--- title: Syntomic Steenrod Algebra url: https://www.emergentmind.com/topics/syntomic-steenrod-algebra type: topic --- # Syntomic Steenrod Algebra Searching arXiv for the cited paper and closely related background papers to ground the article. Search query: arXiv:2507.13471 Search query: "Prismatic Steenrod operations and arithmetic duality on Brauer groups" The syntomic Steenrod algebra is the algebra of stable cohomology operations acting on mod $p$ syntomic cohomology, also called étale-motivic cohomology, of algebraic varieties in characteristic $p$. In the form constructed in "Prismatic Steenrod operations and arithmetic duality on Brauer groups" [2507.13471], it is defined through prismatic and perfectoid methods, organized in a spectral framework, and used to prove arithmetic duality statements for Brauer groups over finite fields. The theory combines prismatic cohomology, the Nygaard filtration, perfectoid nearby cycles, spectral syntomic cohomology, and a category of spectral prismatic $F$-gauges in order to produce explicit operations $P^i$ and $\beta P^i$, establish their Adem and Cartan formulas, compare them with $E_\infty$ operations, and apply them to the Milne–Artin–Tate pairing and higher Brauer groups [2507.13471]. ## 1. Foundational setting in syntomic and prismatic cohomology Mod $p$ syntomic cohomology enters the construction through the Bhatt–Morrow–Scholze definition in terms of prismatic cohomology and the Nygaard filtration. In the affine formal case $\hat X=\operatorname{Spf}(\hat R)$, one has $$ R\Gamma_{\mathrm{syn}}(\hat X;\mathbf Z_p(n)):=\operatorname{Fib}\!\big(\operatorname{Fil}^n_N R\Gamma_{\mathrm{prism}}(\hat X)\to R\Gamma_{\mathrm{prism}}(\hat X)\big), $$ where the map is $\varphi-1$, with the twist by $n$ accounted for via Breuil–Kisin twists and the Nygaard filtration. If $\hat X/\mathcal O$ is smooth of relative dimension $d$, then $$ H^i_{\mathrm{syn}}(\hat X;\mathbf Z_p(n))=0 \quad \text{for } i\notin [0,d+1]. $$ Over schemes $X/\mathbf Z_p$, Bhatt–Lurie define decompleted syntomic cohomology by a derived Cartesian square relating completed syntomic cohomology and étale cohomology. The data further records that multiplication by a compatible system of $p$-power roots of unity $$ \epsilon=(1,\zeta_p,\zeta_{p^2},\dots)\in H^0_{\mathrm{syn}}(\operatorname{Spec}\mathbf Z_p^{\mathrm{cyc}};\mathbf Z_p(1)) $$ intertwines the decompleted theory compatibly with the completed one. Over finite fields $k$ of characteristic $p$, syntomic cohomology is identified with mod $p$ logarithmic de Rham–Witt cohomology. More precisely, the étale sheaves $\nu_n(b)$ are identified with the pushforward of $\mathbf Z/p^n(b)[b]$ from the quasisyntomic site to the étale site, and for smooth $X/k$, Geisser–Levine identify étale-motivic mod $p$ complexes with syntomic cohomology by $$ \mathbf Z(b)^{\mathrm{\acute et}}_X\otimes \mathbf Z/p^n \simeq \nu_n(b)[-b]. $$ This is the point at which syntomic cohomology becomes the arithmetic analogue of motivic cohomology used by the later Steenrod-theoretic construction. The ambient geometry is prismatic. Prismatic cohomology is defined on the absolute or relative prismatic site using prismatic envelopes and $\delta$-ring structures. The Nygaard filtration $N^\bullet$ is aligned with Frobenius $\varphi$, and vanishing estimates