1. Comparison of support theories for finite group schemes

We review a few salient features of four formulations of support theories for modules $M$ for a finite group scheme $G$

\[ M \mapsto \mathbb{P} |G|_M; \quad M \mapsto \mathbb{P} V_r(G)_M; \quad M \mapsto \Pi(G)_M; \quad M \mapsto \mathbb{P} \mathfrak{C}_r(\mathbb{G}_{(r)})_M \]

and establish their close relationships. In Theorem 1.4, we recall that the cohomological support theory $M \mapsto \mathbb {P} |G|_M$ agrees with the infinitesimal $1$-parameter support theory $M \mapsto \mathbb {P} V_r(G)_M$ for an infinitesimal group scheme of height $\leq r$ provided that $M$ is finite dimensional. In Theorem 1.6, we recall that the cohomological support theory $M \mapsto \mathbb {P} |G|_M$ is equivalent to the $\pi$-point support theory $M \mapsto \Pi (G)_M$ for any finite group scheme provided $M$ is finite dimensional. In Theorem 1.7, we make explicit the equivalence between $M \mapsto \Pi (\mathbb {G}_{(r)})_M$ and $M \mapsto \mathbb {P} V_r(\mathbb {G}_{(r)})_M$.

This section concludes with the formulation of the exponential support theory $M \mapsto \mathbb {P} \mathfrak{C}_r(\mathbb {G}_{(r)})_M$ for a Frobenius kernel of a linear algebraic group $\mathbb {G}$ of exponential type. Our definition of a linear algebraic group of exponential type in Definition 1.9 allows the exponential $\mathcal {E}: \mathcal {N}_p(\mathfrak{g}) \times \mathbb {G}_a \to \mathbb {G}$ to be a continuous algebraic map (a rational map with unique specialization at every geometric point). In Theorem 1.13, we demonstrate that the exponential support theory $M \mapsto \mathbb {P} \mathfrak{C}_r(\mathbb {G}_{(r)})_M$ agrees with the infinitesimal $1$-parameter support theory $M \mapsto \mathbb {P} V_r(\mathbb {G}_{(r)})_M$ for Frobenius kernels $\mathbb {G}_{(r)}$ of linear algebraic groups of exponential type. This exponential support theory is an extension of joint work with Suslin and Bendel for infinitesimal group schemes [Reference Suslin, Friedlander and BendelSFB97a, Reference Suslin, Friedlander and BendelSFB97b] which itself extends earlier joint work with Parshall [Reference Friedlander and ParshallFP86]. In some sense, $M \mapsto \mathbb {P} \mathfrak{C}(\mathbb {G})_M$ is the ‘rank variety’ formulation of $M \mapsto \Pi (\mathbb {G})_M$.

The additive group $\mathbb {G}_a$ plays a central role in what follows. The coordinate algebra of $\mathbb {G}_a$ is the polynomial algebra $k[T]$ on one variable, whereas the group algebra $k\mathbb {G}_a$ is the truncated polynomial algebra $k[u_0,\ldots, u_n, \ldots ]/(u_i^p)$ on countably infinite generators each of whose $p$th powers is $0$. Here, $u_r: k[T] \to k$ is the functional sending a polynomial $p(T)$ to its coefficient of $T^{p^r}$. If we denote by $v_i$ the dual basis element to $T^i \in k[\mathbb {G}_a]$ so that $v_i(p(T))$ reads off the coefficient of $T^i$ in $p(T)$, then $u_r = v_{p^r}$. We denote by $\epsilon _r$ the map of $k$-algebras (but not Hopf algebras, unless $r = 1$)

(1.2.1)\begin{equation} \epsilon_r: k\mathbb{G}_{a(1)} \simeq k[t]/t^p \to k\mathbb{G}_a, \quad t \mapsto u_{r-1}. \end{equation}

We also use $\epsilon _r$ to denote the factorization of (1.2.1) through $k\mathbb {G}_{a(r)} \hookrightarrow k\mathbb {G}_a$, $\epsilon _r: k\mathbb {G}_{a(1)} \to k\mathbb {G}_{a(r)}$.

The analysis of $M \mapsto \mathbb {P}|G|_M$ was achieved for the very special case in which $G \simeq \mathbb {Z}/p^{\times r}$ by replacing cohomological support varieties for modules for these elementary abelian $p$-groups by more accessible rank varieties as conjectured by Carlson and proved by Avrunin and Scott [Reference Avrunin and ScottAS62]. This was first generalized to restricted Lie algebras, especially in joint work with Parshall (see [Reference Friedlander and ParshallFP86]).

A fundamental step in extending support varieties to finite group schemes was the extension to all infinitesimal group schemes in joint work with Suslin and Bendel in [Reference Suslin, Friedlander and BendelSFB97a, Reference Suslin, Friedlander and BendelSFB97b] as we next recall.

Theorem 1.4 [Reference Suslin, Friedlander and BendelSFB97b, Theorem 5.2, Corollary 6.8]

Let $G$ be an infinitesimal group scheme of height $\leq r$. There is a natural (in $G$) map $\psi : H^\bullet (G,k) \to k[V_r(G)]$ inducing a universal homeomorphism

(1.4.1)\begin{equation} \Psi: \mathbb{P} V_r(G) \to \mathbb{P} |G|, \quad \text{restricting to} \quad \Psi: \mathbb{P} V_r(G)_M \to \mathbb{P} |G|_M \end{equation}

(where $\mathbb {P} V_r(G)$ denotes $\operatorname {Proj}\nolimits k[V_r(G)]$) for each finite-dimensional $G$-module $M$.

The most general construction of a support theory for finite group schemes is the $\pi$-point theory $M \mapsto \Pi (G)_M$ introduced by Pevtsova and the current author [Reference Friedlander and PevtsovaFP07] which we next recall.

Recollection 1.5 Let $G$ be a finite group scheme over $k$. Elements of $\Pi (G)$ are equivalence classes of flat maps of $K$-algebras for some field extension $K/k$, $\alpha _K: K[t]/t^p \to KG$, which factor through the group algebra of some unipotent abelian subgroup scheme $U_K \subset G_K$. Two such flat maps $\alpha _K:K[t]/t^p \to KG,\ \beta _L:L[t]/t^p \to LG$ are equivalent if there exists a common field extension $\Omega$ of $K, L$ such that $\alpha _\Omega ^*(M_\Omega )$ is free if and only if $\beta _\Omega ^*(M_\Omega )$ is free for any finite-dimensional $G$-module $M$.

We consider subsets of $\Pi (G)$ of the form

\[ \Pi(G)_M \equiv \{[\alpha_K]: \alpha_K^*(M_K) \text{ is not free as a } K[t]/t^p\text{-module}\} \]

for an arbitrary $G$-module $M$, well defined by [Reference FriedlanderFri15, Theorem 6.6]. The closed subsets of $\Pi (G)$ are the subsets of this form with $M$ finite dimensional.

The following theorem tells us that $M \mapsto \mathbb {P} |G|_M$ is equivalent to $M \mapsto \Pi (G)_M$ provided that $M$ is finite dimensional.

Theorem 1.6 [Reference Friedlander and PevtsovaFP07, Theorem 7.4]

There exists a natural scheme structure on $\Pi |G|$ and a scheme-theoretic isomorphism

(1.6.1)\begin{equation} \Phi: \mathbb{P} |G| \stackrel{\sim}{\to} \Pi(G),\quad \text{restricting to} \quad \mathbb{P} |G|_M \stackrel{\sim}{\to} \Pi(G)_M \end{equation}

for any finite-dimensional $G$-module $M$.

