2.1 Overview

Let $p$ be an odd prime number, and let $\mu _p$ be the group of complex $p$th roots of unity. Let $R$ be a commutative ring. Let $k$ be a field of characteristic $p$, and let $F:k\to k$ be the Frobenius map. Let $X$ be a topological space and let $D^+(X,R)$ denote the bounded-below derived category of sheaves of $k$-modules on $X$. We write $X^{\mu _p}$ for $\operatorname {Map}(\mu _p,X)$. Following Steenrod [Reference Steenrod and EpsteinSE62], we construct a functor

\[ St:D^+(X,R)\to D^+_{\mu_p}(X^{\mu_p},R) \]

where $D^+_{\mu _p}(X^{\mu _p},R)$ denotes the bounded-below $\mu _p$-equivariant derived category of sheaves of $R$-modules on $X^{\mu _p}$. This functor is not linear or triangulated, but nonetheless if we take $R=k$, compose with restriction to the diagonal and apply to morphisms between shifted constant sheaves we obtain linear maps

(2.1)\begin{equation} F^*H^n(X,k)\to H^{pn}_{\mu_p}(X,k)\cong \bigoplus_{i+j=pn}H^i(X,k)\otimes H^j({\rm B}\mu_{\rm p},k) \end{equation}

for each $n\geq 0$. Recall that $H^*({\rm B}\mu _{\rm p},k)=k[a,\hbar ]$ is the super-polynomial algebra in one variable $a$ of degree $1$ and one variable $\hbar$ of degree $2$. Here $\hbar$ is the first Chern class of the tautological complex line bundle on ${\rm B}\mu _{\rm p}$ arising from the embedding $\mu _p\subset \mathbb {C}^*$.

The direct sum of the maps of (2.1) is not in the most naive sense an algebra homomorphism. This fact led Steenrod to introduce certain correction factors which make it so; his famous cohomology operations are then defined to be the coefficients of the resulting algebra homomorphism in the monomial basis of $k[a,\hbar ]$. However, the sum of maps of (2.1) does give a homomorphism of super-graded algebras

(2.2)\begin{equation} H^*(X,k)^{(1)}\to H^*_{\mu_p}(X,k)\cong H^*(X,k)[a,\hbar] \end{equation}

where $H^*(X,k)^{(1)}$ denotes the Frobenius twist of $H^*(X,k)$. Naively one might think that this is just the $p$-dilation of $F^*H^*(X,k)$. This is wrong: rather, the natural and correct definition of the Frobenius twist of an algebra $A$ in any symmetric monoidal category over $k$ is as the Tate cohomology

\[ A^{(1)}:= \hat{H}^0_{\mu_p}(A^{\otimes\,\mu_p}) \]

where the symmetric monoidal structure endows $A^{\otimes\, \mu _p}$ with the structure of $\mu _p$-equivariant algebra. In the case of the super-graded $k$-algebra $H^*(X,k)$, the underlying super-graded $k$-module of this construction is the same as the $p$-dilation of $F^*H^*(X,k)$, but the multiplication differs by a sign (at least when $p\equiv 3\text { mod 4}$), removal of which is part of the purpose of Steenrod's correction factors.

We prefer therefore to use Steenrod's operations in their raw form, that is, without the correction factors and packaged as in (2.2). This has the advantage of revealing the following fundamental connection between Steenrod's operations and the Artin–Schreier map, which is obscured by the correction factors.

Fact 2.1 (i) Let $X=BT$ for some complex torus $T$. Then the Picard group of $X$ is canonically isomorphic to the character lattice $\mathbb {X}^\bullet (T)$ of $T$, and the cohomology ring is the polynomial algebra

\[ H^*(X,\mathbb{Z}) = \operatorname{Sym}_{\mathbb{Z}}\mathbb{X}^\bullet(T) \]

with $\mathbb {X}^\bullet (T)$ in degree $2$. This is equal to the ring $\mathcal {O}(\mathfrak {t}_{\mathbb {Z}})$ of polynomial functions on the canonical $\mathbb {Z}$-form of the scheme $\mathfrak {t}=\operatorname {Lie}(T)$. Likewise we have

\[ H^*(X,k) = \mathcal{O}(\mathfrak{t}_{k}) \]

where $\mathfrak {t}_{k}$ denotes the canonical $k$-form of $\mathfrak {t}$.

(ii) Under this identification, the map of (2.2) factors as

\[ \mathcal{O}(\mathfrak{t}_{k})^{(1)}\xrightarrow{AS_\hbar} \mathcal{O}(\mathfrak{t}_{k})[\hbar] \subset \mathcal{O}(\mathfrak{t}_{k})[a,\hbar] \]

where $AS_\hbar$ corresponds, on the level of $\bar {k}$-points, to the $\bar {k}^\times$-equivariant family, parameterized by $\hbar \in \bar {k}$, of additive maps of free $\bar {k}$-modules

\[ \begin{matrix} \bar{k}\otimes\mathfrak{t}_{k} & \to & \bar{k}^{(1)}\otimes\mathfrak{t}_{k}\\ \displaystyle\sum_ix_i\otimes v_i & \mapsto & \displaystyle\sum_i(x_i^p-\hbar^{p-1}x_i)\otimes v_i \end{matrix} \]

for a basis $\{v_i\}$ of $\mathfrak {t}_{\mathbb {F}_{\rm p}}$. This family interpolates between the usual Artin–Schreier map for $\hbar =1$ and the Frobenius map for $\hbar =0$.

Proof. (i) This is a standard fact which boils down to the calculation

\[ H^*(\mathbb{C}\mathbb{P}^{\infty}, \mathbb{Z}) = \mathbb{Z}[[\mathbb{C}\mathbb{P}^1]^*], \]

the free algebra on the degree $2$ cochain $v^*=[\mathbb {C}\mathbb {P}^1]^*$ dual to the degree $2$ chain represented by the embedded $\mathbb {C}\mathbb {P}^1$. An isomorphism $T= (\mathbb {C}^*)^d$ gives an isomorphism

\[ H^*(BT, R) \cong R[v_1^*,\ldots, v_d^*] \]

for any commutative ring $R$, and we will fix one for ease of exposition.

(ii) This is just saying that the map of (2.2), when applied to $X = BT$, sends each generator $v_i^*$ to $(v_i^*)^p-\hbar ^{p-1}v_i^*$. Since we have not actually defined the map (2.2) yet, we postpone the proof until § 2.8. However, notice that since $v_i^*$ has degree $2$ the expected image

\[ (v_i^*)^p-\hbar^{p-1}v_i^* \]

is exactly the generating function of the Steenrod powers of $v_i^*$ (with placeholder variable $\hbar$), up to some sign in the coefficients. Therefore the result will follow from a careful study of Steenrod's normalization factors (or rather, what happens if we don't use them!).

2.2 Steenrod's construction

Recall that $p$ is an odd prime, $\mu _p$ is the group of complex $p$th roots of unity, $R$ is a commutative ring, $k$ is a field of characteristic $p$ and $X$ is a topological space. We denote by $C^+(X,R)$ (respectively, $D^+(X,R)$) the category of bounded-below cochain complexes of sheaves of $R$-modules on $X$ (respectively, the corresponding bounded-below derived category). Recall that the latter is obtained from the former by inverting quasi-isomorphisms (this is important to us, and does not hold in the unbounded case). If $Y$ is a topological space with an action of $\mu _p$, we denote by $C^+_{\mu _p}(Y,R)$, $D^+_{\mu _p}(Y,R)$ the corresponding $\mu _p$-equivariant categories. Since $\mu _p$ is a finite group, these are the same as the (bounded-below) cochain, derived categories of $\mu _p$-equivariant sheaves of $R$-modules on $Y$.

Consider the functor of $p$th external tensor power

\[ C^+(X,R)\xrightarrow{\boxtimes p} C^+(X^{\mu_p},R). \]

It sends quasi-isomorphisms to quasi-isomorphisms, and so descends to a (non-triangulated) functor

\[ D^+(X,R)\xrightarrow{\boxtimes p} D^+(X^{\mu_p},R) \]

by the universal property of derived categories. Notice that the cochain-level functor factors as

\[ \boxtimes p:C^+(X,R)\xrightarrow{St_C}C^+_{\mu_p}(X^{\mu_p},R) \xrightarrow{} C^+(X^{\mu_p},R). \]

To make this explicit, we first choose an isomorphism

\[ \mu_p\cong \mathbb{Z}/p\cong\{1,\ldots,p\}=:[p]; \]

the result will be independent of this choice. Write $\sigma$ for the generator of $\mu _p$ corresponding to $1$ under the isomorphism. Then, for a complex $A^\bullet$, we give the complex $(A^\bullet )^{\boxtimes p}$ with degree $n$ term

\[ \big((A^\bullet)^{\boxtimes p}\big)^n=\bigoplus_{i_1+\cdots+ i_p=n}A^{i_1}\boxtimes\cdots\boxtimes A^{i_p} \]

the $\mu _p$-equivariant structure by letting the generator $\sigma$ act in degree $n$ by the direct sum of the canonical isomorphisms of sheaves

\[ A^{i_1}\boxtimes\cdots\boxtimes A^{i_p}\cong \sigma^*(A^{i_2}\boxtimes\cdots\boxtimes A^{i_p}\boxtimes A^{i_1}) \]

each twisted by the sign $(-1)^{i_1(n-i_1)}$. The sign twist is the natural (Koszul) choice which makes the action of $\mu _p$ commute with the differential. Moreover, given a chain map $f:A^\bullet \to B^\bullet$, $f^{\boxtimes p}$ is automatically a $\mu _p$-equivariant chain map. Since the functor $C^b_{\mu _p}(X^{\mu _p},R)\to C^b(X^{\mu _p},R)$ reflects quasi-isomorphisms, it follows immediately that $St_C$ descends to a functor $St_D$ as below:

\[ \boxtimes p:D^+(X,R)\xrightarrow{St_D}D^+_{\mu_p}(X^{\mu_p},R) \xrightarrow{} D^+(X^{\mu_p},R). \]

Writing $\Sigma$ for the suspension functor, we have $St_D\Sigma \cong \Sigma ^pSt_D$. Also, $St_D$ is not triangulated, nor additive, nor even linear. To control the failure of linearity, and for future reference, we now introduce the following special element of $R[\mu _p]$.

Definition 2.3 The norm element $N\in R[\mu _p]$ is given by

\[ N = \sum_{x\in\mu_p}x. \]

Observe that, given two objects $A, B$ of $D^+_{\mu _p}(X, R)$, we have an action of $\mu _p$ on

\[ {\rm Hom}_{D^+(X, R)}(A, B) \]

where by definition, $x\in \mu _p$ sends the non-equivariant morphism $f$ to $xfx^{-1}$. The following two propositions control the failure of linearity.

Proposition 2.4 Suppose given two parallel morphisms $f,g:A^\bullet \to B^\bullet$ in $D^+(X,R)$. Then the morphism

\[ St_D(f+g)-St_D(f)-St_D(g):St_D(A^\bullet)\to St_D(B^\bullet) \]

is an induced map. That is, there exists some non-equivariant map

\[ h:(A^\bullet)^{\boxtimes\,\mu_p}\to (B^\bullet)^{\boxtimes\,\mu_p} \]

such that the equivariant map

\[ N(h)=\sum_{x\in\mu_p}xhx^{{-}1}: St_D(A^\bullet)\to St_D(B^\bullet) \]

is equal to $St_D(f+g)-St_D(f)-St_D(g)$.