come from the Hodge–Tate comparison: graded pieces of prismatic cohomology, after Frobenius and twist, identify with truncated Breuil–Kisin twists of $\Omega^i_{X/\mathcal O}$. A decisive input is perfectoid nearby cycles. For a rank-one $p$-adic perfectoid field $K$ such as $\mathbf Q_p^{\mathrm{cyc}}$, with $\mathcal O=\mathbf Z_p^{\mathrm{cyc}}$ and residue field $k$, the perfectoid nearby cycles functor $$ \psi:\mathrm{SH}_K\to \mathrm{MS}_k $$ is defined by $$ \psi:=i^*\circ L_{\mathrm{\acute et}}\circ j_*\circ (\text{inclusion }\mathrm{SH}_K\to \mathrm{MS}_K). $$ Its key calculation is $$ \psi(\mathrm{MH}\mathbf Z_p)_{{\mathbf Q}_p^{\mathrm{cyc}}}\simeq (\mathrm{MS}\mathbf Z_p)_k,\qquad \psi(\mathrm{MH}\mathbf Z_p(n))_{{\mathbf Q}_p^{\mathrm{cyc}}}\simeq (\mathrm{MS}\mathbf Z_p(n))_k, $$ with compatibility with Tate twists and sums. This provides the bridge from motivic operations in characteristic $0$ to syntomic operations in characteristic $p$. ## 2. Spectral syntomic cohomology and spectral prismatic $F$-gauges The construction is not formulated only at the level of cohomology groups. It is organized in a category of $p$-complete motivic spectra. Let $\mathrm{MS}_S$ denote the $p$-complete motivic spectra category over a base $S$, constructed as a symmetric $\mathbf P^1$-spectrum category. It admits Tate twists $$ Y(n):=Y\otimes (\Sigma^\infty \mathbf P^1)^{\otimes n}[-2n]. $$ An oriented graded algebra construction is used: a $\mathbf P^1$-preorientation $(E_\bullet,\omega:\mathbf P^1\to E_1)$ determines a lax symmetric spectrum object via $\nu$, and oriented objects thereby give rise to objects of $\mathrm{MS}_S$. Spectral syntomic cohomology is introduced in this framework. One defines $$ S\mathbf Z_p(\bullet)[2\bullet]_S\in \operatorname{Alg}(\mathcal C_S^{\mathbf N}) $$ with syntomic first Chern class $$ c_1^{\mathrm{syn}}:\operatorname{Pic}\to S\mathbf Z_p(1)[2], $$ and then sets $$ (\mathrm{MS}\mathbf Z_p)_S:=\nu(S\mathbf Z_p(\bullet)[2\bullet]_S,\xi^{\mathrm{syn}}). $$ The cohomology theory represented by $(\mathrm{MS}\mathbf Z_p(n))_S$ is syntomic cohomology with weight $n$, and functoriality is absolute in the sense that $$ f^*(\mathrm{MS}\mathbf Z_p)_T\simeq (\mathrm{MS}\mathbf Z_p)_S $$ for any morphism $f:S\to T$. The categorical environment for prismatization is a version of prismatic $F$-gauges. The full subcategory $$ \mathrm{FGauge}(\mathbf F_p)\subset \operatorname{Mod}_{\mathrm{MS}\mathbf F_p}(\mathrm{MS}_k) $$ is generated under colimits, twists, and shifts by $\Sigma^\infty_+X\otimes \mathrm{MS}\mathbf F_p$ for smooth projective $X/k$. There is a canonical symmetric monoidal equivalence $$ \mathrm{FGauge}(\mathbf F_p)\simeq D(\mathrm{FSyn}_{\mathbf F_p}), $$ the derived category of quasicoherent sheaves on the prismatic stack. The spectral enhancement is $$ p\mathrm{FGauge}(\mathbb S):=\operatorname{Mod}_{\psi(\mathrm{Sph}_{\mathbf F_p})}(\mathrm{MS}_k)\times_{\operatorname{Mod}_{\mathrm{MS}\mathbf F_p}(\mathrm{MS}_k)}\mathrm{FGauge}(\mathbf F_p), $$ where $\psi(\mathrm{Sph}_{\mathbf F_p})$ is the syntomic sphere object. There is an adjunction $$ \iota^*:p\mathrm{FGauge}(\mathbb S)\;\dashv\;\mathrm{FGauge}(\mathbf F_p):\iota_*, $$ together with a projection formula, colimit preservation, conservativity of $\iota_*$, and the equivalence $$ \operatorname{Mod}_{\iota_*\mathrm{MS}\mathbf F_p}(p\mathrm{FGauge}(\mathbb S))\simeq \mathrm{FGauge}(\mathbf F_p). $$ This categorical apparatus supports a spectral version of Serre duality. Writing $\mathcal S=\mathrm{FSyn}_{\mathbf F_p}$ for the classical prismatic stack and $\mathcal S^+$ for its spectral enhancement, one has on $\mathcal S$ $$ \omega_{\mathcal S}\simeq \mathcal O_{\mathcal S}[1]. $$ On $\mathcal S^+$, the spectral dualizing object is defined by $$ \omega_{\mathcal S^+}:=\pi^! I, $$ where $I$ is the $p$-completed Brown–Comenetz spectrum, and one sets $$ D_{\mathcal S^+}:=R\!\operatorname{Hom}(-,\omega_{\mathcal S^+}). $$ Compatibility with the classical duality is expressed by $$ \iota_*\circ D_{\mathcal S}\simeq D_{\mathcal S^+}\circ \iota_*. $$ This “spectral Serre duality” is central to the later duality-equivariance of Steenrod actions [2507.13471]. ## 3. Definition of the syntomic Steenrod algebra The syntomic Steenrod algebra is defined as an Ext algebra over the syntomic sphere. Let $\mathrm{MH}\mathbf F_p$ be the mod $p$ motivic cohomology spectrum and $\mathrm{Sph}$ the motivic sphere. Then over the residue field $k$ one sets $$ A_{\mathrm{syn}}^{*,*}:=\operatorname{Ext}^{*,*}_{\operatorname{Mod}_{\psi(\mathrm{Sph}_{\mathbf F_p})}(\mathrm{MS}_k)}\big((\mathrm{MS}\mathbf F_p)_k,(\mathrm{MS}\mathbf F_p)_k\big). $$ Similarly, over $\mathbf Z_p^{\mathrm{cyc}}$, $$ A_{\mathrm{syn},\mathbf Z_p^{\mathrm{cyc}}}^{*,*}:= \operatorname{Ext}^{*,*}_{\operatorname{Mod}_{\Psi(\mathrm{Sph}_{\mathbf F_p})}(\mathrm{MS}_{\mathbf Z_p^{\mathrm{cyc}}})} \big((\mathrm{MS}\mathbf F_p)_{\mathbf Z_p^{\mathrm{cyc}}},(\mathrm{MS}\mathbf F_p)_{\mathbf Z_p^{\mathrm{cyc}}}\big). $$ There is a natural homomorphism from the motivic Steenrod algebra $$ A_{\mathrm{mot}}^{*,*}=\operatorname{Ext}^{*,*}_{\mathrm{SH}_K}(\mathrm{MH}\mathbf F_p,\mathrm{MH}\mathbf F_p) $$ induced by $\psi$ and its enhanced version $\psi^{\mathrm{enh}}$. The resulting algebra is free over $H_{\mathrm{syn}}^{*,*}(\operatorname{Spec}k)$ with basis indexed by admissible compositions. It is generated by power operations $P^i$ and $\beta P^i$ for $i\ge 0$, acting by $$ P^i:H_{\mathrm{syn}}^{a,b}\to H_{\mathrm{syn}}^{a+2i(p-1),\,b+i(p-1)}, $$ and $$ \beta P^i:H_{\mathrm{syn}}^{a,b}\to H_{\mathrm{syn}}^{a+2i(p-1)+1,\,b+i(p-1)}. $$ A basis is given by admissible monomials $$ P^\alpha:=\beta^{\epsilon_r}P^{i_r}\cdots \beta^{\epsilon_1}P^{i_1}\beta^{\epsilon_0}, $$ indexed by $$ \alpha=(r,\epsilon_r,i_r,\dots,\epsilon_1,i_1,\epsilon_0)\in I, $$ with $i_j>0$, $\epsilon_j\in\{0,1\}$, and admissibility condition $$ i_{j+1}\ge p\,i_j+\epsilon_j. $$ The paper states that this mirrors the Voevodsky/HKO decomposition over characteristic $0$ and yields freeness over $H_{\mathrm{syn}}(\operatorname{Spec}k)$. The reduced syntomic Steenrod algebra $$ rA_{\mathrm{syn}}^{*,*}\subset A_{\mathrm{syn}}^{*,*} $$ is the Hopf subalgebra over $\mathbf F_p$ generated by the $P^i$ and $\beta P^i$, and one has $$ A_{\mathrm{syn}}^{*,*}\simeq rA_{\mathrm{syn}}^{*,*}\otimes_{\mathbf F_p} H_{\mathrm{syn}}^*(\operatorname{Spec}k;\mathbf F_p). $$ For $p=2$, the notation is normalized by $$ Sq^{2i}:=P^i,\qquad Sq^{2i+1}:=\beta P^i. $$ A common misconception is that the construction identifies the full ring of stable syntomic cohomology operations. The paper