The isomorphisms $\Psi$ of (1.4.1) and $\Phi$ of (1.6.1) are quite abstract. The natural map $\psi : H^\bullet (G,k) \to k[V_r(G)]$ of [Reference Suslin, Friedlander and BendelSFB97a, Theorem 1.14] determining $\Psi$ for an infinitesimal group scheme of height $\leq r$ uses the universal height $r$ $1$-parameter subgroup for $G$. The isomorphism $\Phi$ of [Reference Friedlander and PevtsovaFP07, Theorem 7.5] entails consideration of a sheafified version of the endomorphisms of the stable module category of finite-dimensional $G$-modules.

The following proposition makes $\Psi, \Phi$ more concrete.

Theorem 1.7 Let $G$ be an infinitesimal group scheme over $k$ of height $\leq r$. Then the composition

\[ \mathbb{P} V_r(G) \stackrel{\Psi}{\to} \mathbb{P} |G| \stackrel{\Phi}{\to} \Pi(G) \]

sends the $K$-point represented by the $1$-parameter subgroup $\eta _K: \mathbb {G}_{a(r),K} \to G_K$ to the $K$-point represented by the $\pi$-point $\eta _{K*} \circ \epsilon _r: K[t]/t^p \to K\mathbb {G}_{a(r),K} \to KG_K$.

Moreover, if $G$ is the Frobenius kernel of the linear algebraic group $\mathbb {G}$ and if $K/k$ is a field extension, then the maps induced by $\Psi$ and $\Phi$ are both $\mathbb {G}(K)$-equivariant with respect to the natural actions induced by conjugation of $\mathbb {G}$ on itself.

Proof. The map $\Psi$ sends a $K$-point of $\mathbb {P} V_r(G)$ represented by $1$-parameter subgroup $\eta _K: \mathbb {G}_{a(r),K} \to G_K$ to

(1.7.1)\begin{equation} ker\{ eval_{\eta_K} \circ \psi: H^\bullet(G,k) \to k[V_r(G)] \to K \}, \end{equation}

where $\psi$ is the map of [Reference Suslin, Friedlander and BendelSFB97a, Theorem 1.14] mentioned previously. By [Reference Friedlander and PevtsovaFP07, Theorem 3.6], the inverse of $\Phi$ (on $K$-points) sends a $\Pi$-point $\alpha _K: K[t]/t^p \to KG_K$ to

(1.7.2)\begin{equation} ker\{H^\bullet(G,k) \to H^\bullet(G_K,K) \stackrel{ \alpha_K^*}{\to} H^\bullet(K[t]/t^p,K) \}. \end{equation}

To identify $\Psi$, we first consider the special case $G = GL_{N(r)}$. In this case, each $K$-point of $\Pi (G)$ is uniquely represented by the composition of $\epsilon _r: K[t]/t^p \to \mathbb {G}_{a(r)}$ and a $1$-parameter subgroup $\eta _K$ of the form $\mathcal {E}_{\underline B}: \mathbb {G}_{a(r),K} \to KG$ by [Reference Suslin, Friedlander and BendelSFB97a, Proposition 1.2] and [Reference Friedlander and PevtsovaFP05, Proposition 3.8]. (See also Definition 1.9(2).) Thus, for $G = GL_{N(r)}$, it suffices to show the equality of the radicals of the kernels of (1.7.1) and (1.7.2) with $\eta _K= \mathcal {E}_{\underline B}$ and $\alpha _K = \mathcal {E}_{\underline B} \circ \epsilon _r$. This follows from [Reference Suslin, Friedlander and BendelSFB97a, Theorem 5.2].

For a general infinitesimal group scheme $G$, we embed $G$ in some $GL_N$ and use [Reference Suslin, Friedlander and BendelSFB97b, Theorem 5.2] which asserts that the map $\psi : H^\bullet (G,k) \to k[V_r(G)]$ has nilpotent kernel and [Reference Suslin, Friedlander and BendelSFB97b, Proposition 4.3] which implies that $H^\bullet (GL_N,k) \to H^\bullet (G,k)$ has image containing all $p^r$th powers so that $\mathbb {P} |G| \to \mathbb {P} |GL_{N(r)}|$ is injective. This injectivity together with the commutativity of the diagram (implied by the naturality of $\Psi$ and $\Phi )$

(1.7.3)

implies that the identification of $\Phi \circ \Psi$ for $GL_N$ implies the ‘same’ identification for $G$.

For $G = \mathbb {G}_{(r)}$, the $\mathbb {G}(K)$-equivariance of the maps on $K$-points determined by $\Psi$ and $\Phi$ follows directly from the explicit descriptions of these maps given previously.

The abstract formulation of $M \mapsto \Pi (G)_M$ does not easily lead to computations. Following [Reference Suslin, Friedlander and BendelSFB97a], we consider the following affine $k$-varieties, worthy of study in their own right.

Definition 1.8 Let $\mathbb {G}$ be a linear algebraic group with Lie algebra $\mathfrak{g}$. We denote by $\mathcal {N}_p(\mathfrak{g}) \subset \mathfrak{g}$ the $p$-nilpotent variety of $\mathfrak{g}$, the (reduced) closed subvariety of the affine space $\operatorname {Spec}\nolimits S(\mathfrak{g}^*) \simeq \mathbb {A}^d$ whose $K$-points are the elements $X \in \mathfrak{g}_K$ such that $X^{[p]} = 0$ for any field extension $K/k$. For any $r > 0$, we define

\[ \mathfrak{C}_r(\mathbb{G}) \equiv \mathfrak{C}_r(\mathcal{N}_p(\mathfrak{g})) \subset \mathcal{N}_p(\mathfrak{g})^{\times r} \]

to be the (reduced) closed subvariety of $\mathcal {N}_p(\mathfrak{g})^{\times r}$ whose $K$-points are $r$-tuples $\underline B = (B_0,\ldots,B_{r-1}) \in \mathcal {N}_p(\mathfrak{g}_K)^{\times r}$ satisfying the condition the $[B_i,B_j] = 0$ for all $i,j$ with $0 \leq i < j < r$.

The following definition of a linear algebraic group of exponential type is a slight modification of that of [Reference FriedlanderFri15] in that we allow $\mathcal {E}$ to be a continuous algebraic map and we explicitly require $\mathcal {E}$ to be $\mathbb {G}$-equivariant with respect to the adjoint action of $\mathbb {G}$ on $\mathfrak{g}$ and the conjugation action of $\mathbb {G}$ on itself. Following [Reference FriedlanderFri11], we define a continuous algebraic map $f: X \dashrightarrow Y$ to be a ‘roof’ $X \stackrel {p}{\leftarrow } \tilde X \stackrel {g}{\to } Y$ of $k$-schemes such that $p: \tilde X \to X$ is a universal homeomorphism (i.e. a finite, surjective map such that $k(\tilde x)$ is purely inseparable of $k(p(\tilde x))$ for all points $\tilde x \in \tilde X$). A typical example is a rational map from $X$ to $Y$ (i.e. a morphism with domain a dense open set) whose graph in $X\times Y$ projects to $X$ via a universal homeomorphism. A bicontinuous algebraic map of irreducible varieties $X, Y$, $f: X \stackrel {\sim }{\dashrightarrow } Y$, is given by a finite, purely separable extension $k(X)$ over $k(Y)$ inducing a bijection between the $K$-points of $X$ and the $K$-points of $Y$ for any algebraically closed extension $K$ of $k$. Examples include the Frobenius map $F: \mathbb {G} \to \mathbb {G}^{(1)}$ and the canonical map $\tilde N(sl_2) \to N(sl_2)$ from the normalization of the nilpotent cone of the Lie algebra $sl_2$ to the nilpotent cone itself.