Proof. The maps $f, g, h$ appearing in the statement are morphisms of derived categories. Let us right away replace $A^\bullet ,B^\bullet$ by quasi-isomorphic complexes so that $f,g$ may be realized as genuine maps of complexes. Let $\underline {f},\underline {g}$ denote the constant functions $\mu _p\to \{f,g\}$ with respective values $f,g$. Then $\mu _p$ acts freely on $\{f,g\}^{\mu _p}-\{\underline {f},\underline {g}\}$; choose a set $\{h_1,\ldots ,h_d\}$ of orbit representatives $(d=(2^p-2)/p)$. Then each $h_i$ determines a non-equivariant map $(A^\bullet )^{\boxtimes \,\mu _p}\to (B^\bullet )^{\boxtimes \,\mu _p}$, hence so does their sum $h$. Then we have

\[ (f+g)^{\boxtimes\,\mu_p}-f^{\boxtimes\,\mu_p}-g^{\boxtimes\,\mu_p}=\sum_{x\in\mu_p} xhx^{{-}1} \]

where, by definition, $xhx^{-1}$ is the composition

\[ xhx^{{-}1} : (A^\bullet)^{\boxtimes\,\mu_p}\cong x^*(A^\bullet)^{\boxtimes\,\mu_p} \xrightarrow{x^*(h)} x^*(B^\bullet)^{\boxtimes\,\mu_p}\cong (B^\bullet)^{\boxtimes\,\mu_p} \]

where the two isomorphisms are given by the equivariant structures.

Let $Fr: R\to R$ denote the $p$th-power morphism of commutative monoids (using the multiplication on $R$ and forgetting the addition).

Proposition 2.5 $St_D$ is Frobenius-multiplicative with respect to the action of the multiplicative monoid $R$ on hom-sets. That is, $St_D$ determines a functor

\[ St_D:\operatorname{Ind}_R^RD^+(X,R)\to D^+_{\mu_p}(X^{\mu_p},R) \]

which respects multiplication by $R$. Here the category on the left is obtained from $D^b(X,R)$ by regarding each hom-set as a set with an action of the multiplicative monoid $R$ and inducing along the $p$th-power map of monoids $R\to R$.

Note that $\operatorname {Ind}_R^RD^b(X,R)$ is not an additive category in general. However, suppose that $R=k$ and $k$ is perfect. In that case, the Frobenius map of monoids is actually a map of rings $F$, and is, moreover, bijective. Write $M:k\operatorname {-mod}\to k\operatorname {-set}$ for the forgetful functor, where $k\operatorname {-set}$ denotes the category of sets with action of the multiplicative monoid $k$. We have the following lemma.

Lemma 2.6 Suppose that $k$ is a perfect field of characteristic $p$. Then we have

\[ M\circ F^*\cong \operatorname{Ind}_k^k\circ M. \]

It follows that if $k$ is a perfect field of characteristic $p$, we have produced a $k$-multiplicative functor

\[ St_D:F^*D^+(X,k)\to D^+_{\mu_p}(X^{\mu_p},k). \]

Since the source $F^*D^+(X,k)$ is triangulated, we find this statement somewhat nicer than the version for general $R$. However, the functor $St_D$ itself is still not linear, triangulated etc.

2.3 Localization

The category $D^+_{\mu _p}(X^{\mu _p},R)$ is enriched, in a triangulated sense, over $H^*_{\mu _p}(X^{\mu _p},R)$. That is, the monoidal structure of $D^+_{\mu _p}(X^{\mu _p},R)$ gives maps of $R$-modules

\[ H^n_{\mu_p}(X^{\mu_p},R)\cong \operatorname{Hom}_{D^+_{\mu_p}(X^{\mu_p},R)}(R,\Sigma^nR)\to \operatorname{Hom}_{D^+_{\mu_p}(X^{\mu_p},R)}({\rm id},\Sigma^n) \]

for each $n\geq 0$, whose sum is a map of algebras. In particular, $D^+_{\mu _p}(X^{\mu _p},R)$ is enriched in the same sense over

\[ H^*_{\mu_p}(*,R)\cong H^*({\rm B}\mu_{\rm p},R)\cong R\otimes_{\mathbb{Z}}^L (\mathbb{Z}[\hbar]/p\hbar). \]

Here $\hbar$ is the first Chern class of the tautological line bundle on ${\rm B}\mu _{\rm p}$ corresponding to the embedding $\mu _p\to \mathbb {C}^*$. In particular, this super-commutative ring receives a map from $R\otimes _{\mathbb {Z}}^L\mathbb {Z}[\hbar ] = R[\hbar ]$ so that $D^+_{\mu _p}(X^{\mu _p},R)$ is enriched over $R[\hbar ]$. Thus we may consider the 2-periodic $R$-linear triangulated category $D^+_{\mu _p}(X^{\mu _p},R)[\hbar ^{-1}]$, which is enriched over

\[ R\otimes_{\mathbb{Z}}^L (\mathbb{Z}[\hbar]/p\hbar)[\hbar^{{-}1}]\cong R\otimes_{\mathbb{Z}}^L \mathbb{F}_{\rm p}[\hbar^{\pm1}]. \]

The degree $0$ component of this ring is $R/p$, and the natural map from $R$ to here is the modular reduction map. In particular, there is a Frobenius map of rings

\[ F: R\to R\otimes_{\mathbb{Z}}^L \mathbb{F}_{\rm p}[\hbar^{\pm1}] \]

In this way, it makes sense to ask whether a functor from a triangulated category enriched over $R$ to one enriched over $R\otimes _{\mathbb {Z}}^L \mathbb {F}_{\rm p}[\hbar ^{\pm 1}]$ is Frobenius-linear.

Proposition 2.7 The composition

\[ St_D':D^+(X,R)\xrightarrow{St_D}D^+_{\mu_p}(X^{\mu_p},R)\to D^+_{\mu_p}(X^{\mu_p},R)[\hbar^{{-}1}] \]

is exact, Frobenius-linear and preserves direct sums.

Proof. First, we prove that $St_D'$ preserves direct sums. Let $A^\bullet$, $B^\bullet$ be complexes in $D^+(X,R)$. We argue as in the proof of Proposition 2.4 that we have

\[ St_D(A^\bullet{\oplus} B^\bullet)\cong St_D(A^\bullet)\oplus St_D(B^\bullet)\oplus \operatorname{Ind}_1^{\mu_p}C^\bullet \]

for some complex $C^\bullet$ in $D^b(X^{\mu _p},R)$. Here $\operatorname {Ind}_1^{\mu _p}$ is the averaging functor

\[ \operatorname{Ind}_1^{\mu_p}: D^+(X^{\mu_p},R)\to D^+_{\mu_p}(X^{\mu_p},R) \]

bi-adjoint to the restriction functor $\operatorname {Res}_1^{\mu _p}$. Therefore, it suffices to prove that the composition

\[ D^+(X^{\mu_p},R)\xrightarrow{\operatorname{Ind}_1^{\mu_p}} D^+_{\mu_p}(X^{\mu_p},R)\to D^+_{\mu_p}(X^{\mu_p},R)[\hbar^{{-}1}] \]

is isomorphic to $0$. This follows by adjunction from the fact that $Res_1^{\mu _p}(\hbar )=0$.

Next, we prove Frobenius-linearity. By Proposition 2.5, it suffices to prove that $St_D'$ respects addition of parallel morphisms. By Proposition 2.4, it is enough to see that the image of an induced morphism in $D^+_{\mu _p}(X^{\mu _p},R)$ in the localized category $D^+_{\mu _p}(X^{\mu _p},R)[\hbar ^{-1}]$ is $0$. This we have shown in the previous paragraph.

Finally, we prove exactness. First we must specify an exact structure, that is, an isomorphism $e:\Sigma St_D'\cong St_D'\Sigma$, which makes the image under $St_D'$ of any triangle a triangle. Note that since $p$ is odd, there is a morphism in $D^+_{\mu _p}(X^{\mu _p},R)$ of functors $\hbar ^{(p-1)/2}:\Sigma \xrightarrow {}\Sigma ^p$ which becomes an isomorphism when $\hbar$ is inverted. Already on the level of complexes we have a canonical isomorphism $\Sigma ^pSt_C\cong St_C\Sigma$. The exact structure is taken to be the composition

\[ e:\Sigma St_D'\xrightarrow{((p-1)/2)!\hbar^{(p-1)/2}}\Sigma ^pSt_D'\cong St_D'\Sigma. \]

This is indeed an isomorphism since the localized category is enriched over $\mathbb {F}_{\rm p}$. The reason for the factor $((p-1)/2)!$ will be explained shortly. Thus, given a triangle

\[ B^\bullet\xrightarrow{g}C^\bullet\xrightarrow{h}\Sigma A^\bullet\xrightarrow{-\Sigma f}\Sigma B^\bullet \]

in $D^+(X,R)$, we have a triangle

in $D^+_{\mu _p}(X^{\mu _p},R)[\hbar ^{-1}]$. We must show that this is exact whenever the original triangle is. Since any exact triangle in a derived category is isomorphic to a semi-split one, we may assume that $f$ is a chain map and $C^\bullet$ is the usual mapping cone satisfying

\[ C^n = A^{n+1}\oplus B^n \]

with differential

\[ \begin{pmatrix} -d & 0\\ f & d \end{pmatrix}. \]

So we have

\[ St_C(C^\bullet) = \bigg(\bigoplus_{i_1+\cdots+ i_p={\bullet}}(A^{i_1+1}\oplus B^{i_1})\boxtimes\cdots\boxtimes(A^{i_p+1}\oplus B^{i_p}) \bigg)^\bullet \]

with some differential (whose precise form is not important). This complex has a $(p+1)$-step equivariant increasing filtration

\[ St_C(B^\bullet) = F_0St_C(C^\bullet)\subset\cdots\subset F_pSt_C(C^\bullet) = St_C(C^\bullet) \]

where $F_iSt_C(C^\bullet )$ consists of the subcomplex of $St_C(C^\bullet )$ in which at most $i$ summands $A^?$ are taken in the expansion of the external tensor product. The inclusion of the zeroth piece of this filtration is equal to $St_C(g)$, while the quotient map

\[ St_C(C^\bullet)\twoheadrightarrow St_C(C^\bullet)/F_{p-1}St_C(C^\bullet) \cong St_C(\Sigma A^\bullet) \]

is equal to $St_C(h)$. Furthermore, arguing as in Proposition 2.4, we see that $F_iSt_C(C^\bullet )/F_{i-1}St_C(C^\bullet )$ is an induced complex for each $1\leq i\leq p-1$. Therefore, the map

\[ St_C(C^\bullet)/St_C(B^\bullet)\twoheadrightarrow St_C(\Sigma A^\bullet) \]

becomes an isomorphism in $D^b_{\mu _p}(X^{\mu _p},R)[\hbar ^{-1}]$. Consider now the commutative diagram

of equivariant chain complexes. Here $\alpha ,\beta$ are the usual chain maps which make the image of the top row in $D^+_{\mu _p}(X,R)$ an exact triangle, $\gamma$ is the quotient map, whose image in $D^+_{\mu _p}(X,R)[\hbar ^{-1}]$ is an isomorphism, and $St_C\Sigma$ has been identified with $\Sigma ^pSt_C$. Comparing with the definition of the triangle obtained by applying $St_D'$ to $B^\bullet \xrightarrow {g}C^\bullet \xrightarrow {h}\Sigma A^\bullet \xrightarrow {-\Sigma f}\Sigma B^\bullet$, we see that it is enough to prove that the diagram