explicitly states a more limited claim: it identifies a Hopf subalgebra with the “correct” behavior, but does not claim completeness. This suggests that the presently constructed algebra is best viewed as the explicitly controlled Steenrod-theoretic core of a potentially larger operation algebra [2507.13471]. ## 4. Structure: Adem relations, Cartan formula, and comparison with $E_\infty$ operations The algebraic structure is controlled by explicit Adem and Cartan formulas. For odd $p$ and $0b$, then $$ P^i=\tau^{(p-1)(i-b)}Pe^i: H_{\mathrm{syn}}^{a,b}\to H_{\mathrm{syn}}^{a+2i(p-1),\,b+i(p-1)}. $$ If $b\ge i$, then $$ Pe^i=\tau^{(p-1)(b-i)}P^i: H_{\mathrm{syn}}^{a,b}\to H_{\mathrm{syn}}^{a+2i(p-1),\,pb}. $$ In particular, when $b=i$, the two operations agree. The paper also states that $Pe^i$ vanishes for $2i>a$ and $i\ge b$, and over $k$ one has $\tau^{p-1}=0$, so certain $Pe^i$ or $P^i$ vanish modulo $\tau$. In the special case $H_{\mathrm{syn}}^{2i,i}$, $P^i$ acts by $p$th power. These formulas are the computational mechanism that allows passage between the weight-shifting operation $P^i$ and the weight-multiplying operation $Pe^i$ [2507.13471]. ## 5. Duality, anti-involution, and arithmetic consequences Spectral Serre duality is used to relate Steenrod operations to arithmetic pairings. The prismatized Steenrod algebra is $$ sA_{\mathrm{syn}}:=R\!\operatorname{Hom}_{\mathcal S^+}(\iota_*\mathcal O_{\mathcal S},\iota_*\mathcal O_{\mathcal S}), $$ and duality produces an anti-involution $$ \sigma:sA_{\mathrm{syn}}\to sA_{\mathrm{syn}}^{op} $$ characterized by $$ \sigma^*\iota_*\mathcal O_{\mathcal S}[1]\simeq R\!\operatorname{Hom}_{\mathcal S^+}(\iota_*\mathcal O_{\mathcal S},\omega_{\mathcal S^+}). $$ This anti-involution is compatible with the product and coproduct structure on $sA_{\mathrm{syn}}$. The spectral Serre duality statement asserts compatibility of Serre duality with Steenrod actions. For dualizable $s\mathcal F\in \operatorname{Perf}(\mathcal S^+)$ and $c\mathcal F:=\iota^*s\mathcal F$, one has $$ \sigma^*\iota_*D_{\mathcal S}(c\mathcal F)\simeq \iota_*c\mathcal F^\vee[1] $$ as $sA_{\mathrm{syn}}^{op}$-modules in $D(\mathcal S^+)$, and consequently $$ \sigma^*H^{*,*}(\mathcal S;D_{\mathcal S}(c\mathcal F)) \simeq H^{*,*}(\mathcal S;c\mathcal F^\vee[1]). $$ These duality compatibilities yield a prismatized pushforward $$ \varphi^{\mathrm{prism}}:H^{*,*}(\mathcal S;c\mathcal F\otimes c\mathcal G)\to H^{*,*}(\mathcal S;c\mathcal F)\otimes H^{*,*}(\mathcal S;c\mathcal G)[1], $$ lifting the usual $\varphi_*$. The main structural theorem is that $\varphi_*$ is equivariant for the action of $rA_{\mathrm{syn}}^{*,*}$, so $\varphi_*$ commutes with syntomic Steenrod operations. The principal arithmetic application concerns the Milne–Artin–Tate pairing on the Brauer group of a smooth, proper, geometrically connected surface over a finite field. In characteristic $2$, the diagonal value is expressed by $$ \langle u,u\rangle_{\mathrm{MAT}}=\int_X Pe^1(\beta u),\qquad u\in \operatorname{Br}(X)[2]. $$ The comparison theorem then relates this to syntomic operations and Wu formulas, leading to the alternation statement. The paper proves: - For a smooth, proper, geometrically connected surface $X$ over a finite field of characteristic $2$, $\#\operatorname{Br}(X)_{nd}$ is a perfect