As remarked in [Reference FriedlanderFri15, Remark 1.7], if the linear algebraic group $\mathbb {G}$ admits the structure $(\mathbb {G},\mathcal {E})$ of an algebraic group of exponential type, two such structures are isomorphic via an automorphism of $\mathcal {N}_p(\mathfrak{g})$.

We are much indebted to P. Sobaje for guiding us to the following results in the literature.

In the following definition, we consider $p$-nilpotent linear operators on $K[\mathbb {G}]$ of the form $\mathcal {E}_{B*}(u_r)$. We recall that $u_r:k[\mathbb {G}_a] \to k$ reads off the coefficient of $T^{p^r}$ of a polynomial in $k[T] = k[\mathbb {G}_a]$ and $\mathcal {E}_{B*}(u_r)$ is the composition $u_r \circ \mathcal {E}_B^*: K[\mathbb {G}] \to K[\mathbb {G}_a] \to K$.

Definition 1.11 Let $(\mathbb {G},\mathcal {E})$ be a linear algebraic group of exponential type. For any $\mathbb {G}$-module $M$, we define the Jordan type of $M$ at a $K$-point $\mathcal {E}_{\underline B}$ of $\mathfrak{C}_r(\mathcal {N}_p(\mathfrak{g}))$ to be the block sum decomposition

\[ JT(M)_{\mathcal{E}_{\underline B}} \equiv [1]^{\oplus m_1} \oplus \cdots \oplus [p]^{m_p},\quad 0 \leq m_i \leq \infty \]

for the action on $M_K$ of the $p$-nilpotent element $\sum _{s \geq 0} (\mathcal {E}_{B_s})_*(u_s) \in K\mathbb {G};$ we further define the stable Jordan type of $M$ at $\mathcal {E}_{\underline B}$ to be

\[ sJT(M)_{\mathcal{E}_{\underline B}} \equiv [1]^{\oplus m_1} \oplus \cdots \oplus [p-1]^{m_{p-1}},\quad 0 \leq m_i \leq \infty. \]

We define $\mathfrak{C}_r(\mathcal {N}_p(\mathfrak{g}))_M \subset \mathfrak{C}_r(\mathcal {N}_p(\mathfrak{g}))$ to be the subspace whose set of $K$-points is the subset of those $K$-points $\underline B$ of $\mathfrak{C}_r(\mathcal {N}_p(\mathfrak{g}))$ with the property that the stable Jordan type $sJT(M)_{\mathcal {E}_{\underline B}}$ is not $0$.

The natural grading on the affine scheme $V_r(\mathbb {G}_{(r)})$ for $(\mathbb {G},\mathcal {E})$ a linear algebraic group of exponential type corresponds to the monoid action

\[ V_r(\mathbb{G}_{(r)}) \times \mathbb{A}^1 \to V_r(\mathbb{G}_{(r)}), \quad (\mathcal{E}_{\underline B}, a) \mapsto \mathcal{E}_{a \bullet \underline B}, \]

where $\mathcal {E}_{a \bullet \underline B} = \mathcal {E}_{\underline B}(a \cdot t)$. As $(\mathcal {E}_B\circ F^s)(a\cdot t) =( \mathcal {E}_{a^{p^s} \cdot B} \circ F^s)(t)$, we conclude that $V_r(\mathbb {G}_{(r)}) \times \mathbb {A}^1 \to V_r(\mathbb {G}_{(r)})$ is given by

(1.11.1)\begin{equation} (\mathcal{E}_{\underline B},a) \mapsto \mathcal{E}_{(a\bullet \underline B)}, \quad a \bullet (B_0,\ldots B_{r-1}) \equiv (a\cdot B_0,\ldots,a^{p^{r-1}}\cdot B_{r-1}). \end{equation}

Then $\mathbb {P} V_r(\mathbb {G}_{(r)})$, the projective variety associated to $V_r(\mathbb {G}_{(r)})$ with this grading, agrees with the more general notation $\mathbb {P} V_r(G)$ appearing in Theorem 1.4.

We denote by $\Lambda _r: \mathfrak{C}_r(\mathbb {G}) \to \mathfrak{C}_r(\mathbb {G})$ the isomorphism sending $(B_0,\ldots,B_{r-1})$ to $(B_{r-1},\ldots,B_{0})$. We introduce a grading on $\mathfrak{C}_r(\mathbb {G})$ which enables bicontinuous algebraic maps $\rho _r \circ \Lambda _r: \mathbb {P} \mathfrak{C}_r(\mathbb {G}) \to \mathbb {P} V_r(\mathbb {G}_{(r)})$. Namely, we define the monoid action $\mathfrak{C}_r(\mathbb {G}) \times \mathbb {A}^1 \to \mathfrak{C}_r(\mathbb {G})$ as the restriction of the monoid action $\mathfrak{g}^{\times r} \times \mathbb {A}^1 \to \mathfrak{g}^{\times r}$ given by

(1.11.2)\begin{equation} (\underline B,a) \mapsto a * \underline B, \quad a* (B_0,\ldots B_{r-1}) \equiv (a^{p^{r-1}}\cdot B_0,\ldots,a \cdot B_{r-1}). \end{equation}

We denote by $\mathbb {P} \mathfrak{C}_r(\mathbb {G})$ the associated projective variety. For any $\mathbb {G}$-module $M$ and any algebraically closed field extension $K/k$, we define the $K$-points of $\mathbb {P}\mathfrak{C}_r(\mathbb {G})_M$ to be the image $K$-points of $\mathfrak{C}_r(\mathbb {G})_M \backslash \{ 0 \}$.

Definition 1.12 Let $(\mathbb {G},\mathcal {E})$ be a linear algebraic group of exponential type. We denote by

(1.12.1)\begin{equation} \rho_r: \mathfrak{C}_r(\mathbb{G}) \dashrightarrow V_r(\mathbb{G}_{(r)}). \end{equation}

the continuous algebraic map associating to a $K$ point $\underline B$ of $\mathfrak{C}_r(\mathbb {G})$ the $K$-point $\mathcal {E}_{\underline B}: \mathbb {G}_{a(r),K} \to \mathbb {G}_{(r),K}$ of $V_r(\mathbb {G}_{(r)})$; so defined, $\rho _r$ is a bicontinuous algebraic map and, thus, a universal homeomorphism. Here, we have abused notation by using $\mathcal {E}_{\underline B}$ to denote both the $1$-parameter subgroup $\mathbb {G}_{a,K} \to \mathbb {G}_K$ and its restriction $(\mathcal {E}_{\underline B})_{(r)}: \mathbb {G}_{a(r),K} \to \mathbb {G}_{(r),K}$.

Thus (see [Reference FriedlanderFri15, Definition 4.4]), the $K$-point $\underline B$ is a $K$-point of $\mathfrak{C}_r(\mathcal {N}_p(\mathfrak{g}))_M$ if and only if $(\mathcal {E}_{\Lambda _r(\underline B)}) \circ \epsilon _r)^*(M_K)$ is not free as a $K[t]/t^p$-module.

Theorem 1.13 Let $(\mathbb {G}, \mathcal {E})$ be an algebraic group of exponential type and let $M$ be a $\mathbb {G}$-module. The bicontinuous algebraic map given as the composition

(1.13.1)\begin{equation} \rho_r \circ \Lambda_r:\mathfrak{C}_r(\mathbb{G}) \stackrel{\sim}{\to} \mathfrak{C}_r(\mathbb{G}) \stackrel{\sim}{\dashrightarrow} V_r(\mathbb{G}_{(r)}) \end{equation}

commutes with the gradings of $\mathfrak{C}_r(\mathbb {G})$ given by (1.11.2) and of $V_r(\mathbb {G}_{(r)})$ given by (1.11.1).

Furthermore, $\rho _r \circ \Lambda _r$ satisfies the following properties.

