prismatic cohomology, \(X^N\) computes Nygaard-filtered prismatic cohomology, and \(X^{dR,+}\) computes Hodge-filtered de Rham cohomology. The stacky comparison theorem is expressed by a commutative square \[ \begin{tikzcd} Z_p^{dR}\ar[r]\ar[d] & Z_p^{dR,+}\ar[d] \\ Z_p^\prism\ar[r] & Z_p^N \end{tikzcd} \] which is an almost pushout up to \(p\)-isogeny; for \(E\in \Perf(Z_p^N)\), the induced square on cohomology after inverting \(p\) is a pullback, and it is integral when the Hodge–Tate weights of \(E\) are all at least \(-p\) [2505.03546]. This stacky formalism also enlarges the coefficient theory. The natural coefficient category for Nygaard-filtered prismatic cohomology is \[ \Gauge_\prism(X):=D(X^N), \] whose objects are called gauges. For a gauge \(E\in D(X^N)\), \[ \Fil^\bullet_N R\Gamma_\Delta(X,E):=t_{X,*}(E). \] The comparison with Hodge-filtered de Rham cohomology then extends from the structure sheaf to arbitrary perfect gauges on smooth proper \(p\)-adic formal schemes, with the same weight-dependent integral range [2505.03546]. A geometric refinement appears in positive characteristic. For a perfect ring \(k\) of characteristic \(p>0\), the parameter stack is \[ k^\nyg \simeq \Spf W(k)\langle u,t\rangle/(ut-p)\,/\,\mathbf G_m, \] and the Rees algebra of Nygaard filtered prismatic cohomology is \[ \rees(Fil_\nyg F^*_{R/k}) = \widehat{\bigoplus_{i\in \mathbf Z} Fil^i_\nyg F^*_{R/k}\, t^{-i}}. \] The Nygaard filtered prismatization \(R^\nyg\) is then identified with the relative spectrum of this Rees algebra over \(k^\nyg\): \[ R^\nyg \simeq \uspf_{k^\nyg}\big(\rees(Fil_\nyg F^*_{R/k})\big). \] Its specializations recover Frobenius-twisted prismatic cohomology, prismatic cohomology, Hodge-filtered derived de Rham cohomology, conjugate-filtered Hodge–Tate cohomology, Hodge–Tate cohomology, and Hodge cohomology [2512.19348]. ## 3. Higher, logarithmic, and relative variants One higher analogue is the \(r\)-Nygaard filtration on absolute prismatic cohomology. For an animated ring \(S\), the \(r\)-th Nygaard piece is defined by iterating the pullback of the inclusion and divided Frobenius maps: \[ \mathcal N_r^{\ge i}\Delta_S\{i\} := \mathcal N^{\ge i}\Delta_S\{i\} \times_{\Delta_S\{i\}} \cdots \times_{\Delta_S\{i\}} \mathcal N^{\ge i}\Delta_S\{i\}. \] This literal “gluing \(r\) copies of the usual Nygaard filtration” produces canonical maps \[ \iota:\mathcal N_r^{\ge i}\Delta_S\{i\}\hookrightarrow \Delta_S\{i\}, \qquad \varphi_{r,i}:\mathcal N_r^{\ge i}\Delta_S\{i\}\to \Delta_S\{i\}, \] and recovers the usual Nygaard filtration at \(r=1\). For quasiregular-semiperfectoid \(S\), \[ \mathcal N_r^{\ge i}\Delta_S \simeq \{x\in \Delta_S\mid \varphi^r(x)\in d_r^i\Delta_S\}, \] while for a perfectoid ring \(R_0\), \[ \mathcal N_r^{\ge i}\mathbb A_{\inf}(R_0)= \begin{cases} \xi_r^i\,\mathbb A_{\inf}(R_0), & i\ge 0,\\ \mathbb A_{\inf}(R_0), & i<0. \end{cases} \] Moreover, \[ \mathcal N_r^0\Delta_S\simeq W_r(S), \] so truncated Witt vectors occur as the \(0\)-th graded pieces of the \(r\)-Nygaard filtration [2412.00914]. In logarithmic prismatic cohomology, the appropriate object is the Frobenius-twisted derived logarithmic prismatic complex \[ \Prism_{(R,P)/(A,M_A)}^{(1)} = \Prism_{(R,P)/(A,M_A)}\widehat\otimes^L_{A,\phi_A}A. \] The derived Nygaard filtration is a decreasing multiplicative filtration by \((p,I)\)-complete objects on this Frobenius twist, characterized locally by the divisibility condition \[ \Fil_N^i \Prism_{R/A_0}^{(1)} = \{x\in \Prism_{R/A_0}^{(1)} \mid \phi_{R/A_0}(x)\in I^i\Prism_{R/A_0}\}, \] and globally by left Kan extension from free pre-log rings. Its graded pieces are identified with the conjugate filtration on the Hodge–Tate specialization: \[ \operatorname{gr}_N^i \Prism_{(R,P)/(A,M_A)}^{(1)} \cong \Fil_i\,\overline{\Prism}_{(R,P)/(A,M_A)}\{i\}. \] Under Cartier type hypotheses, Frobenius factors through \(L\eta_I\), and one has \[ \Prism_{(R,P)/(A,M_A)}^{(1)}\xrightarrow{\sim} L\eta_I\,\Prism_{(R,P)/(A,M_A)}, \] which yields the logarithmic de Rham comparison after reducing modulo \(I\) [2306.00364]. A relative display-theoretic variant occurs over a PD-thickening \(S\to R\). There the relative Nygaard complex \(N^r_{\mathrm{rel}/R}W\Omega^\bullet_{X_S/S}\) modifies the de Rham–Witt Nygaard complex by adjoining terms involving the logarithmic Teichmüller ideal, and the relative Frobenius-divisibility is encoded by replacing multiplication by \(p\) with the map \(T(a+V\xi)=a+pV\xi\). The hypercohomology of these relative Nygaard complexes gives the filtered pieces of a relative display over the relative Witt frame [1711.09940]. ## 4. Comparison with Hodge, conjugate, Hodge–Tate, and Sen structures A central theme of the modern theory is that Nygaard filtration is the integral object whose associated graded or special fibers recover Hodge-theoretic structures. In the stacky prismatic formulation, the Hodge-filtered de Rham map \[ i_{dR,+}:Z_p^{dR,+}\to Z_p^N \] is an almost isomorphism over the locus \(t=0\) up to \(p\)-isogeny. For \(E\in \Perf(Z_p^N)\), \[ R\Gamma((Z_p^N)_{t=0},E)[1/p] \cong R\Gamma(Z_p^{Hod},i_{Hod}^*E)[1/p], \] and this is integral when the Hodge–Tate weights are all at least \(-(p-1)\). The key operator is \[ uD-i:\Fil_iV\to \Fil_iV, \] which acts by multiplication by \(-i\) on \(\gr_iV\). This is what yields the \(p\)-isogeny comparison between Nygaard truncations and Hodge truncations. The same framework identifies the special fiber \((X^N)_{t=0}\) with conjugate-filtered diffracted Hodge data together with the Sen operator [2505.03546]. An integral representation-theoretic version appears in the theory of Breuil–Kisin modules. For an effective isogeny \((A,d,\varphi,M)\), the decreasing Nygaard filtration is \[ \mathrm{Fil}^i M^\varphi := M^\varphi \cap d^i M. \] It induces the Hodge filtration on \[ M^{\mathrm{dR}}:=M^\varphi/dM^\varphi \] through \[ \mathrm{Fil}^i M^{\mathrm{dR}} = \mathrm{Fil}^i M^\varphi / d\,\mathrm{Fil}^{i-1} M^\varphi, \] and the conjugate filtration on \[ M^{\mathrm{HT}}:=M/dM \] through the map \[ \frac{\mathrm{Fil}^i