\[ \begin{array}{ccl} St_D(C^\bullet)/St_D(B^\bullet) & \xrightarrow{~~~~~~~~~~~\beta~~~~~~~~~~~} & \Sigma St_D(B^\bullet)\hfill\\ \Big\downarrow{\gamma} & & \Big\downarrow{{((p-1)/2)!\hbar^{(p-1)/2}}} \\ \Sigma^pSt_D(A^\bullet) & \xrightarrow{~~~~~~-\Sigma^pSt_D(f)~~~~~~} & \Sigma^pSt_D(B^\bullet)\hfill\\ \end{array} \]

commutes in $D^+_{\mu _p}(X,R)[\hbar ^{-1}]$. Here we have written $\beta ,\gamma$ for their own images in $D^+_{\mu _p}(X,R)$. Let $D^\bullet$ be the cocone of $A^\bullet \xrightarrow {{\rm id}}A^\bullet$, so that $f$ induces a map $D^\bullet \to C^\bullet$. We have a commutative diagram

\[ \begin{matrix} St_C(D^\bullet)/St_C(A^\bullet) & \xrightarrow{\epsilon} & \Sigma St_C(A^\bullet)\\ \Big\downarrow{\delta} & & \Big\downarrow{\Sigma St_C(f)} \\ St_C(C^\bullet)/St_C(B^\bullet) & \xrightarrow{\beta} & \Sigma St_C(B^\bullet)\\ \end{matrix} \]

where $\epsilon$ is the standard boundary map, and the vertical arrows are induced by $f$ (no signs necessary). Now $\gamma \delta$ becomes an isomorphism in $D^b_{\mu _p}(X^{\mu _p},R)$ for the same reason that $\gamma$ does; therefore so does $\delta$. So it is enough to show that the composition of these two diagrams is commutative. By functoriality of $\hbar$, the resulting composition equals

\[ \begin{array}{ccl} St_D(D^\bullet)/St_D(A^\bullet) & \xrightarrow{\epsilon} & \Sigma St_D(A^\bullet)\hfill\\ \Big\downarrow{\gamma\delta} & & \Big\downarrow{\Sigma^p St_D(f)\circ{((p-1)/2)!\hbar^{(p-1)/2}}} \\ \Sigma^pSt_D(A^\bullet) & \xrightarrow{~~~~~~-\Sigma^pSt_D(f)~~~~~~} & \Sigma^pSt_D(B^\bullet)\hfill\\ \end{array} \]

Let

\[ \mathfrak{p} = (R\to R[\mu_p]\xrightarrow{\sigma-1}R[\mu_p]\xrightarrow{N}\cdots\xrightarrow{\sigma-1}R[\mu_p]) \]

be the equivariant resolution of the trivial $R[\mu _p]$-module supported in degrees $(1-p),\ldots ,0$. Here $N$ is the norm element as defined in Definition 2.3. We have the standard chain maps $\mathfrak {p}\to R$, which is an isomorphism in the equivariant derived category, and $\mathfrak {p}\to \Sigma ^{p-1}R$, which equals $\hbar ^{(p-1)/2}$ by definition. Now $\epsilon$ is a chain map, which is an isomorphism in the equivariant derived category but not in the equivariant complex category. However, there is a chain map

\[ \mathfrak{p}\otimes\Sigma St_C(A^\bullet)\xrightarrow{\zeta} St_C(D^\bullet)/St_C(A^\bullet) \]

such that $\epsilon \zeta$ is induced by the standard chain map $\mathfrak {p}\to R$ (i.e. counit in degree $0$). To see this, it is enough to do the case $A^\bullet =R$ and then tensor on the right with $St_C(A^\bullet )$. In that case, we are looking for an equivariant chain map from the complex

\[ R\to R[\mu_p]\xrightarrow{\sigma-1}R[\mu_p]\xrightarrow{N}\cdots\xrightarrow{\sigma-1}R[\mu_p] \]

supported in degrees $-p,\ldots ,-1$ to the complex $E^\bullet$ satisfying

\[ E^i=\bigoplus_{\substack{S\subset [p]\\ |S|={-}i}}R_S \]

where $R_S$ is a copy of $R$, and with differential sending $R_S$ to $\bigoplus _{s\in S}R_{S-\{s\}}$ by $(1,-1,1,\ldots )$. Let us write $1_S$ for the canonical generator of $R_S$. One example of such a map is the map which sends the element $1$ in the degree $-(2i+1)$ copy of $R[\mu _p]$ to the term

\[{-}i!\sum_{\substack{T\subset \{2,\ldots,p\}\\ |T|=2i\\ \text{even block lengths }}} 1_{\{1\}\cup T} \]

and sends the element $1$ in the degree $-(2i+2)$ copy of $R[\mu _p]$ to the term

\[{-}i!\sum_{\substack{T\subset \{3,\ldots,p\}\\ |T|=2i\\ \text{even block lengths}}}1_{\{1\}\cup\{2\}\cup T}. \]

The sign is chosen so that $\epsilon \zeta$ is induced by the standard chain map $\mathfrak {p}\to R$. We compute

\[ \gamma\delta\zeta ={-}((p-1)/2)!\hbar^{(p-1)/2}. \]

Therefore, the two paths $St_D(D^\bullet )/St_D(A^\bullet )\to \Sigma ^pSt_D(B^\bullet )$ in this composed diagram are equalized by $\zeta$. Since $\zeta$ is an isomorphism in $D^+_{\mu _p}(X^{\mu _p},R)[\hbar ^{-1}]$, they coincide in the localized category as required.

Corollary 2.8 Suppose $R=k$ is a field of characteristic $p$. Then we have a triangulated $k$-linear functor

\[ St_D':F^*D^+(X,k)\to D^+_{\mu_p}(X^{\mu_p},k)[\hbar^{{-}1}]. \]

2.4 Six functors

We shall henceforth restrict ourselves to the geometric context of [Reference Bernstein and LuntsBL06]. Thus from now on every topological space $X$ will be the Borel quotient $EG\frac {\times }{G}Y$ of a complex algebraic variety $Y$ by the action of some affine algebraic group $G$, every coefficient ring $R$ will be a Noetherian ring of finite homological dimension, and we will restrict attention to the full subcategory

\[ D^+_G(Y, R) \subset D^+(X, R) \]

spanned by those objects which descend from $Y$.

For the rest of this paper, we will cut down our scope even further (perhaps unnecessarily) to the constructible equivariant derived category

\[ D^b_C{(c, G)}(Y, R). \]

This is the full subcategory of $D^+_G(Y, R)$ spanned by those complexes $\mathcal {F}$ which have only finitely many non-zero cohomology sheaves, and all of them are algebraically constructible, that is, locally constant along every stratum of some algebraic stratification of $Y$ (depending on $\mathcal {F}$).

The constructions of the previous section preserve constructibility, so we have a Steenrod construction

\[ St_D:D^b_{c,G}(Y,R)\to D^b_{c,G^{\mu_p}\rtimes\,\mu_p}(Y^{\mu_p},R). \]

Recall that we have the deep ‘six-functor formalism’ for constructible derived categories; see Bernstein and Lunts [Reference Bernstein and LuntsBL06]. We assume that the reader is familiar with this material, but remind him/her of the standard notation: for a $G$-equivariant algebraic map $f:Y\to Y'$, we have the adjoint pairs of exact functors

\begin{gather*} {f^*\negmedspace: D^b_{c,G}(Y',R)\rightleftarrows D^b_{c,G}(Y,R):\negmedspace f_*},\\ {f_!\negmedspace: D^b_{c,G}(Y,R)\rightleftarrows D^b_{c,G}(Y',R):\negmedspace f^!} \end{gather*}

and also a pair of bi-exact bifunctors

\begin{gather*} (-)\otimes(-):D^b_{c,G}(Y,R)\times D^b_{c,G}(Y,R)\to D^b_{c,G}(Y,R),\\ \mathcal{H}om(-,-):D^b_{c,G}(Y,R)^{op}\times D^b_{c,G}(Y,R)\to D^b_{c,G}(Y,R) \end{gather*}

related by a tensor-hom adjunction. There is also a Verdier duality functor $\mathbb {D}$, and an exceptional tensor product $\otimes ^!$, which can be written in terms of the other functors, as can the external tensor product $\boxtimes$. We call the collection of all of these functors the six plus functors. Notice that ${G^{\mu _p}\rtimes\,\mu _p}$ is also an affine algebraic group, so the six-functor formalism exists for the target category of $St_D$. Also if $f:Y\to Y'$ is $G$-equivariant then $f^{\mu _p}:Y^{\mu _p}\to (Y')^{\mu _p}$ is $G^{\mu _p}\rtimes\, \mu _p$-equivariant. The following fact is essentially a consequence of the same fact for $^{\boxtimes p}$:

Proposition 2.9 Steenrod's construction is compatible with the six-functor formalism. That is, we have canonical isomorphisms

\begin{align*} (f^{\mu_p})^*St_D & \cong St_Df^* ,\\ (f^{\mu_p})_*St_D & \cong St_Df_* ,\\ (f^{\mu_p})_!St_D & \cong St_Df_! ,\\ (f^{\mu_p})^!St_D & \cong St_Df^! ,\\ St_D(-)\otimes St_D(-) & \cong St_D(-{\otimes}{-}) ,\\ St_D(-)\otimes^! St_D(-) & \cong St_D(-{\otimes}^! -) ,\\ St_D(-)\boxtimes St_D(-) & \cong St_D(-\boxtimes -) ,\\ \mathcal{H}om(St_D(-),St_D(-)) & \cong St_D\mathcal{H}om(-,-) \end{align*}

commuting with any and all adjunction morphisms of the six-functor formalism.

We have the same compatibilities with functors $St_D'$.

Proof. That the claim for $St_D'$ follows from the claim for $St_D$, is a direct consequence of the $H^*_{\mu _p}(*)$-linearity of the six functors in the $\mu _p$-equivariant context. So let us focus on $St_D$. It is known (cf. [Reference Bernstein and LuntsBL06]) that the six functors intertwine with external tensor product, so it is just a matter of showing that the $\mu _p$-equivariant structures are carried along. This is easy for $f^*$: it is defined on the cochain level, and strictly intertwines with $St_C$ on that level (no homotopy involved). It follows by adjointness that we have a canonical map

\[ St_Df_* \to (f^{\mu_p})_*St_D \]

which is an isomorphism upon forgetting equivariance, hence itself an isomorphism.

We may argue similarly for $f_!$, $f^!$. Indeed, the thrust of the argument for $f^*$ was to lift both $f^*$ and $St_D$ to the concrete model of bounded-below cochain complexes and observe strict intertwining there. There is an alternative concrete model which works just as well for the purposes of defining $St_D$, namely the category of bounded-below cochain complexes of soft sheaves. But $f_!$ is defined on the level of the soft model, and the intertwining holds strictly there. We deduce the desired claim for $f^!$ in the same way we deduced it for $f_*$ from $f^*$.

The canonical isomorphism

\[ St_D(-)\boxtimes St_D(-) \cong St_D(-\boxtimes -) \]

is a tautological consequence of the cochain-level construction of $St_D$, and intertwining for the other two tensors follows by composing with $*$- and $!$- restriction along the diagonal. Finally, the canonical isomorphism

\[ \mathcal{H}om(St_D(-),St_D(-)) \cong St_D\mathcal{H}om(-,-) \]

can be deduced using the hom-tensor adjunction.