1. (1) For all $\mathbb {G}$-modules $M$, $\rho _r \circ \Lambda _r$ determines a bicontinuous algebraic map

   (1.13.2)\begin{equation} \rho_r \circ \Lambda_r: \mathbb{P}\mathfrak{C}_r(\mathbb{G}) \stackrel{\sim}{\dashrightarrow} \mathbb{P} V_r(\mathbb{G}_{(r)}), \quad \text{restricting to}\ \ \mathbb{P}\mathfrak{C}_r(\mathbb{G})_M \stackrel{\sim}{\dashrightarrow} \mathbb{P} V_r(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}} \end{equation}
   where $M_{|\mathbb {G}_{(r)}}$ denotes the restriction of $M$ to $\mathbb {G}_{(r)}$.

2. (2) If $M$ is finite dimensional, then $\rho _r \circ \Lambda _r$ restricts to a universal homeomorphism from $\mathbb {P}\mathfrak{C}_r(\mathbb {G})_M \subset \mathbb {P}\mathfrak{C}_r(\mathbb {G})$ to $\mathbb {P} V_r(\mathbb {G}_{(r)})_{M_{|\mathbb {G}_{(r)}}} \subset \mathbb {P} V_r(\mathbb {G}_{(r)})$.

3. (3) The adjoint action of $\mathbb {G}$ on $\mathfrak{g}$ induces a $\mathbb {G}(K)$-action on the $K$-points of $\mathbb {P} \mathfrak{C}(\mathbb {G})$ for any field extension $K/k$. This action stabilizes the $K$-points of $\mathbb {P}\mathfrak{C}_r(\mathbb {G})_M$ for any $\mathbb {G}$-module $M$.

4. (4) The composition $\Psi \circ (\rho _r \circ \Lambda _r): \mathbb {P} \mathfrak{C}_r(\mathbb {G}) \simeq \mathbb {P} |\mathbb {G}_{(r)}|$ is $\mathbb {G}(K)$ equivariant when evaluated at $K$-points for any field extension $K/k$.

2. Support theories $M \mapsto \Pi (\mathbb {G})_M, M \mapsto \mathfrak{C}(\mathbb {G})_M$

In this section, we consider support theories for $\mathbb {G}$-modules for a linear algebraic group. Our first theory, $M \mapsto \Pi (\mathbb {G})_M$ is a natural extension of the $\pi$-point support theory for finite group schemes recalled in Recollection 1.5. Although simple to define and good for establishing general properties, $M \mapsto \Pi (\mathbb {G})_M$ does not lend itself to computation. With this in mind, we also consider the natural extension $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ of the exponential theory for Frobenius kernels of linear algebraic groups of exponential type formulated in Definition 1.8.

For notational convenience, we frequently denote the restriction $M_{|\mathbb {G}_{(r)}}$ of a $\mathbb {G}$-module $M$ to some Frobenus kernel $\mathbb {G}_{(r)} \subset \mathbb {G}$ by $M$.

We utilize $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ to verify various properties of $M \mapsto \Pi (\mathbb {G})_M$ for $\mathbb {G}$ a linear algebraic group of exponential type. We also use $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ in order to consider $\mathbb {G}$-modules of bounded exponential degree in Proposition 2.12.

We begin with an observation about closed embeddings of infinitesimal group schemes. This observation is contrary to the behavior of cohomological support varieties for finite groups. For example, if $\tau$ is a finite group and $\kappa \subset \tau$ is a $p$-Sylow subgroup, then $|\kappa | \to |\tau |$ is rarely injective.

Proposition 2.1 justifies the constructions of the following definition of $M \mapsto \Pi (\mathbb {G})_M$ as a colimit with respect to $r$ of $M \mapsto \Pi (\mathbb {G}_{(r)})_M$. One can view $M \mapsto \Pi (\mathbb {G})_M$ as a support theory for modules for the hyperalgebra

\[ k\mathbb{G} \equiv \varinjlim_r k\mathbb{G}_{(r)}. \]

Definition 2.2 Let $\mathbb {G}$ be a linear algebraic group over $k$. We define $\Pi (\mathbb {G})$ to be the topological space $\Pi (\mathbb {G}) \equiv \varinjlim _r \Pi (\mathbb {G}_{(r)})$, the colimit with the colimit topology whose connecting maps $\Pi (\mathbb {G}_{(r)}) \to \Pi (\mathbb {G}_{(r+1)})$ are given by sending a $\pi$-point $\alpha _K: K[t]/t^P \to K\mathbb {G}_{(r)}$ to its composition with the flat map $K\mathbb {G}_{(r)} \to K\mathbb {G}_{(r+1)}$ induced by the closed embedding $\mathbb {G}_{(r)} \hookrightarrow \mathbb {G}_{(r+1)}$.

For a $\mathbb {G}$-module $M$, the $\Pi$-support space $\Pi (\mathbb {G})_M$ is defined to be

\[ \Pi(\mathbb{G})_M \equiv \varinjlim_r \Pi(\mathbb{G}_{(r)})_M \subset \varinjlim_r \Pi(\mathbb{G}_{(r)}) \equiv (\mathbb{G}). \]

The fact that $\Pi (\mathbb {G}_{(r)}) \hookrightarrow \Pi (\mathbb {G}_{(r+1)})$ restricts to $\Pi (\mathbb {G}_{(r)})_M \hookrightarrow \Pi (\mathbb {G}_{(r+1)})_M$ for $M$ a finite-dimensional $\mathbb {G}_{(r+1)}$-module is immediate from the definition of the equivalence relation on $\pi$-points. For arbitrary $\mathbb {G}_{(r+1)}$-modules $M$, one appeals to [Reference Friedlander and PevtsovaFP07, Theorem 4.6] which asserts that equivalence of $\pi$-points implies strong equivalence.

We recall the definition of a mock injective module for a linear algebraic group $\mathbb {G}$ introduced in [Reference FriedlanderFri15, Definition 4.3]. The subcategory of $\mathit {Mod}(\mathbb {G})$ consisting of mock injective modules plays a key role in our formulation of stable module categories in § 4. Interesting examples of such modules, even for $\mathbb {G}_a$, are constructed in [Reference FriedlanderFri15, Reference Hardesty, Nakano and SobajeHNS17].

The properties for our support theory $M \mapsto \Pi (\mathbb {G})_M$ stated in the following theorem are the basic properties required of a theory of support. Perhaps it is worth mentioning that support theories do not determine functors from categories of $\mathbb {G}$-modules to subsets of some space. For example, one could consider a $\mathbb {G}$-module $M$ with non-trivial support and an embedding $M \hookrightarrow I$ of $M$ into an injective $\mathbb {G}$-module; in this case, there is no conceivable map from the support of $M$ to the empty set (which is the support of $I$).

As $M \to \Pi (\mathbb {G})_M$ is the colimit of $M \to \Pi (\mathbb {G}_{(r)})_{M_{|Gr}}$, the properties of $M \mapsto \Pi (\mathbb {G})_M$ stated in Theorem 2.5 are immediate consequences of the corresponding properties for each Frobenius kernel $\mathbb {G}_{(r)}$ of $\mathbb {G}$ as established in [Reference Friedlander and PevtsovaFP07] together with the definition of mock injective modules.