M^\varphi}{\mathrm{Fil}^{i+1}M^\varphi} \xrightarrow{\,1/d^i\,} M/dM. \] The resulting graded pieces satisfy \[ \operatorname{gr}^i M^{\mathrm{dR}} \simeq \operatorname{gr}^{\mathrm{conj}}_i M^{\mathrm{HT}}. \] In the same setting, filtered integral Sen theory shows that, for the amplified Sen operator \(\Theta\), \[ (\Theta-na)\bigl(\mathrm{Fil}^{\mathrm{conj}}_n M^{\mathrm{HT}}\bigr) \subset \mathrm{Fil}^{\mathrm{conj}}_{n-1} M^{\mathrm{HT}}, \] and \(\Theta\) acts on the \(n\)-th conjugate graded piece by multiplication by \(na\) [2411.11084]. These comparison statements show that Nygaard filtration is not merely parallel to Hodge or conjugate filtrations. It is the filtered source from which both can be recovered, either through associated graded constructions, through special fibers of Rees objects, or through explicit quotient and image filtrations. ## 5. Arithmetic, homotopical, and geometric applications The \(r\)-Nygaard filtration provides the prismatic interpretation of topological restriction homology. For a quasisyntomic ring \(S\), the motivic filtrations on \(\TR^r(S;\mathbf Z_p)^{hS^1}\) and \(\TR^r(S;\mathbf Z_p)\) have even graded pieces \[ {}^{i,\mathrm{even}}\TR^r(S;\mathbf Z_p)^{hS^1}\simeq \mathcal N_r^{\ge i}\Delta_S\{i\}[2i], \qquad {}^{i,\mathrm{even}}\TR^r(S;\mathbf Z_p)\simeq \mathcal N_r^i\Delta_S\{i\}[2i]. \] For perfectoid \(R_0\), \[ \pi_* \TR^r(R_0;\mathbf Z_p)^{hS^1} \simeq \mathbb A_{\inf}(R_0)[u_r,v_r]/(u_rv_r-\xi_r), \] and \[ \pi_* \TR^r(R_0;\mathbf Z_p)\simeq W_r(R_0)[u_r]. \] This makes the \(r\)-Nygaard filtration the organizing structure for the motivic theory of \(\TR^r\), just as the ordinary Nygaard filtration organizes the motivic theory of \(\TC^-\) and \(\THH\) [2412.00914]. In the arithmetic of schemes over finite fields, a Nygaard-style proof of Milne’s formula expresses the \(p\)-adic absolute value of the special value \(C(X,n)\) through the syntomic complex and the Nygaard quotient: \[ |C(X,n)|_{p}^{-1} = \chi(X,Z_p(n),e)\chi(X,W\Omega_X/\mathrm N^{\geq n}W\Omega_X) = \chi(X,Z_p(n),e)p^{\chi(X,\mathcal O_X,n)}. \] Here the quotient \(W\Omega_X/\mathrm N^{\ge n}W\Omega_X\) is identified with the correction-factor object, and its finite Nygaard filtration reduces the computation to truncated de Rham complexes [2404.18367]. In positive characteristic, the first Nygaard piece on crystalline cohomology yields the exact triangle \[ \R\Gamma_{\mathrm{fppf}}(X,\mathbf Z(1)) \to \Nyg\Rcris(X/W(k)) \xrightarrow{F/p-1} \Rcris(X/W(k)), \] which recovers Illusie’s comparison with the slope \(1\) part after inverting \(p\). The same Frobenius-divisibility formalism, combined with perfection or quasisyntomic-style descent, is used to revisit infinitesimal and fppf comparison statements and to compute explicit examples such as ordinary abelian varieties and products of supersingular elliptic curves [2507.08568]. In the theory of displays, Frobenius-divisibility induced by the Nygaard filtration on the relative de Rham–Witt complex equips crystalline cohomology with a higher display structure. For \(r