Corollary 2.11 Let $R$ denote the constant sheaf and $\omega _R$ denote the dualizing complex (see [Reference Chriss and GinzburgCG97] for details and evidence of its ubiquity in geometric representation theory). We have canonical isomorphisms

\[ St_D R \cong R \]

and

\[ St_D \omega_R \cong \omega_R. \]

2.5 Tate's construction

Note that the object $St_D(\Sigma ^nR)$ is canonically isomorphic to $\Sigma ^{pn}R$ with its trivial $\mu _p$-equivariant structure. Here $R$ is the constant sheaf. Since a degree $n$ cohomology class is just a morphism $R\to \Sigma ^nR$ in $D^b(X,R)$, we thus obtain a Frobenius-multiplicative map of multiplicative $R$-sets:

\[ St^H:H^n(X,R)\to H^{pn}_{\mu_p}(X^{\mu_p},R). \]

This map is not Frobenius-linear, but as in Proposition 2.4, its deviation from additivity is by a class induced from $H^{pn}(X^{\mu _p},R)$. To see how these maps interact with multiplication, we make the following definition.

Definition 2.12 Let $(\mathcal {C},*,\mathbb{1},e,a,s)$ be a symmetric monoidal abelian category enriched over some commutative ring $R$, that is, $\mathcal {C}$ is an abelian category, $*$ is a bi-exact $R$-linear functor $\mathcal {C}\times \mathcal {C}\to \mathcal {C}$, $\mathbb {1}$ is an object of $\mathcal {C}$, $e$ is a pair of equivalences $\mathbb {1}*{\rm id}\cong {\rm id}\cong {\rm id}*\mathbb {1}$, $a$ is an associativity constraint for $*$ and $s$ is a commutativity constraint for $*$, satisfying natural compatibilities. Let $A$ be an object of $\mathcal {C}$. Then $s$ determines an action of $\mu _p$ on $A^{*\,\mu _p}$, and we define

\[ A^{(1)} := \hat{H}_{\mu_p}^0(A^{\mu_p}) := \ker_{A^{*\,\mu_p}}(1-\sigma)/\operatorname{im}_{A^{*\,\mu_p}}(N). \]

Here $N$ is the norm element defined in Definition 2.3. This is the so-called Tate construction; see [Reference Cartan and EilenbergCE99] for more details. For a morphism $f:A\to B$ the morphism $f^{*\,\mu _p}:A^{*\,\mu _p}\to B^{*\,\mu _p}$ is $\mu _p$-equivariant and so induces a morphism $A^{(1)}\to B^{(1)}$, so that $(-)^{(1)}$ becomes a functor.

Proof. We essentially rehash the proof of Propositions 2.4 and 2.7. First we show that $(-)^{(1)}$ is linear over $\mathbb {Z}$. Suppose $f,g:A\to B$ are two parallel morphisms in $\mathcal {C}$. Let $\underline {f},\underline {g}$ denote the constant functions $\mu _p\to \{f,g\}$ with respective values $f,g$. Then $\mu _p$ acts freely on $\{f,g\}^{\mu _p}-\{\underline {f},\underline {g}\}$; choose a set $\{h_1,\ldots ,h_n\}$ of orbit representatives ($n=(2^p-2)/p$). Then each $h_i$ determines a non-equivariant map $A^{*\,\mu _p}\to B^{*\,\mu _p}$, and we have

\[ (f+g)^{*\,\mu_p}-f^{*\,\mu_p}-g^{*\,\mu_p}=\sum_{x\in\mu_p}\sum_{i=1}^n xh_ix^{{-}1}. \]

Restricting to $\ker _{A^{*\,\mu _p}}(1-\sigma )$, this becomes

\[ ((f+g)^{*\,\mu_p}-f^{*\,\mu_p}-g^{*\,\mu_p})_{\ker_{A^{*\,\mu_p}}(1-\sigma)}=\sum_{x\in\mu_p}\sum_{i=1}^n xh_i=N\sum_{i=1}^n h_i \]

which factors through $\operatorname {im}_{B^{*\,\mu _p}}(N)$ as required.

Next we show that $(-)^{(1)}$ preserves direct sums. Let $\underline {A},\underline {B}$ denote the constant functions $\mu _p\to \{A,B\}$ with respective values $A,B$. Then $\mu _p$ acts freely on $\{A,B\}^{\mu _p}-\{\underline {A},\underline {B}\}$; choose a set $\{C_1,\ldots ,C_n\}$ of orbit representatives. Then each $C_i$ determines an object of $\mathcal {C}$, and as a $\mu _p$-module in $\mathcal {C}$ we have

\[ (A\oplus B)^{*\,\mu_p}\cong A^{*\,\mu_p}\oplus B^{*\,\mu_p}\oplus \bigoplus_{i=1}^nC_i[\mu_p]. \]

The result then follows from the fact that $\hat {H}^0_{\mu _p}(k[\mu _p])=0$ in $R\operatorname {-mod}$.

Let $\operatorname {SVect}_k$ denote the symmetric monoidal category of $\mathbb {Z}/2$-super-graded $k$-vector spaces.

Now suppose that $A,B$ are objects of $\mathcal {C}$. We have a $\mu _p$-equivariant isomorphism $(A*B)^{*\,\mu _p}\cong (A^{*\,\mu _p}*B^{*\,\mu _p})$. We have also the natural inclusions

\begin{gather*} \ker_{A^{*\,\mu_p}}(1-\sigma)*\ker_{B^{*\,\mu_p}}(1-\sigma)\to \ker_{A^{*\,\mu_p}*B^{*\,\mu_p}}(1-\sigma),\\ \operatorname{im}_{A^{*\,\mu_p}}(N)*\ker_{B^{*\,\mu_p}}(1-\sigma)\to \operatorname{im}_{A^{*\,\mu_p}*B^{*\,\mu_p}}(N),\\ \ker_{A^{*\,\mu_p}}(1-\sigma)*\operatorname{im}_{A^{*\,\mu_p}}(N)\to \operatorname{im}_{A^{*\,\mu_p}*B^{*\,\mu_p}}(N) \end{gather*}

which induce a map $A^{(1)}*B^{(1)}\to (A*B)^{(1)}$. Suppose that $(A,\mathbb {1}_A,m_A)$ is a unital ring in $\mathcal {C}$. Then $A^{(1)}$ still has a multiplication

\[ m_{A^{(1)}}:A^{(1)}*A^{(1)}\to (A*A)^{(1)}\xrightarrow{m_A^{(1)}}A^{(1)}. \]

Also, there is a canonical isomorphism $\ker _{\mathbb {1}^{*\,\mu _p}}(1-\sigma )\cong \mathbb {1}$, hence a canonical surjection $\mathbb {1}\to \mathbb {1}^{(1)}$ which determines a map

\[ \mathbb{1}_{A^{(1)}}:\mathbb{1}\to \mathbb{1}^{(1)}\xrightarrow{\mathbb{1}_A^{(1)}}A^{(1)}. \]

One may check that this makes $A^{(1)}$ into a ring, and, moreover, that $A^{(1)}$ is associative or commutative if $A$ is. The following lemma explains how this looks in the main example.

Lemma 2.15 Suppose $A=\bigoplus _{i\in \mathbb {Z}/2}A_i$ is a unital ring in $\mathcal {C}=\operatorname {SVect}_k$. Then the ring structure on $A^{(1)}$ corresponds under the identification $A^{(1)}=k\otimes _FA$ to the ring structure with unit $1\otimes _F\mathbb {1}_A$ and multiplication

\[ m_{A^{(1)}}(r\otimes a,r'\otimes a')=({-}1)^{ij{\binom{p}{2}}}rr'\otimes m_A(a,a') \]

for $a\in A_i,a'\in A_j$ and $r,r'\in k$.

Proof. The isomorphism of $k\otimes _FA_{i}$ with $(A^{(1)})_{i}$ sends the element $r\otimes a$ to the class of ${r}.\underbrace {a\otimes \cdots \otimes a}_ {p\ {\rm times}}$. The natural map

\[ (A^{(1)})_{i}\otimes (A^{(1)})_{j}\cong \hat{H}^0_{\mu_p}((A_i)^{*\,\mu_p})\otimes \hat{H}^0_{\mu_p}((A_j)^{*\,\mu_p})\to \hat{H}^0_{\mu_p}((A_i\otimes A_j)^{*\,\mu_p}) \]

sends the class of $\underbrace {a\otimes \cdots \otimes a}_ {p\text { times}}\otimes \underbrace {a'\otimes \cdots \otimes a'}_ {p\text { times}}$ to the class of $(-1)^{ij{\binom {p}{2}}}\underbrace {(a\otimes a')\otimes \cdots \otimes (a\otimes a')}_ {p\text { times}}$, since it entails permuting the $x$th copy of $A_i$ with the $y$th copy of $A_j$ for every $p\geq x>y\geq 1$.

Arguing the same way, we have the following proposition.

Proposition 2.16 Let $A$ be a Hopf algebra in $\operatorname {SVect}_k$. Then $A^{(1)}$ is naturally a Hopf algebra in $\operatorname {SVect}_k$. It has multiplication and unit given as in Lemma 2.15, comultiplication given by

\[ \Delta_{A^{(1)}}(r\otimes a)=r\otimes ({-}1)^{{\binom{p}{2}}\deg\otimes\deg}\Delta_{A}(a), \]

counit given by $\epsilon _ {A^{(1)}}(r\otimes a) = r(\epsilon _A(a))^p$ and antipode given by $S_ {A^{(1)}}(r\otimes a) = r\otimes S_A(a)$. Moreover, the functor $(-)^{(1)}$ on $\operatorname {SVect}_k$ upgrades to functor

\[ (-)^{(1)}:A\operatorname{-comod}\to A^{(1)}\operatorname{-comod}. \]

For an $A$-comodule $M$, the $A^{(1)}$-comodule structure on $M^{(1)}$ is given by

\[ \Delta_{M^{(1)}}(r\otimes m) = r\otimes({-}1)^{{\binom{p}{2}}\deg\otimes\deg}\Delta_M(m). \]

Example 2.17 Suppose $\mathcal {O}$ is a commutative Hopf algebra in $\operatorname {SVect}_k$. Then the monoidal category $\mathcal {C}$ of $\mathcal {O}$-comodules is symmetric. Taking $p$th powers gives a (Frobenius) map of Hopf algebras $F_ {\mathcal {O}}:\mathcal {O}^{(1)}\to \mathcal {O}$. Then Tate's construction on $\mathcal {C}$ factors as

\[ \mathcal{O}\operatorname{-comod}\xrightarrow{(-)^{(1)}}\mathcal{O}^{(1)}\operatorname{-comod}\xrightarrow{F_{\mathcal{O}}^*}\mathcal{O}\operatorname{-comod}. \]

For instance, we could take $\mathcal {O}$ to be the ring of functions $\mathcal {O}(\mathbb {G}_m)$ on the multiplicative group $\mathbb {G}_m$ over $k$, concentrated in degree $0\in \mathbb {Z}/2$. Then $\mathcal {O}(\mathbb {G}_m)\operatorname {-comod}$ contains as a full subcategory over $\operatorname {SVect}_k$ the category of $\mathbb {Z}$-super-graded vector spaces, and Tate's construction there is isomorphic to the functor which applies $k\otimes _F(-)$ and multiplies degrees by $p$.

Recall we have the Frobenius-multiplicative maps of multiplicative $k$-sets

\[ St^H:H^n(X,k)\to H^{pn}_{\mu_p}(X^{\mu_p},k). \]

We view cohomology rings as commutative ring objects of $\mathbb {Z}$-super-graded vector spaces; in particular, we can apply functor $(-)^{(1)}$ to them. By Lemma 2.6, if $k$ is perfect then it gives a map of $\mathbb {Z}$-super-graded $k$-sets

\[ St_{\mathrm{ex}}:H^*(X,k)^{(1)}\to H^{*}_{\mu_p}(X^{\mu_p},k). \]

The following fact is immediate from the constructions.