To establish further properties and to compute examples, we consider the colimit of the exponential support theory $M \mapsto \mathbb {P} \mathfrak{C}_r(\mathbb {G})_M$ of Definition 1.11. We observe that the natural embedding $\mathfrak{C}_r(\mathbb {G}) \hookrightarrow \mathfrak{C}_{r+1}(\mathbb {G})$ (determined by sending a $K$-point $\underline B = (B_0,\ldots,B_{r-1})$ to $(\underline B,0) \equiv (B_0,\ldots,B_{r-1},0)$ multiplies the degrees of homogeneous elements (specified by the $\mathbb {A}^1$-action given in (1.11.2)) by $p$ and, thus, induces $\mathbb {P} \mathfrak{C}_r(\mathbb {G}) \hookrightarrow \mathbb {P} \mathfrak{C}_{r+1}(\mathbb {G})$. Moreover, for $(\mathbb {G},\mathcal {E})$ an algebraic group of exponential type, one checks by inspection the equality for any $K$-point $\underline B$ of $\mathfrak{C}_r(\mathbb {G})$

(2.5.1)\begin{equation} \mathcal{E}_{\Lambda_r(\underline B)*} \circ \epsilon_r = \mathcal{E}_{\Lambda_{r+1}((\underline B,0))*} \circ \epsilon_{r+1}: K[t]/t^p \to K\mathbb{G}_K. \end{equation}

Theorem 2.6 Let $(\mathbb {G}, \mathcal {E})$ be an algebraic group of exponential type and consider the bicontinuous algebraic map $\rho _r \circ \Lambda _r: \mathbb {P}\mathfrak{C}_r(\mathbb {G}) \stackrel {\sim }{\dashrightarrow } \mathbb {P} V_r(\mathbb {G}_{(r)})$ of Theorem 1.13 for each $r > 0$. Then the diagram

(2.6.1)

commutes, where the horizontal maps are bicontinuous algebraic maps and the vertical maps are the natural embeddings. For any field extension $K/k$, (

2.6.1

) determines a $G(K)$-equivariant commutative diagram of $K$-points.

Moreover, for every $\mathbb {G}$-module $M$ and every $r > 0$, the composition $\Phi \circ \Psi \circ (\rho _r \circ \Lambda _r): \mathbb {P} \mathfrak{C}_r(\mathbb {G}) \to \Pi (\mathbb {G})$ restricts to a bijection

(2.6.2)\begin{equation} \Phi \circ \Psi\circ (\rho_r \circ \Lambda_r): \mathbb{P} \mathfrak{C}_r(\mathbb{G})_M \stackrel{\sim}{\to} \Pi(\mathbb{G}_{(r)})_M. \end{equation}

Taking colimits with respect to $r$ determines a homeomorphism derived from bicontinuous algebraic maps

(2.6.3)\begin{equation} \Phi: \mathbb{P} \mathfrak{C}(\mathbb{G}) \stackrel{\sim}{\to} \Pi(\mathbb{G}), \quad \text{restricting to} \quad \mathbb{P} \mathfrak{C}(\mathbb{G})_M \stackrel{\sim}{\to} \Pi(\mathbb{G})_M \end{equation}

for any $\mathbb {G}$-module $M$.

For each field extension $K/k$, the adjoint action of $\mathbb {G}(K)$ on the $K$-points of each $\mathbb {P}\mathfrak{C}_r(\mathbb {G})$ determines an adjoint action of $\mathbb {G}(K)$ on the $K$-points of $\mathbb {P}\mathfrak{C}(\mathbb {G})$.

To formulate certain functoriality properties (with respect to $\mathbb {G}$) of $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$, we require the following definition.

The following proposition is complementary to Proposition 2.8. One major difference is $F^r: \mathbb {G} \to \mathbb {G}$ is far from smooth; in fact, its associated differential map on Lie algebras is the 0-map. Another difference is that Proposition 2.10 gives an explicit relationship between elements of the support varieties of $M^{[r]}$ and of $M$.

Proof. The action $(\mathcal {E}_B)_*(u_{r+s})$ on $M_K^{[r]}$ is given by the composition

\[ M_K \to M_K \otimes K[\mathbb{G}] \stackrel{1\otimes F^r}{\to} M_K \otimes K[\mathbb{G}] \stackrel{(\mathcal{E}_B)^*}{\to} M_K \otimes K[t] \stackrel{1 \otimes u_s}{\to} M_K. \]

Our condition on $(\mathbb {G},\mathcal {E})$ as the restriction (defined over $\mathbb {F}_{p^r}$) of the exponential structure on $GL_N$ implies that the composition $F^r \circ \mathcal {E}_B: \mathbb {G}_a \to \mathbb {G} \to \mathbb {G}$ equals $\mathcal {E}_{B^{(r)}} \circ F^r: \mathbb {G}_a \to \mathbb {G}_a \to \mathbb {G}$. (See [Reference FriedlanderFri15, Proposition 1.11].) Thus, the equality of the actions of $(\mathcal {E}_{B^{(r)}})_*(u_s)$ on $M_K$ and $(\mathcal {E}_B)_*(u_{r+s})$ on $M^{[r]}_K$ and the triviality of the actions of $(\mathcal {E}_B)_*(u_{s})$ on $M^{[r]}_K$ for $s < r$ both follow from the identification of $u_{s}$ with $F^r_*(u_{r+s})$.

This immediately implies the identification of $K$-points of $\mathfrak{C}(\mathbb {G})_{M^{[r]}}$ with $\delta _r^{-1}(\mathfrak{C}(\mathbb {G})_M)$ and, thus, the asserted identification of $\mathbb {P} \mathfrak{C}(\mathbb {G})_{M^{[r]}}$.

The following is an immediate consequence of Proposition 2.10 and Theorem 2.5(3) and (7).

Let $\mathbb {G}$ be a linear algebraic group of exponential type. A $\mathbb {G}$-module $M$ has exponential degree $< p^r$ if every $u_s \in K\mathbb {G}_a$ with $s \geq r$ acts trivially on $\mathcal {E}_B^*(M)$ for every $B$ a geometric point of $\mathcal {N}_p(\mathfrak{g})$ (see [Reference FriedlanderFri15, Definition 4.5]).

As we show in Proposition 2.12(1), $\mathbb {G}$-modules of bounded exponential degree have a strong condition on their $\mathbb {P} \mathfrak{C}_r(\mathbb {G})$-support. Observe that other $\mathbb {G}$-modules also satisfy this condition. For example, if $M$ has exponential degree $< p^r$ and if $M \hookrightarrow J$ is an embedding of $M$ in a mock injective $\mathbb {G}$-module $J$, then $J/M$ has the same condition on its support as does $M$ by the ‘two out of three’ property of Theorem 2.5(4).

3. Examples of $M \mapsto \mathbb {P} \mathfrak{C}(\mathbb {G})_M$

In the examples that follow, we frequently encounter projectivized versions of the projection $\pi _r: \mathfrak{C}(\mathbb {G}) \to \mathfrak{C}_r(\mathbb {G})$ sending a $K$-point $(B_0,\ldots,B_n,\ldots )$ to $(B_0,\ldots,B_{r-1})$. Starting with a subspace $X \subset \mathbb {P}\mathfrak{C}_r(\mathbb {G})$, we abuse notation by writing $W = \pi _r^{-1}(X) \subset \mathbb {P}\mathfrak{C}(\mathbb {G})$ determined by the condition that a $K$-point $\underline B$ of $\mathfrak{C}(\mathbb {G})$ represents a $K$-point of $W$ if and only if $(B_0,\ldots,B_{r-1})$ represents a $K$-point of $X$. For examples, see Proposition 2.12(1).