square. - For a smooth, proper, geometrically connected surface $X$ over a finite field of characteristic $p$, including $p=2$, the Milne–Artin–Tate pairing on $\operatorname{Br}(X)_{nd}$ is symplectic, meaning alternating and non-degenerate. The same framework is stated to yield symplectic structures on higher Brauer groups of even-dimensional varieties over finite fields. In this sense, the syntomic Steenrod algebra is not merely a formal extension of motivic Steenrod theory; it serves as the cohomological mechanism through which arithmetic duality is made explicit [2507.13471]. ## 6. Functoriality, comparisons, and unresolved questions The theory has strong functoriality properties. The syntomic motivic spectrum $(\mathrm{MS}\mathbf Z_p(n))_S$ is absolute under pullback. The perfectoid nearby cycles functor $\psi$ is compatible with Tate twists, sums, and base change via $\iota^*$. The category $\mathrm{FGauge}(\mathbf F_p)\simeq D(\mathrm{FSyn}_{\mathbf F_p})$ is symmetric monoidal, and the Künneth formula for syntomic cohomology lifts to its monoidal structure. The functors $\iota^*$ and $\iota_*$ satisfy a projection formula. The coproduct on $A_{\mathrm{syn}}^{*,*}$ matches the Cartan formula on cup products, and cup products on sheaves are compatible with Steenrod actions after passage to $\mathcal S^+$. The comparison with classical motivic Steenrod theory is explicit but not tautological. Over characteristic $0$ fields, the motivic Steenrod algebra decomposes by work of Voevodsky and Hoyois–Kelly–Østvær, and $\psi$ transports this structure to syntomic operations. In characteristic $p$, Annala–Elmanto construct motivic Steenrod operations with Adem and Cartan formulas, and étale sheafification recovers the syntomic results recorded as Theorem 6.1 in the paper. This suggests that the syntomic Steenrod algebra is best understood as a characteristic-$p$ arithmetic realization of a broader motivic pattern, but with genuinely new weight behavior coming from syntomic and prismatic geometry. The instability behavior is partially controlled. The Bockstein $\beta$ has bi-degree $(1,0)$, and $\beta P^i$ has degree $(2i(p-1)+1,i(p-1))$. Corollary 10.3 gives the vanishing of $Pe^i$ if $2i>a$ and $i\ge b$ over $k$, and similarly $P^i$ vanishes if $2i>a$ and $i>b$. The data notes that Nishida nilpotence is not explicitly addressed. Several open directions are identified. A complete identification of the full syntomic Steenrod algebra as stable cohomology operations remains open. A motivic version $$ \Psi^{\mathrm{mot}}:\mathrm{SH}_K\to \mathrm{MS}_{\mathcal O} $$ would identify $\Psi^{\mathrm{mot}}(\mathrm{MH}\mathbf Z_p)_K\simeq (\mathrm{MH}\mathbf Z_p)_{\mathcal O}$ and recover syntomic structure after étale sheafification, but this requires further development in motivic cohomology in mixed characteristic. Further computations beyond the line $b=i$, more systematic instability criteria, and potential Nishida-type results in syntomic settings are also listed as open problems. A plausible implication is that the present construction provides a stable and computable core from which a fuller operation theory may eventually be extracted, but the paper does not claim that this extraction has yet been achieved [2507.13471]. Source: https://www.emergentmind.com/topics/syntomic-steenrod-algebra