2.6 Borel–Moore homology

We return to the setting of § 2.4. By Corollary 2.11, we have a canonical isomorphism

\[ St_D(\omega)\cong\omega. \]

Here the $\omega$ on the left-hand side denotes the $G$-equivariant dualizing complex on $Y$ with coefficients in $R$, while the $\omega$ on the left-hand side denotes the $G^{\mu _p}\rtimes\, \mu _p$-equivariant dualizing complex on $Y^{\mu _p}$ with coefficients in $R$. By definition, the $G$-equivariant Borel–Moore homology of $Y$ is

\[ H_n^{\mathrm{BM},G}(Y,R):=\operatorname{Hom}_{D^b_G(Y,R)}(R,\Sigma^n\omega). \]

See § 3.10 for more about this. Altogether $H_n^{\mathrm {BM},G}(Y,R)$ form a $\mathbb {Z}$-super-graded $H^*_G(Y,R)$-module; in particular, it is a module for $H^*_G(*,R)$. By functoriality we have the nonlinear maps

\[ St^{\mathrm{BM}}:H_n^{\mathrm{BM},G}(Y,R)\to H_{pn}^{\mathrm{BM},G^{\mu_p}\rtimes\,\mu_p}(Y^{\mu_p},R). \]

This is a map of $St^H$-monoids. Its discrepancy from additivity it averaged from $H_ {pn}^{\mathrm {BM},G^{\mu _p}}(Y^{\mu _p},R)$. If $R$ is a perfect field $k$ of characteristic $p$, we can say that we have a nonlinear graded map of $St_ {\mathrm {ex}}$-monoids:

\[ St^{\mathrm{BM}}_{\mathrm{ex}}:H_*^{\mathrm{BM},G}(Y,k)^{(1)}\to H_{*}^{\mathrm{BM},G^{\mu_p}\rtimes\,\mu_p}(Y^{\mu_p},k). \]

2.7 Steenrod operations

For simplicity let us assume $k$ to be perfect from now on. Let us compose $St_ {\mathrm {ex}}$ with the restriction map to the diagonal:

\[ St_{\mathrm{in}}:H^*(X,k)^{(1)}\xrightarrow{St_{\mathrm{ex}}} H^{*}_{\mu_p}(X^{\mu_p},k)\xrightarrow{\Delta^*} H^{*}_{\mu_p}(X,k)\cong H^*(X,k)[a,\hbar]. \]

This is again a map of multiplicative $k$-monoids. Tautologically we have $St_ {\mathrm {in}}(x)=x^p\text { mod }(a,\hbar )$. Also $St_ {\mathrm {in}}$ is compatible with pullback maps in cohomology in the natural way. Since induction commutes with restriction, the difference between $St_ {\mathrm {in}}(x+y)$ and $St_ {\mathrm {in}}(x)+St_ {\mathrm {in}}(y)$ is induced from a cohomology class $z\in H^\bullet (X;k)$. Since $\mu _p$ acts trivially on $X$, that means that it is equal to $pz=0$, so $St_ {\mathrm {in}}$ is linear. That is, we have a map of super-commutative $k$-algebras

\[ St_{\mathrm{in}}:H^*(X,k)^{(1)}\to H^*(X,k)[a,\hbar]. \]

Remark 2.20 The coefficients of $\hbar ^m,a\hbar ^m$ in $St_ {\mathrm {in}}$ are not the Steenrod operations. More precisely, they are the Steenrod operations only up to some non-zero scalars. Even more precisely, let $x\in H^n(X,k)$ and let $p=2q+1$. Consider

\[ ({-}1)^{qn(n-1)/2}(q!)^{{-}n}St_{\mathrm{in}}(x). \]

where $x$ is viewed as a degree $pn$ element of $H^*(X,k)^{(1)}$. The coefficient of $\hbar ^m$ in this expression vanishes unless $m=\frac {1}{2}(p-1)(n-2s)$ for some $s$ such that $2s\leq n$, in which case that coefficient is equal to $(-1)^sP^s(x)$ where $P^s$ is the $s$th Steenrod operation. Similarly, the coefficient of $a\hbar ^m$ in that expression vanishes unless $m=\frac {1}{2}(p-1)(n-2s)-1$ for some $s$ such that $2s\leq n$, in which case that coefficient is equal to $(-1)^{s+1}\beta P^s(x)$ where $\beta$ is the Bockstein operation. A careful comparison of $St_ {\mathrm {in}}$ with the sum of Steenrod operations as defined in [Reference Steenrod and EpsteinSE62] shows the constructions are indeed identical except that N. Steenrod deliberately inserted exactly these invertible ‘correction factors’. For him, as far as we can tell, the purpose was to ensure that

1. (i) $P^0={\rm id}$, and

2. (ii) the total Steenrod, $\sum _ {s=0}^{\infty }P^s$, is a ring endomorphism of $H^*(X)$.

On the one hand, from the perspective of the results of this paper, these two constraints are pure obfuscation; the latter denies the reality that the natural source of the algebra homomorphism is the Tate construction $H^*(X)^{(1)}$ rather than $H^*(X)$, while the former masks the resemblance (not a coincidence!) to the Artin–Schreier morphism. On the other hand, from the perspective of the author, it is rather fortunate as the core innovation of this work would surely have been discovered sooner otherwise.

2.8 Artin–Schreier

We indicate how the Artin–Schreier map comes naturally out of the above considerations. First note that if $n$ is even then the number

\[ ({-}1)^{qn(n-1)/2}(q!)^{{-}n} \]

boils down to $(-1)^{n/2}$. It is a standard fact that on a degree $2$ class $x$ we have $P^0(x)=x,P^1(x)=x^p$, and higher powers vanish. Therefore

\[ St_{\mathrm{in}}(x) = x^p - \hbar^{p-1}x + \hbar^{p-2}\beta(x). \]

Let $X=BT$ for some compact torus $T$. Since its cohomology is supported in even degrees, the Bockstein operator acts as zero and $St_ {\mathrm {in}}$, on the level of $k$-cohomology, is exactly the $\hbar$-Artin–Schreier map

\[ \mathcal{O}(\mathfrak{t}_{k})^{(1)}\xrightarrow{AS_\hbar} \mathcal{O}(\mathfrak{t}_{k})[\hbar] \subset \mathcal{O}(\mathfrak{t}_{k})[a,\hbar] \]

as defined in Fact 2.1.

Recall that if $G$ is a compact Lie group with maximal torus $T$, and $p$ is large enough with respect to the Weyl group of $G$, then the projection $BT\to BG$ induces an inclusion

\[ H^*(BG,k)\to H^*(BT,k) \]

which is identified with

\[ \mathcal{O}(\mathfrak{t}_{k}/{/}W)\to \mathcal{O}(\mathfrak{t}_{k}). \]

The $\hbar$-Artin–Schreier map induces a map on subspaces

\[ \mathcal{O}(\mathfrak{t}_{k}/{/}W)^{(1)}\xrightarrow{AS_\hbar} \mathcal{O}(\mathfrak{t}_{k}/{/}W)[\hbar] \]

which is also important in the theory of Frobenius-constant quantization. The point is that this map is also induced by $St_ {\mathrm {in}}$, since it is compatible with pullbacks.

It is entertaining to show more directly how $AS_\hbar$ arises, without relying on any outside facts about Steenrod operations. We can reduce to the rank $1$ case $T=S^1$. Let $b$ denote the degree $2$ generator (first Chern class of tautological line bundle) of $BS^1$; we need to show that $St_ {\mathrm {in}}(b)=b^p-\hbar ^{p-1}b$. Let $C_p\subset S^1$ denote the cyclic group of order $p$, considered as distinct from $\mu _p$. Consider the projection

\[ BC_p\to BS^1. \]

It induces an injective map

\[ k[b]\to k[s,b] \]

in cohomology, where $s$ is a degree $1$ generator. By functoriality it is enough to prove the equality when $b$ is regarded as a cohomology class of $BC_p$. Note that amongst degree $2p$ elements of $k[b,\hbar ]$, the desired element $b^p-\hbar ^{p-1}b$ is the unique one which gives $0$ when we set $b$ to any multiple in $\mathbb {F}_{\rm p}$ of $\hbar$, and gives $b^p$ when we set $\hbar =0$. The latter statement is automatic, so we have to check the former. So fix some $t\in \mathbb {F}_{\rm p}$. Having chosen an isomorphism $\mu _p\cong C_p$, $t$ determines a group homomorphism $\mu _p\to C_p$.

The constant sheaf $\Sigma ^nk$ of $BC_p$ is contained in the full subcategory

\[ D^b(k[C_p]\operatorname{-mod})=D_{C_p}(*)\subset D(BC_p). \]

Our coefficients are $k$, which we drop from the notation. It is easier for our purpose to work in $D^b(k[C_p]\operatorname {-mod})$. Compatible with the functor $St_D$ out of $D(BC_p)$ we have the functor

\[ St_D:D^b(k[C_p]\operatorname{-mod})\to D^b(k[\mu_p\ltimes(C_p)^{{\times} p}]\operatorname{-mod}). \]

This is then composed with the diagonal restriction

\[ D^b(k[\mu_p\ltimes(C_p)^{{\times} p}]\operatorname{-mod})\xrightarrow{\Delta^*} D^b(k[\mu_p\times C^p]\operatorname{-mod}). \]

By definition $St_ {\mathrm {in}}(b)$ is given by applying that composition to the morphism $k\xrightarrow {b}k[2]$, where $k$ is the trivial $C_p$-module. We want further to set $b=t\hbar$; this corresponds to restricting along the map

\[ \mu_p\xrightarrow{{\rm id}\times t}\mu_p\times C_p. \]

Write

\[ ({\rm id}\times t)^*:D^b(k[\mu_p\times C_p]\operatorname{-mod})\to D^b(k[ \mu_p]\operatorname{-mod}) \]

for the corresponding restriction map. We need to show that $({\rm id}\times t)^*\circ \Delta ^*\circ St(b)=0$. But actually there is an isomorphism of functors

\[ ({\rm id}\times t)^*\circ\Delta^*\circ St_D\cong St_D\circ i^* \]

where $i^*$ is the forgetful functor $D^b(k[C_p]\operatorname {-mod})=D_ {C_p}(*)\to D(*)$. Indeed, for an object $A^\bullet$ of $D^b(k[C_p]\operatorname {-mod})$, the underlying complex of both functors is $(A^\bullet )^{\otimes\, \mu _p}$, and the automorphism which sends each summand

\[ A^{i_1}\otimes\cdots\otimes A^{i_p} \]

to itself by $1\otimes \sigma ^t\otimes \sigma ^{2t}\otimes \cdots \otimes \sigma ^{(p-1)t}$ intertwines the two actions of $\mu _p$. Here $\sigma$ is some generator of $\mu _p$. But the functor $St\circ i^*$ kills $b$, since $i^*$ does.

3.1 Prelude: Frobenius-constant quantizations

Let $k$ be a field of characteristic $p$ and let $\mathcal {C}$ be a symmetric monoidal category over $k$. The reader may assume that $\mathcal {C}$ is the category of comodules of some commutative Hopf algebra in $\operatorname {SVect}_k$. Let $A$ be a commutative (and associative) algebra in $\mathcal {C}$. Let

\[ F:A^{(1)}\to A \]

be the Frobenius map. Let $Q$ be an augmented commutative algebra in $\mathcal {C}$ with augmentation $\epsilon :Q\to k$. Following [Reference Bezrukavnikov and KaledinBK08], we make the following definition.