Example 3.2 Let $\mathbb {G}$ be a split, reductive algebraic group which admits an embedding $(\mathbb {G},\mathcal {E}) \hookrightarrow (GL_N,exp)$ of exponential type. Denote by $\mathbb {B} \hookrightarrow \mathbb {G}$ the closed subgroup given by the intersection of $\mathbb {G}$ with the Borel subgroup $B_N \subset GL_N$ of upper triangular matrices. For any $s > 0$, let $St_s$ be the $\mathbb {G}$-module given by

\[ St_s \equiv L((p^s-1)\rho) = H^0(\mathbb{G}/\mathbb{B},(p^s-1)\rho)). \]

The restriction of $St_s$ to $\mathbb {G}_{(s)}$ is both irreducible and injective. Consequently, the only $K$-point of $\mathfrak{C}_s(\mathbb {G})_{St_s}$ is $0$. As $St_s$ is a block of the $\mathbb {G}$-module $k\mathbb {G}_{(s)}$, $u_r$ acts trivially on $St_s$ for $r \geq s$. Thus, $\mathbb {P} \mathfrak{C}(\mathbb {G})_{St_s} = \pi _s^{-1}(\{ \emptyset \})$; in other words, $\mathbb {P} \mathfrak{C}(\mathbb {G})_{St_s}$ is the center of the projection $\pi _s: \mathbb {P}\mathfrak{C}(\mathbb {G}) \dashrightarrow \mathbb {P} \mathfrak{C}_s(\mathbb {G})$.

Following [Reference Suslin, Friedlander and BendelSFB97b, Proposition 7.8], we can use this computation to determine $\mathbb {P}\mathfrak{C}(\mathbb {G})_{H^0(\lambda )}$ for any dominant weight $\lambda$ of the form $n\rho$.

Example 3.3 Let $\mathbb {U}_3$ denote the Heisenberg group, the unipotent radical of a split Borel subgroup of $GL_3$. The computations in [Reference FriedlanderFri19] give the identification $\mathfrak{C}_r(\mathbb {U}_3) = Y_r \times \mathbb {A}^r \subset \mathbb {A}^{2r} \times \mathbb {A}^1,$ where $Y_r$ is the closed subvariety of $\mathbb {A}^{2r}$ with generators $x_{1,2}(\ell ),\ x_{2,3}(\ell ^\prime ),\ 0 \leq \ell,\ \ell ^\prime < r$ subject to the relations $x_{1,2}(\ell )\cdot x_{2,3}(\ell ^\prime ) = x_{2,3}(\ell )\cdot x_{1,2}(\ell ^\prime ), \ 0 \leq \ell < \ell ^\prime$. In fact, $Y_r$ is shown to be a rational variety of dimension $r+1$, smooth outside of the origin [Reference FriedlanderFri19, Proposition 5.2].

Thus, $\mathbb {P} \mathfrak{C}_r(\mathbb {U}_3) = (\mathbb {P} Y_r) \# \mathbb {P}^{r-1} \subset \mathbb {P}^{3r-1}$. Here, the linear join $(\mathbb {P} Y_r) \# \mathbb {P}^{r-1}$ of the disjoint closed subvarieties $\mathbb {P} Y_r,\ \mathbb {P}^{r-1}$ of $\mathbb {P}^{3r-1}$ is the closed subvariety whose points are points in $\mathbb {P}^{3r-1}$ lying on some projective line from a point on $\mathbb {P} Y_r$ to a point on $\mathbb {P}^{r-1}$. Furthermore, the colimit $\mathbb {P} \mathfrak{C}(\mathbb {U}_3)$ is identified with $(\mathbb {P} Y_\infty ) \# \mathbb {P}^\infty$.

Observe that the restriction to $\mathfrak{C}_r(\mathbb {U}_3)$ of the adjoint action of $\mathbb {U}_3$ on $\mathcal {N}_p(\mathfrak{u}_3)^{\times r}$ is non-trivial. As the adjoint action must stabilize subsets of the form $\mathbb {P}\mathfrak{C}(\mathbb {U}_3)_M$ for a $\mathbb {U}_3$-module $M$ by Theorem 1.13(3), this constrains which subspaces are of the form $\mathbb {P}\mathfrak{C}(\mathbb {U}_3)_M$.

One class of examples of $\mathbb {U}_3$-modules is given by inflation $pr^*(N)$ of $\mathbb {G}_a^{\times 2}$-modules $N$ along the projection homomorphism $pr: \mathbb {U}_3 \to \mathbb {G}_a^{\times 2}$. This projection induces the projection map

\[ p_r: \mathbb{P}\mathfrak{C}_r(\mathbb{U}_3) \simeq (\mathbb{P} Y_r) \# \mathbb{P}^{r-1} \dashrightarrow \mathbb{P} Y_r \hookrightarrow \mathbb{P}^{2r-1}. \]

Consequently, for any (Zariski) closed subspace $W \subset \mathbb {P} Y_r$, we may realize $p_r^{-1}(W) \subset \mathbb {P}\mathfrak{C}_r(\mathbb {U}_3)$ as $\mathbb {P}\mathfrak{C}(\mathbb {U}_3)_{p_r^*(N)}$ for some finite-dimensional $\mathbb {U}_3/Z$-module $N$ by Example 3.1.

Interesting infinite-dimensional examples are given by induction from the central subgroup $\mathbb {G}_a \simeq Z \subset \mathbb {U}_3$. For example, let $M = \operatorname{\textit{ind}}\nolimits _Z^{\mathbb {U}_3} k \simeq k[\mathbb {U}_3/Z]$. The action of $\mathbb {U}_3$ on $M$ is given by the projection to $\mathbb {U}_3/Z$ followed by the left regular representation of $\mathbb {U}_3/Z$. Thus, if $\mathcal {E}_{\underline B} \in \mathfrak{C}_r(\mathbb {U}_3)$ factors through $Z$, then the action of $(\mathcal {E}_{\underline B})_*(u_{r-1})$ on $M$ is trivial and, therefore, $\underline B \in \mathfrak{C}_r(\mathbb {U}_3)$ lies in $\mathfrak{C}_r(\mathbb {U}_3)_M$. Otherwise, the composition $pr \circ \mathcal {E}_{\underline B}: \mathbb {G}_a \to \mathbb {U}_3 \to \mathbb {U}_3/Z$ represents a non-zero point of $\mathfrak{C}_r(\mathbb {U}_3/Z)$; because $\mathfrak{C}(\mathbb {U}_3/Z)_{k[\mathbb {U}_3/Z]} = 0$, the action at the $1$-parameter subgroup $\mathcal {E}_{\underline B} \to \mathbb {U}_3$ on $M$ can be identified with the action at $p_r \circ \mathcal {E}_{\underline B} \to \mathbb {U}_3/Z$ for the regular representation of $\mathbb {U}_3/Z$ which is free. We conclude that $Z \hookrightarrow \mathbb {U}_3$ induces a homeomorphism $\mathbb {P}\mathfrak{C}(Z) \stackrel {\sim }{\to } \mathbb {P}\mathfrak{C}(\mathbb {U}_3)_M$.

Example 3.4 Let $\mathbb {G}$ be a simple algebraic group over an algebraically closed field $k$ with Coxeter number $h< p$. Assuming the Lusztig character formula is valid for all restricted dominant weights $\lambda$, Drupieski, Nakano and Parshall showed in [Reference Drupieski, Nakano and ParshallDNP12, Theorem 3.3] for any restricted dominant weight $\lambda$ that

\[ \mathfrak{C}_1(\mathbb{G})_{L(\lambda)} = \mathbb{G} \cdot \mathfrak{u}_{J(\lambda)}, \]

where $\mathfrak{u}_{J(\lambda )}$ is the Lie algebra of the unipotent radical of a parabolic subgroup $\mathbb {P}_{J(\lambda )}$ explicitly determined by $\lambda$ and $L(\lambda )$ is the irreducible $\mathbb {G}$-module of high-weight $\lambda$.

For $\mathbb {G}$ a classical simple algebraic group of rank $\ell$ satisfying $p > {\ell ^2}/{2} + 1$, Sobaje in [Reference SobajeSob13] explicitly computed $\mathfrak{C}_r(\mathbb {G})_{L(\lambda )}$ for any dominant weight $\lambda = \sum _{i \geq 0}p^i\lambda _i$ (with each $\lambda _i$ a restricted dominant weight). The form of Sobaje's determination is

\[ \mathbb{P}\mathfrak{C}_r(\mathbb{G})_{L_\lambda} = \{ \underline B \in \mathbb{P}\mathfrak{C}_r(\mathbb{G}): B_i \in \mathbb{G}\cdot \mathfrak{u}_{J(\lambda_i)} \}. \]

Assume further that $\lambda = \sum _{i=0}^m p^i\lambda _i$ satisfies the condition $2\sum _{j=1}^\ell \langle \lambda _i,\omega _j^\vee \rangle < p(p-1)$ for each $i$. Then, as argued in [Reference FriedlanderFri15, Proposition 5.1], we can identify the $K$-points of $\mathfrak{C}(\mathbb {G})_M$ for $M = H^0(\lambda _0) \otimes H^0(\lambda _1) \otimes \cdots \otimes H^0(\lambda _m)$ as those $K$-points $\mathcal {E}_{\underline B}$ of $\mathfrak{C}(\mathbb {G})$ such that $B_i$ is a $K$-point of $\mathbb {G}\cdot \mathfrak{u}_{J(\lambda _i)}$.

4. Stable module categories

For a finite group scheme $G$, a natural domain for support theory is the stable module category $\mathit {StMod}(G)$, with triangulated category structure introduced by Happel [Reference HappelHap88]. This triangulated structure utilizes the fact that the group algebra $kG$ is self-injective, implying that projective objects are the same as injective objects in the abelian category $\mathit {Mod}(G)$. Our primary objective in this section is to introduce in Definition 4.7 the triangulated categories $\mathit {StMod}(\mathbb {G})$ and $\mathit {stmod}(\mathbb {G})$ associated to the triangulated structures of the homotopy categories of (cochain) complexes $\mathcal {K}^b(\mathit {Mod}(\mathbb {G}))$ and $\mathcal {K}^b(mod(\mathbb {G}))$.

We begin by recalling various constructions associated to the abelian category $\mathit {Mod}(G)$ of $G$-modules and the full abelian subcategory $mod(G)$ of finite-dimensional $G$-modules for an affine group scheme $G$ over $k$. We denote by $CH(\mathit {Mod}(G))$ the abelian category of (cochain) complexes of $G$-modules and by $\mathcal {K}(\mathit {Mod}(G))$ the associated homotopy category with respect to cochain homotopy equivalence. The additive category $\mathcal {K}(\mathit {Mod}(G))$ has the natural structure of a triangulated category with basic exact triangles $C^\bullet \to D^\bullet \to E^\bullet \to C^\bullet [1]$ associated to short exact sequences $0 \to C^\bullet \to D^\bullet \to E^\bullet \to 0$ in $CH(\mathit {Mod}(G))$. We are primarily interested in the analogous structures $CH^b(\mathit {Mod}(G)),\ \mathcal {K}^b(\mathit {Mod}(G))$ whose objects are bounded complexes of $G$-modules and their full subcategories $CH^b(mod(G)),\ \mathcal {K}^b(mod(G))$ of bounded complexes of finite-dimensional $G$-modules,

We say that a bounded complex $C^\bullet$ of $G$-modules in $CH^b(\mathit {Mod}(G))$ has length $\leq d$ if there exists some integer $m$ such that $C^i = 0$ for $i < m$ or $i > m+d$. In particular, complexes $M[n]$ which have the $G$-module $M$ in some degree $n$ and 0 in all other degrees have length 0.

We recall that the stable module category $\mathit {StMod}(G)$ of a finite group scheme $G$ is the additive category whose objects are $G$-modules and whose abelian group of maps $\mathit {Hom}_{\mathit {StMod}(G)}(M,N)$ is the quotient of $\mathit {Hom}_{\mathit {Mod}(G)}(M,N)$ by the subgroup of maps $M \to N$ which factor through an injective $G$-module. One equips $\mathit {StMod}(G)$ with a triangulated structure in which short exact sequences in $\mathit {Mod}(G)$ determine exact triangles in $\mathit {StMod}(G)$. The +1 shift for this triangulated structure is given by sending $M$ to the cokernel $\Omega ^{-1}(M)$ of a minimal embedding $M \hookrightarrow I_M$ with $I_M$ an injective $G$-module, whereas the $-$1 shift is given by sending $M$ to the kernel $\Omega ^1(M)$ of a minimal surjection $P_M \twoheadrightarrow M$ with $P_M$ a projective $G$-module. We define $\mathit {stmod}(G) \subset \mathit {StMod}(G)$ to be the tensor triangulated subcategory whose objects are finite-dimensional $G$-modules.

If $\mathfrak{C} \hookrightarrow \mathfrak{D}$ is an embedding of triangulated categories (which implies by definition that the image of $\mathfrak{C}$ is a full subcategory of $\mathfrak{D}$), then the Verdier quotient $\mathfrak{D} \to \mathfrak{D}/\mathfrak{C}$ is the universal triangulated map from $\mathfrak{D}$ to a triangulated category $\mathcal {E}$ with kernel containing $\mathfrak{C}$. If $\mathfrak{C}$ is a thick subcategory (i.e. containing any summand in $\mathfrak{D}$ of an object of $\mathfrak{C}$), then the kernel of $\mathfrak{D} \to \mathfrak{D}/\mathfrak{C}$ equals $\mathfrak{C}$. The Verdier quotient is constructed as the category whose objects are the same as objects of $\mathfrak{D}$ and whose maps $X \to Y$ in $\mathfrak{D}/\mathfrak{C}$ are equivalence classes of ‘roofs’: triples $X \leftarrow E \to Y$ in $\mathfrak{D}$ such that the cone of $E \to X$ is an object of $\mathfrak{C}$. We refer the reader to [Reference NeemanNee01] for a detailed discussion of Verdier quotients of triangulated categories.

Granted the construction of the Verdier quotient, we readily observe that $\mathfrak{D} \to \mathfrak{D}/\mathfrak{C}$ is the localization $\mathfrak{D} \to \mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$, where $mor_\mathfrak{C}$ is the class of maps $X \to Y$ in $\mathfrak{D}$ whose cone lies in $\mathfrak{C}$. Namely, $mor_\mathfrak{C}$ is a ‘multiplicative system’ as considered for example in [Reference WeibelWei94, 10.3.4]. As $\mathfrak{D} \to \mathfrak{D}/\mathfrak{C}$ sends each $X \to Y$ in $mor_\mathfrak{C}$ to an invertible map, $\mathfrak{D} \to \mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$ is well defined and factors $\mathfrak{D} \to \mathfrak{D}/\mathfrak{C}$; on the other hand, granted the existence of $\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$, the fact that $\mathfrak{C}$ lies in the kernel of $\mathfrak{D} \to \mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$ provides the inverse functor $\mathfrak{D}/\mathfrak{C} \to \mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$ factoring $\mathfrak{D} \to \mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$.

We remind the reader that the bounded derived category of an abelian category $\mathcal {A}$, $D^b(\mathcal {A})$, is the Verdier quotient of the homotopy category $\mathcal {K}^b(\mathcal {A})$ of bounded chain complexes of $\mathcal {A}$ by the thick triangulated subcategory $Quasi^b(\mathcal {A})$ of bounded complexes quasi-isomorphic to 0 (i.e. which are exact). If $\mathcal {A}$ has enough injective objects (for example, if $\mathcal {A}$ is the category of $G$-modules for an affine group scheme), then $D^b(\mathcal {A})$ can be defined more simply by first explicitly defining the derived category $D^+(\mathcal {A})$ of complexes bounded below using injective resolutions and then restricting to objects which are bounded complexes. If $f: I^\bullet \to J^\bullet$ is a quasi-isomorphism (i.e. induces an isomorphism in cohomology) of bounded complexes of injective $G$-modules, then $f$ is a chain homotopy equivalence. Consequently, the homotopy category $\mathcal {K}^b(\mathit {Inj}(\mathit {Mod}(G)) )$ of bounded complexes for the full additive subcategory $\mathit {Inj}(\mathit {Mod}(G)) \subset \mathit {Mod}^b(G)$ embeds as a full subcategory of $D^b(\mathit {Mod}(G))$.