The main example for us is as follow. We take $K$ to be some $\mathbb {G}_m$-equivariant algebraic group in $\operatorname {Vect}_k$, and view $\mathcal {O}:=\mathcal {O}(K\rtimes \mathbb {G}_m)$ as a Hopf algebra in $\operatorname {SVect}_k$ concentrated in degree $0$. We take $\mathcal {C}=\mathcal {O}\operatorname {-comod}$. Let $\hbar$ be a basis vector of the one-dimensional representation of $K\rtimes \mathbb {G}_m$ in which $K$ acts trivially and $\mathbb {G}_m$ acts with weight $2$. Let $Q=k[\hbar ]$. In this case, we will call a $Q$-quantization simply an $\hbar$-quantization, or just a quantization if the meaning is clear.

Fact 3.2 (i) Let $X$ be a smooth affine algebraic variety over $k$. Then the ring of asymptotic crystalline differential operators, $\mathcal {D}_\hbar (X)$, is a canonical $\hbar$-quantization of $\mathcal {O}(T^*X)$. Here $\mathbb {G}_m$ acts trivially on $\mathcal {O}(X)$ and on vector field with weight $2$. Let $\partial$ be a vector field on $X$. Then $\partial ^p$ acts as a derivation on $\mathcal {O}(X)$, so that $\partial ^p-\partial ^{[p]}$ annihilates $\mathcal {O}(X)$ for a unique vector field $\partial ^{[p]}$. Then $\mathcal {D}_\hbar (X)$ has a canonical Frobenius-constant structure determined by

\[ \begin{matrix}F_\hbar: & x & \mapsto & x^p\hfill & \hspace{-1.2pc}x\in\mathcal{O}(X)\hfill\\ & \partial & \mapsto & \partial^p-\hbar^{p-1}\partial^{[p]} & \partial\in \!\textit{Vect}(X).\end{matrix} \]

(ii) Let $J$ be a smooth algebraic group over $k$. Then $F_\hbar$ as above is $K=J\times J$-equivariant (induced by left and right regular actions). In particular, if we take invariants for the left factor, we obtain a Frobenius-constant structure for the quantization $\mathcal {U}_\hbar (J)$ of $\mathcal {O}(\operatorname {Lie}(J)^*)$.

(iii) Let $T$ be a complex torus and let $T^\vee$ be the Langlands dual split torus over $k$, that is,

\[ T^\vee{=}\operatorname{Spec}(k[\mathbb{X}_\bullet(T)]) \]

where $\mathbb {X}_\bullet (T)$ is the cocharacter lattice of $T$ and $k[\mathbb {X}_\bullet (T)]$ is its group algebra. We have canonical identifications

\begin{gather*} \mathcal{O}((\mathfrak{t}^\vee)^*)=\mathcal{O}(\mathfrak{t}_k),\\ \mathcal{U}_\hbar(\mathfrak{t}^\vee)=\mathcal{O}(\mathfrak{t}_k\times\mathbb{G}_a). \end{gather*}

If we take $\operatorname {Spec}$ of the Frobenius-constant structure we recover the $\hbar$-Artin–Schreier map

\[ F_\hbar=AS_\hbar:\mathfrak{t}_k\times\mathbb{G}_a\to \mathfrak{t}_k^{(1)} \]

of Fact 2.1.

3.2 Interlude

The next eight subsections are intended to be a self-contained description of an extension of the theory of Beilinson–Drinfeld Grassmannians, an amazing invention first described in the canonical [Reference Beĭlinson and DrinfeldBD91]. Our extension required a little more geometric machinery than was present in [Reference Beĭlinson and DrinfeldBD91], but we were fortunately able to find what was required in Raskin's excellently concise paper [Reference RaskinRas15] on placid ind-schemes. This was actually originally written with $\mathcal {D}$-modules in mind, but the underlying geometry is the same. These eight subsections are also intended to be a self-contained account of the necessary details of [Reference RaskinRas15], and we have attempted to present things in a slightly more geometric and less categorical way, but highly recommend reading the paper. We have tried to make it clear in the text what is new and what is essentially to be found in [Reference Beĭlinson and DrinfeldBD91] and [Reference RaskinRas15]. What follows is an overview.

We give the following very brief and incomplete overview of the purpose of Beilinson–Drinfeld Grassmannians.

1. (i) If $G$ is a complex algebraic group, its affine Grassmannian $Gr_1$ is the algebro-geometric incarnation of the $E_2$-group $\Omega ^1K$ of based loops in a compact form $K$ of $G$.

2. (ii) $Gr_1$ comes naturally equipped with a ‘convolution diagram’ which is the algebro-geometric incarnation of the pointwise multiplication structure on $\Omega ^1K$.

3. (iii) The Beilinson–Drinfeld Grassmannian $Gr_2$ of $G$ is both a deformation of $Gr_1$ and the algebro-geometric incarnation of the loop-concatenation structure on $\Omega ^1K$.

4. (iv) The Beilinson–Drinfeld Grassmannian fits inside a deformation of the aforementioned ‘convolution diagram’, called the ‘global convolution diagram’ (see [Reference Beĭlinson and DrinfeldBD91, Reference Mirković and VilonenMV07]), which is the algebro-geometric incarnation of the Eckmann–Hilton argument.

Now we review in similar terms what we do that is new. (i) We construct a ‘global convolution diagram’ in the context of the ‘variety of triples’ (which is the geometric basis of the quantum Coulomb branch; see [Reference Braverman, Finkelberg and NakajimaBFN16]). This construction is also summarized in the appendix to [Reference Braverman, Finkelberg and NakajimaBFN17]. The Beilinson–Drinfeld Grassmannian was already constructed in this wider context in [Reference Braverman, Finkelberg and NakajimaBFN16], but the global convolution diagram (and hence the Eckmann–Hilton argument) had proved elusive.

(ii) We construct a geometric analogue of the Kudo–Araki–Dyer–Lashof operations, in the same way that the Beilinson–Drinfeld Grassmannian itself is a geometric analogue of the loop-concatenation structure of loop groups. We do this for the variety of triples,but it is new even for the ordinary Grassmannian.

(iii) We produce a gadget that is to our geometric analogue of the Kudo–Araki–Dyer–Lashof operations as the global convolution diagram is to the Beilinson–Drinfeld Grassmannian (for the variety of triples as well as the basic Grassmannian case). This provides the geometric incarnation of a principle which is similar in spirit to the Eckmann–Hilton argument, but hitherto unknown, namely, that any $S^1$-framed $E_3$-algebra over $\mathbb {F}_{\rm p}$ defines a Frobenius-constant quantization (to be elucidated in a forthcoming work; the topological insight is that one can move a ball past – or rather, through – a donut in an $S^1$-equivariant way).

3.3 Formal neighborhoods

Let $X$ be a smooth complex curve and let $S$ be a finite set. Given a commutative ring $R$ and an $R$-point $x$ of $X^S$, we denote the coordinates of $x$ by $x_s$ ($s\in S$), write $\Gamma (x_s)$ for the graph of $x_s$ in $X_R$ and write $I(x_s)$ for its ideal. We write $\Delta _ {S}(x)$ for the formal neighborhood of the union of the graphs of $x_s$ ($s\in S$). That is, $\Delta _S(x)$ is the direct limit in affine schemes over $X$:

\[ \Delta_S(x)=\mathop {\rm colim }\limits_{\mathrel{\mathop{\kern0pt\longrightarrow}\limits_i} } \Delta_{S,i}(x) \]

where

\[ \Delta_{S,i}(x) = \operatorname{Spec}\bigg(\mathcal{O}_{X_R}\bigg/\prod_{s\in S}I(x_s)^i\bigg). \]

Given a subset $S'\subset S$ and an $R$-point $x$ of $X^S$, we will write

\[ \Delta_S^{S'}(x) \]

for the $S'$-punctured formal neighborhood, that is, the complement of the union of the graphs of $x_s$ ($s\in S'$) in $\Delta _S(x)$. As a sheaf of algebras on $\Delta _S(x)$, $\mathcal {O}(\Delta _S^{S'}(x))$ has an exhaustive increasing filtration:

\[ F^j\mathcal{O}(\Delta_S^{S'}(x)) = \mathcal{O}(\Delta_S(x)).\prod_{s\in S'}I(x_s)^{{-}j}. \]

Suppose we have $S''\subset S'\subset S$ and $x\in X^S(R)$. The inclusion $S'\subset S$ defines a projection $f:X^S\to X^{S'}$, and we will occasionally write

\[ \Delta_{S'}^{S''}(x) \]

for $\Delta _ {S'}^{S''}(f(x))$. We have a closed embedding $\Delta _ {S'}^{S''}(x)\to \Delta _S^{S''}(x)$. Note, however, that this is in a sense non-uniform in $x$: for instance, if for every $s\in S$ there exists an $s'\in S'$ such that $x_s=x_ {s'}$, then the embedding is an isomorphism; and conversely. This is essentially the fact underlying Beilinson and Drinfeld's ‘fusion’ Grassmannian [Reference Beĭlinson and DrinfeldBD91]. We will make more of this when we discuss co-placid morphisms; see Example 3.15.

For notational simplicity, we frequently remove commas and braces from $S$, $S'$, and also drop the part $(x)$, when it is clear which point we refer to. So, for example, the expression

\[ \Delta_{\{1,2\}}^{\{1\}}(x) \]

becomes

\[ \Delta_{12}^1. \]

3.4 Global groups; pro-smoothness

Now fix an affine algebraic group $G$ over $\mathbb {C}$. Consider the following functor from commutative rings to groups over $X^S$:

\[ G_{S}(R):=\{(x,f)|x\in X^S(R),f:\Delta_S(x)\to G\}. \]

Then $G_ {S}$ is represented by the limit of an inverse system of smooth affine group schemes over $X^S$,

\[ G_S{ = }\mathop {\lim }\limits_{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_i} } G_{S,i,} \]

such that each transition morphism is a smooth homomorphism. Here $G_ {S,i}$ may be taken to represent the functor

\[ G_{S,i}(R)=\{(x,f)|x\in X^S(R),f:\Delta_{S,i}(x)\to G\}. \]

Later, the notation $G_ {S,i}$ may represent a piece of some other cofiltered system presenting $G_S$; we will refer to the specific group above as $\operatorname {Map}(\Delta _ {S,i},G)$. The fact that each transition morphism is smooth is directly verified using the valuative criterion. Indeed, let $\operatorname {Spec}(\tilde {R})$ be a square-zero thickening of $\operatorname {Spec}(R)$. A commutative diagram

\[ \begin{matrix}\operatorname{Spec}(R) & \to & G_{S,i+1}\\ \downarrow & & \downarrow\\ \operatorname{Spec}(\tilde{R}) & \to & G_{S,i}\end{matrix} \]

is the same thing as a point $\tilde {x}\in X^S(\tilde {R})$, with residue $x\in X^S(R)$, and a commutative diagram

\[ \begin{matrix}\Delta_{S,i}(x) & \to & \Delta_{S,i+1}(x)\\ \downarrow & & \downarrow\\ \Delta_{S,i}(\tilde{x}) & \to & G.\end{matrix} \]

This determines a morphism $P\to G$ where $P$ is the appropriate pushout in affine schemes. Since $\Delta _ {S,i}(x)$ is equal to the intersection of $\Delta _ {S,i+1}(x)$ with $\Delta _ {S,i}(\tilde {x})$ inside $\Delta _ {S,i+1}(\tilde {x})$, and $\Delta _ {S,i+1}(\tilde {x})$ is a square-zero thickening of $\Delta _ {S,i+1}({x})$, it follows that $\Delta _ {S,i+1}(\tilde {x})$ is a square-zero thickening of $P$. Therefore since $G$ is smooth we can extend $P\to G$ to $\Delta _ {S,i+1}(\tilde {x})\to G$, as required. Note that $G_ {S,0}=X^S$ so, in particular, each $G_ {S,i}$ is smooth over $X^S$.

Now fix $x\in X^S(\mathbb {C})$. It partitions $S$ into subsets $S_1,\ldots , S_n$ according to coincidence amongst the coordinates. Write $y_m$ for the coordinate $x_s$ for any $s\in S_m$, and $z_m$ for the $\mathbb {C}$-point of $X^{S_m}$ with coordinates $y_m$. We have

\[ \begin{matrix} \Delta_{S,i}(x) & = & \operatorname{Spec}\big(\mathcal{O}_X/\prod_{m=1}^nI(y_m)^{i|S_m|}\big)\\ & = & \coprod_{m=1}^n\operatorname{Spec}\big(\mathcal{O}_X/I(y_m)^{i|S_m|}\big).\end{matrix} \]

Therefore we have

\[ G_{S,i}\times_{X^S}\{x\} = \prod_{m=1}^{n}G_{S_m,i}\times_{X^{S_m}}\{z_m\} = \prod_{m=1}^{n}G_{\{m\},i|S_m|}\times_{X^{\{m\}}}\{y_m\}. \]

The smooth transition map $G_ {\{m\},(i+1)|S_m|}\times _ {X^{\{m\}}}\{y_m\}\to G_ {\{m\},i|S_m|}\times _ {X^{\{m\}}}\{y_m\}$ is surjective for all $i\geq 0$ and has a unipotent kernel for all $i\geq 1$. It follows that $G_ {S,i+1}\to G_ {S,i}$ has the same property. Thus $G_S$ is a prosaic affine group scheme over $X^S$ in the following sense.

From now on, ‘groupoid’ will mean ‘affine pro-smooth covering groupoid’, unless it is clear from the context that this is not the case. All examples of groupoids will actually be prosaic.

3.5 Beilinson–Drinfeld Grassmannians; reasonableness

We also consider the functor

\[ G_{S}^{S'}(R):=\{(x,f)|x\in X^S(R), f:\Delta_S^{S'}\to G\}. \]

Then $G_ {S}^{S'}$ is represented by an ind-affine ind-scheme, formally smooth over $X^S$. It is a group in ind-schemes over $X^S$, but not an inductive limit of group schemes. It is a reasonable ind-scheme in the following sense (taken from [Reference DrinfeldDri06]).

Example 3.8 (i) Let $T$ be a reasonable ind-scheme and let $U\to T$ be either ind-f.p. or an ind-flat cover. Then $U\to T$ is co-reasonable.

(ii) In the case of $G_S^{S'}$, one reasonable presentation is given as follows. Fix a finite set $\{a_1,\ldots ,a_n\}$ of generators of $\mathcal {O}(G)$. Then set $\mathcal {J}=\mathbb {Z}_ {\geq 0}$ and set $G_S^{S',j}$ to be the closed subscheme of $G_S^{S'}$ which on the level of $R$-points is given by

\[ G_S^{S',j}(R) = \{(x,f)|x\in X^S(R), f:\Delta_S^{S'}\to G,a_k\circ f\in H^0 F^j\mathcal{O}(\Delta_S^{S'})\}. \]

Here we have taken $G_S^{S',0}=G_S$. The left- and right-regular actions of the subgroup $G_S$ preserve the inductive structure, meaning that each $G_S^{S',j}$ has a free action on both sides by $G_S$ over $X^S$, even though it is not itself a group. Moreover, the fpqc quotient $G_S^{S',j}/G_S$ is of finite type over $X^S$, and flat, although generally quite singular. The result is that the fpqc quotient

\[ G_S^{S'}/G_S \]

has the structure of ind-finite-type ind-flat ind-scheme over $X^S$. In particular, it is reasonable, and $G_S^{S'}\to G_S$ is an ind-flat cover and thus co-reasonable.

On $R$-points, we may identify

\[ G_S^{S'}/G_S(R) = \left\{(x,\mathcal{E},f) \,\left|\, \begin{aligned} &x\in X^S(R)\hfill\\ &\mathcal{E}\text{ a principal }G\text{-bundle over }\Delta_S(x)\hfill\\ &f\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S^{S'}(x)\hfill\end{aligned} \right.\right\}/{\sim}. \]

Here the symbol ‘$/\sim$’ means ‘taken up to isomorphism’, that is, we identify two $R$-points

\[ (x,\mathcal{E},f)\sim (x',\mathcal{E}',f') \]

if $x=x'$ and there exists an isomorphism of $\mathcal {E}$ with $\mathcal {E}'$ which intertwines $f,f'$. Such an isomorphism is unique if it exists. The following fact is due to [Reference Beĭlinson and DrinfeldBD91].

We may reidentify the $R$-points of $G_S^{S'}$ in a way more compatible with the above identification of $G_S^{S'}/G_S(R)$:

\[ G_S^{S'}(R) = \left\{(x,\mathcal{E},f,g) \,\left|\, \begin{aligned} &x\in X^S(R)\hfill\\ &\mathcal{E}\text{ a principal }G\text{-bundle over }\Delta_S(x)\hfill\\ &f\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S^{S'}(x)\hfill\\ &g\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S(x)\end{aligned} \right.\right\}/{\sim}. \]

Notice that the inclusion $S'\subset S$ induces a closed embedding $\Delta _ {S'}^{S''}\to \Delta _S^{S''}$ for any $S''\subset S'$. This in turn induces restriction homomorphisms

\[ G_S^{S''}\to G_{S'}^{S''}. \]

These maps are co-reasonable. One readily checks by looking at points that the induced maps

\[ G_S^{S''}/G_S\to X^S\times_{S'}G_{S'}^{S''}/G_{S'} \]

are isomorphisms. In particular, we have $G_S^{S''}/G_S\xrightarrow {\sim } X^S\times _ {S''}G_ {S''}^{S''}/G_ {S''}$.

3.6 Jet bundles; placidity

We will use the following notion, due to Raskin [Reference RaskinRas15].

Remark 3.13 (i) If the placid ind-scheme $T$ maps to some base $B$ of finite type over $\mathbb {C}$, then the placid presentation may be taken over $B$.

(ii) Let $T$ be a placid ind-scheme and let $T=\underrightarrow {\rm colim } _ {j\in \mathcal {J}}(T_j)$ be a reasonable presentation of $T$. Then each $T_j$ is a placid scheme, so that this reasonable presentation can be extended to a placid presentation.

(iii) Any two placid presentations of the placid ind-scheme $T$ admit a common refinement (which is again a placid presentation). Thus the collection of placid presentations of $T$ forms a filtered category $\mathcal {P}(T)$.

(iv) Suppose that $T$ is a placid ind-scheme and $U$ is an ind-scheme with an ind-f.p. map $f:U\to T$. Then $U$ is automatically placid. The short explanation is ‘by Noetherian approximation’. We spell it out: given any reasonable presentation

\[ T=\mathop {\rm colim }\limits_{\mathrel{\mathop{\kern0pt\longrightarrow}\limits_{j\in\mathcal{J}}}}(T^j) \]

of $T$, we set $U_j:=U\times _TT_j$ and obtain the reasonable presentation

\[ U=\mathop {\rm colim }\limits_{\mathrel{\mathop{\kern0pt\longrightarrow}\limits_{j\in\mathcal{J}}} } (U^j) \]

of $U$. Then, given any placid presentation

\[ T^j=\mathop {\lim }\limits_{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_{i\in(\mathcal{I}^j)^{op}}} } T^j_i \]

of $T^j$, there exists some index $a$ of $\mathcal {I}^j$ such that there is a $T^j_a$-scheme $U^j_a$ fitting into a Cartesian diagram

\[ \begin{matrix}U^j & \to & T^j\\ \downarrow & & \downarrow\\ U^j_a & \to & T^j_a . \end{matrix}\]

Moreover, since $T^j\to T^j_a$ is a covering, the choice of $T^j_a$-scheme $U^j_a$ is unique. Thus if we replace $\mathcal {I}^j$ by its final subcategory based at $a$, we can present $U^j\to T^j$ as the limit of a cofiltered system of f.p. morphisms

\[ (U^j_i\to T^j_i)_{i\in(\mathcal{I}^j)^{op}} \]

such that for each $i\to i'$ in $\mathcal {I}$, the square

\[ \begin{matrix} U^j_{i'} & \to & T^j_{i'}\\ \downarrow & & \downarrow\\ U^j_i & \to & T^j_i\end{matrix} \]

is Cartesian. Placid presentations of this form will be called Cartesian.

(v)The product (over $B$) of placid ind-schemes is placid.

(vi) Consider placid presentations

\[ T=\mathop {\rm colim }\limits_{\mathrel{\mathop{\kern0pt\longrightarrow}\limits_{j\in \mathbb{Z}_{\geq0}}}} \mathop {\lim }\limits_{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_{i\in ( \mathbb{Z}_{{\geq} j})^{op}}} } (T^j_i) \]

of $T$ with the property that $T^j_i$ is formed out of $T^{j'}_i$ as in point (iv) whenever $i\geq j'\geq j$. We call such placid presentations neat. This is possibly a technically useless notion. But every placid presentation admits a refinement which is neat up to replacing the indexing categories $\mathcal {J},\mathcal {I}^j$ by final subcategories. Thus many constructions on placid ind-schemes can be phrased in terms of neat presentations.

Note that for any morphism $f:U\to T$ of placid schemes and any placid presentations $U=\underleftarrow {\lim } _ {i\mathcal {I}_U^{op}}U_i$, $T=\underleftarrow {\lim } _ {i\mathcal {I}_T^{op}}T_i$, for any $i\in \mathcal {I}_T$ there exist $i'\in \mathcal {I}_U$ and a unique map $U_ {i'}\to T_i$ making the square

\[ \begin{matrix}U & \to & T\\ \downarrow & & \downarrow\\ U_{i'} & \to & T_i\end{matrix} \]

commutative. Thus, by changing the indexing sets appropriately we can choose placid presentations of $U$, $T$ with a common indexing set $\mathcal {I}$ and write $f=\underleftarrow {\lim } _ {i\in \mathcal {I}}(U_i\to T_i)$. Such a presentation will be called compatible. This notion extends immediately to morphisms of placid ind-schemes. A Cartesian presentation is a compatible presentation in which all appropriate squares are Cartesian. If $\mathcal {G}$ is an affine groupoid scheme over some base $B$ of finite type over $\mathbb {C}$ and $f$ is $\mathcal {G}$-equivariant (over $B$), then we can find a $\mathcal {G}$-equivariant compatible presentation. If in addition $f$ is f.p. so that it admits a Cartesian presentation, then this can also be chosen to be $\mathcal {G}$-equivariant.

The notion of a co-placid morphism has been defined previously in the literature: see [Reference RaskinRas15], where they are called simply ‘placid’. As noted in [Reference RaskinRas15], it is not a relative version of placidity for ind-schemes. This is the reason for the present renaming.

Example 3.15 (i) Let $T$ be a placid ind-scheme. Let $U\to T$ be either ind-smooth or an ind-pro-smooth cover. Then it is co-placid.

(ii) $G_S^{S'}$ is a placid ind-scheme, and the fpqc quotient map

\[ G_S^{S'}\to G_S^{S'}/G_S \]

is an ind-pro-smooth ind-affine affine cover of a placid (indeed, ind-finite type) ind-scheme, so is co-placid.

(iii) Given $S''\subset S'\subset S$, the morphism

\[ f:G_S^{S''}\to X^S\times_{X^{S'}}G_{S'}^{S''} \]

is co-placid. Indeed, consider $G_ {S,i}:=\operatorname {Map}(\Delta _ {S,i},G)$. We have $G_S=\underleftarrow {\lim } _ {i\in \mathbb {Z}_{\geq 0}}G_ {S,i}$ and $X^S\times _ {X^{S'}}G_ {S'}=\underleftarrow {\lim } _ {i\in \mathbb {Z}_{\geq 0}}X^S\times _ {X^{S'}}G_ {S,i}$ and the morphism

\[ G_{S,i}\to X^{S}\times_{X^{S'}}G_{S',i} \]

induced by the closed embeddings $\Delta _ {S',i}(x)\subset \Delta _ {S,i}(x)$ for any $x\in X^S(R)$. This is a smooth covering, by the same argument of § 3.4 for the prosaicness of $G_S$. This shows that

\[ g:G_S\to X^S\times_{X^{S'}}G_{S'} \]

is co-placid. To conclude, note that the morphism $f$ is an ind-locally trivial $g$-bundle over the ind-finite type ind-scheme $G_S^{S''}/G_S=X^S\times _ {X^{S'}}G_ {S'}^{S''}/G_ {S'}$.

Warning 3.16 Morphism $g$ is not a pro-smooth cover, and $f$ is not an ind-pro-smooth cover. It is tempting to imagine that it is the quotient map by some group scheme $\ker (G_S\to X^S\times _ {X^{S'}}G_ {S'})$, but there is no such group scheme. To see this, fix some section $S\to S'$ and consider the corresponding ‘multi-diagonal’ embedding $X^{S'}\to X^S$. Then $g$ is an isomorphism over $X^{S'}$. However, over a generic point of $X^S$, $g$ is a non-trivial projection from $G(\mathcal {O})^S\to G(\mathcal {O})^{S'}$; see § 3.9 for the notation.

(iv) Fix a representation $N$ of $G$ of dimension $d$. Let $\boldsymbol {N}:=\operatorname {Spec}(\operatorname {Sym}(N^*))$ be the corresponding $G$-module, that is, vector space in the category of schemes over $\mathbb {C}$ with $G$-action. Then $\boldsymbol {N}_S$ is a $G_S$-module; indeed, each $\operatorname {Map}(\Delta _ {S,i},\boldsymbol {N})$ is a $\operatorname {Map}(\Delta _ {S,i},G)$-module and the transition maps are $G_S$-equivariant. We have the placid ind-scheme

\[ \widetilde{\mathcal{T}}_S^{S'}:=G_S^{S'}\times_{X^S}{\boldsymbol{N}_S}. \]

This is an ind-pro-smooth covering of its fpqc quotient (relative to $X^S$) ind-scheme

\[ \mathcal{T}_S^{S'}:=G_S^{S'}\frac{\times_{X^S}}{^{G_S}}\boldsymbol{N}_S. \]

This is an infinite-dimensional vector bundle over $G_S^{S'}/G_S$. On the level of $R$-points we identify

\[ \mathcal{T}_S^{S'}(R)=\left\{(x,\mathcal{E},f,\tilde{v}) \,\left|\, \begin{aligned} &x\in X^S(R)\hfill\\ &\mathcal{E}\text{ a principal }G\text{-bundle over }\Delta_S(x)\hfill\\ &f\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S^{S'}(x)\hfill\\ &\tilde{v}\text{ an }\boldsymbol{N}\text{-section of }\mathcal{E}\hfill\end{aligned} \right.\right\}/{\sim}. \]

Here by ‘an $\boldsymbol {N}$-section of $\mathcal {E}$’ we mean a section of the associated $\boldsymbol {N}$-bundle. Of course $\mathcal {T}_S^{S'}$ is the inverse limit of vector bundles over $G_S^{S'}/G_S$,

\[ \mathcal{T}_S^{S'}=\mathop {\lim }\limits_{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_i} } \mathcal{T}_{S,i}^{S'}, \]

where $\mathcal {T}_ {S,i}^{S'}$ is the associated bundle of $\boldsymbol {N}_ {S,i}$, a vector bundle of rank $di|S|$. In particular, $\mathcal {T}_S^{S'}$ is a placid ind-scheme, with $\mathcal {T}_S^{S',j}$ being the infinite-dimensional vector bundle $\mathcal {T}_S^{S'}|_ {G_S^{S',j}/G_S}$ over $G_S^{S',j}/G_S$. We identify the $R$-points of $\widetilde {\mathcal {T}}_S^{S'}$ compatibly as follows:

\[ \widetilde{\mathcal{T}}_S^{S'}(R)=\left\{(x,\mathcal{E},f,g,\tilde{v}) \,\left|\, \begin{aligned} &x\in X^S(R)\hfill\\ &\mathcal{E}\text{ a principal }G\text{-bundle over }\Delta_S(x)\hfill\\ &f\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S^{S'}(x)\hfill\\ &g\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S(x)\hfill\\ &\tilde{v}\text{ an }\boldsymbol{N}\text{-section of }\mathcal{E}\hfill\end{aligned}\right.\right\}/{\sim}.\]

(v) $\boldsymbol {N}_S^{S'}$ is a $G_S^{S'}$-module (in ind-schemes). Therefore multiplication gives a map between the placid ind-schemes

\[ \mathcal{T}_S^{S'}\to\boldsymbol{N}_S^{S'}. \]

We define $\mathcal {R}_S^{S'}$ to be the fiber product

\[ \mathcal{R}_S^{S'}:=\mathcal{T}_S^{S'}\times_{\boldsymbol{N}_S^{S'}}\boldsymbol{N}_S. \]

This is an ind-scheme over $G_S^{S'}/G_S$, with $\mathcal {R}_S^{S',j}:=\mathcal {T}_S^{S',j}\times _ {\boldsymbol {N}_S^{S'}}\boldsymbol {N}_S = \mathcal {R}_S^{S'}|_ {G_S^{S',j}/G_S}$. Moreover, $\mathcal {R}_S^{S'}$ is a vector space over $G_S^{S'}/G_S$, but unlike $\mathcal {T}_S^{S'}$ it is not a vector bundle because the fibers jump. Furthermore, $\mathcal {R}_S^{S',j}$ contains $\ker (\mathcal {T}_ {S}^{S',j}\to \mathcal {T}_ {S,i}^{S',j})$ for $i$ large enough (depending on $j$), that is, we have a diagram

\[ \ker(\mathcal{T}_{S}^{S',j}\to \mathcal{T}_{S,i}^{S',j})\subset \mathcal{R}_S^{S',j}\subset \mathcal{T}_S^{S',j} \]

of vector spaces over $G_S^{S',j}/G_S$. Therefore $\mathcal {R}_S^{S'}$ is placid: we may take $\mathcal {R}_ {S,i}^{S',j}$ to be the image in $\mathcal {T}_ {S,i}^{S',j}$ of $\mathcal {R}_ {S}^{S',j}$, for $i$ large enough. Also, $\mathcal {R}_S^{S'}$ is of ind-finite codimension in $\mathcal {T}_S^{S'}$, that is, it is an ind-f.p. closed sub-ind-scheme. On the level of $R$-points, we have

\[ \mathcal{R}_S^{S'}(R)=\left\{(x,\mathcal{E},f,{v}) \,\left|\, \begin{aligned} &x\in X^S(R)\hfill\\ &\mathcal{E}\text{ a principal }G\text{-bundle over }\Delta_S(x)\hfill\\ &f\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S^{S'}(x)\hfill\\ &{v}\text{ an }\boldsymbol{N}\text{-section of }\mathcal{E}\text{ such that }f(v)\text{ extends to }\Delta_S(x)\hfill\end{aligned}\right.\right\}/{\sim}.\]

Here $f(v)$ is a section of the trivial $\boldsymbol {N}$-bundle on $\Delta _S^{S'}(x)$, and we require that it extends to a section of the trivial $\boldsymbol {N}$-bundle over $\Delta _S(x)$. Such an extension is unique if it exists. We denote the preimage of $\mathcal {R}_S^{S'}$ in $\widetilde {\mathcal {T}}_S^{S'}$ as $\widetilde {\mathcal {R}}_S^{S'}$. It is an ind-f.p. closed sub-ind-scheme of $\widetilde {\mathcal {T}}_S^{S'}$, an ind-pro-smooth cover of $\mathcal {R}_S^{S'}$, its fpqc quotient (locally on $X^S$) by $G_S$, and on $R$-points we identify

\[ \widetilde{\mathcal{R}}_S^{S'}(R)=\left\{(x,\mathcal{E},f,g,{v}) \,\left|\, \begin{aligned} &x\in X^S(R)\hfill\\ &\mathcal{E}\text{ a principal }G\text{-bundle over }\Delta_S(x)\hfill\\ &f\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S^{S'}(x)\hfill\\ &g\text{ a trivialization of }\mathcal{E}\text{ over }\Delta_S(x)\hfill\\ &{v}\text{ a section of the trivial }\boldsymbol{N}\text{-bundle on }\Delta_S(x)\\ &\quad \text{such that }fg^{{-}1}(v)\text{ extends to }\Delta_S(x)\hfill\end{aligned}\right.\right\}/{\sim}.\]

(vi) The product (over $B$) of co-placid maps is co-placid.

Remark 3.17 (i) Suppose $X=\mathbb {G}_a$ with parameter $t$. Then $I(x_s)$ is trivialized by $t - t_ {x_s}$. There is an isomorphism from $\mathcal {T}_S^{S'}$ to the kernel of the covering $\mathcal {T}_S^{S'}\to \mathcal {T}_ {S,i}^{S'}$, given on $R$-points by

\[ (x,f,\mathcal{E},\tilde{v})\mapsto \biggl(x,f,\mathcal{E},\prod_{s\in S}(t - t_{x_s})\tilde{v}\biggr). \]

We call this isomorphism the fiberwise shift map of $\mathcal {T}$.

(ii) The placid ind-scheme $T=G_S^{S'}/G_S,\mathcal {T}_S^{S'},\mathcal {R}_S^{S'}$ is special in that one may take the smooth affine covering maps $T^j_i\to T^j_ {i'}$ to be vector bundles. But placidity seems to be the more flexible definition; for instance, I do not know if point (iii) of Remark 3.13 holds if we replace ‘placid’ by ‘special’.

3.7 Equivariance

(i) Note that $G_S^{S'}/G_S,\mathcal {T}_S^{S'},\mathcal {R}_S^{S'}$ are all acted on by $G_S$, and the various maps between them are $G_S$-equivariant. In fact, each ‘approximation’ $G_S^{S',j}/G_S$, $\mathcal {T}^j_i$, $\mathcal {R}^j_i$ is acted on by some quotient $G_ {S,i'}$ of $G_S$, and the transition morphisms are all $G_S$-equivariant.

(ii) Suppose that $X=\mathbb {G}_a$. Then we have the action of $\mathbb {G}_m$ on $X$ by multiplication. It also acts diagonally on $X^S$. Therefore we may consider $\mathbb {C}^*\times X^S$ as a smooth groupoid over $X^S$. The group $G_S^{S'}$ over $X^S$ is $\mathbb {C}^*\times X^S$-equivariant, so we may form the semidirect product