The following well-known theorem (see, for example, [Reference RickardRic89, Theorem 2.1]) provides motivation for our constructions. We recall that a perfect complex of $G$-modules for a finite group scheme $G$ is a bounded complex of finite-dimensional $G$-modules quasi-isomorphic to a bounded complex of finite-dimensional injective $G$-modules. Thus, the subcategory $\mathit {Perf}(G) \subset D^b(mod(G))$ of perfect complexes is the essential image in $D^b(mod(G))$ of the homotopy category of bounded complexes of finite-dimensional injective $G$-modules, $\mathcal {K}^b(\mathit {Inj}(mod(G))) \subset D^b(mod(G))$.

Theorem 4.1 If $G$ is any finite group scheme, then the natural map $mod(G) \to \mathcal {K}^b(mod(G))$ induces an equivalence of tensor triangulated categories

\[ \Psi: \mathit{stmod}(G) \stackrel{\sim}{\to} D^b(mod(G))/\mathit{Perf}(G). \]

This theorem is somewhat surprising, even in the case of $\mathbb {G}_{a(1)}$ (i.e. $k[t]/t^p$-modules). Consider the complex $C^\bullet$ with $C^0 = k[t]/t^p$, $C^1 = k[t]/t^p$ and $d: C^0 \to C^1$ given by multiplication by $t$; thus, $C^\bullet \not = 0$ in $D^b(mod(G))$ but is 0 in $\mathit {stmod}(G)$. As a possible insight into the proof of Theorem 4.2, the construction of $\tilde \Phi : D^b(mod(G)) \to \mathit {stmod}(G)$ applied to the complex $C^0 \stackrel {t}{\to } C^1$ gives the cone (with respect to the triangulated structure of $\mathit {stmod}(G)$) of $\tilde \Phi (C^0) \to \tilde \Phi (C^1)$ which is 0 in $\mathit {stmod}(G)$.

We shall utilize the truncation functors

\[ \tau_{\leq n}, \tau_{\geq m}: CH(\mathit{Mod}(G)) \to CH(\mathit{Mod}(G)),\quad C^\bullet \mapsto \tau_{\leq n}(C^\bullet) \hookrightarrow C^\bullet,\quad C^\bullet \twoheadrightarrow \tau_{\geq m}(C^\bullet). \]

Here, $\tau _{\leq n}(C^\bullet )$ is the subcomplex of $C^\bullet$ such that $\tau _{\leq n}(C^\bullet )^i = 0$ for $i > n$, $\tau _{\leq n}(C^\bullet )^n = ker\{d^n\}$, and $\tau _{\leq n}(C^\bullet )^i = C^i$ for $i < n$; and $\tau _{\geq m}(C^\bullet )$ is the quotient complex of $C^\bullet$ such that $\tau _{\geq m}(C^\bullet )^n = 0$ for $n < m$, $\tau _{\geq m}(C^\bullet )^m = C^m/im\{ d^{m-1}\}$, and $\tau _{\geq m}(C^\bullet )^n = C^n$ for $n > m$. Thus defined, if $C^\bullet \to D^\bullet$ is a quasi-isomorphism, then $\tau _{\leq n}(C^\bullet ) \to \tau _{\leq n}(D^\bullet )$ and $\tau _{\geq m}(C^\bullet ) \to \tau _{\geq m}(D^\bullet )$ are also quasi-isomorphisms.

In the following theorem, we consider the thick tensor triangulated subcategory $\mathcal {I}\mathit {nj}^b (\mathit {Mod}(G)) \subset D^b(\mathit {Mod}(G))$ of complexes quasi-isomorphic to bounded complexes of injective $G$-modules. This is the ‘essential image’ of $\mathcal {K}^b(\mathit {Inj}(\mathit {Mod}(G))) \subset D^b(\mathit {Mod}(G))$.

Theorem 4.2 (see [Reference RickardRic89, Theorem 2.1])

For any finite group scheme $G$, the natural map $\mathit {Mod}(G) \to \mathcal {K}^b(\mathit {Mod}(G))$ (sending the $G$-module $M$ to the chain complex $M[0]$ concentrated in degree $\textit{0}$) induces an equivalence of tensor triangulated categories

\[ \Psi: \mathit{StMod}(G) \stackrel{\sim}{\to} D^b(\mathit{Mod}(G))/\mathcal{I}\mathit{nj}^b(\mathit{Mod}(G)). \]

Furthermore, we construct $\tilde \Phi : D^b(\mathit {Mod}(G)) \to \mathit {StMod}(G)$ inducing the inverse

(4.2.1)\begin{equation} \Phi = \Psi^{-1}: D^b(\mathit{Mod}(G))/\mathcal{I}\mathit{nj}^b(\mathit{Mod}(G)) \stackrel{\sim}{\to} \mathit{StMod}(G) \end{equation}

sending $M[n]$ to $\Omega ^{-n}(M)$.

Proof. The composition $\mathit {Mod}(G) \to \mathcal {K}^b(\mathit {Mod}(G)) \to D^b(\mathit {Mod}(G))/\mathit {Inj}^b(\mathit {Mod}(G))$ is an additive functor; this functor sends any map $f: M \to N$ in $\mathit {Mod}(G)$ which factors through an injective $G$-module to the zero map in $D^b(\mathit {Mod}(G))/\mathit {Inj}^b(\mathit {Mod}(G))$. Thus, this composition determines the additive functor $\Psi : \mathit {StMod}(G) \to D^b(\mathit {Mod}(G))/\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))$.

As explained in Rickard's proof of [Reference RickardRic89, Theorem 2.1], an exact triangle $X \to Y \to Z \to X[1]$ in the triangulated category $\mathit {StMod}(G)$ arises from a pushout diagram in $\mathcal {K}^b(\mathit {Mod}(G))$ from the short exact sequence $0 \to X \to I \to X[1] \to 0$ (with $I$ injective) along $X \to Y$ to the short exact sequence $0\to Y \to Z \to X[1] \to 0$. Consequently, the result of applying $\Psi$ to such an exact triangle in $\mathit {StMod}(G)$, $\Psi (X) \to \Psi (Y) \to \Psi (Z) \to \Psi (X[1])$, is isomorphic in $D^b(\mathit {Mod}(G))/\mathit {Inj}^b(\mathit {Mod}(G))$ to the image of the exact triangle in $\mathcal {K}^b(\mathit {Mod}(G))$ arising from the short exact sequence $0 \to X \to Y\oplus I \to ( Y\oplus I)/X \to 0$ in $\mathit {Mod}(G)$. As the functor $\mathit {Mod}(G) \to \mathcal {K}^b(\mathit {Mod}(G))$ sends short exact sequences to exact triangles, $\Psi$ is exact (in other words, it sends exact triangles to exact triangles). Necessarily, $\Psi (\Omega ^{n}(M)) = M[-n] \in D^b(\mathit {Mod}(G))/\mathit {Inj}^b(\mathit {Mod}(G))$.

To show that $\Psi$ is a full functor, we first show that $\mathit {Mod}(G) \to D^b(\mathit {Mod}(G))$ is full, then observe that $D^b(\mathit {Mod}(G)) \to D^b(\mathit {Mod}(G))/\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))$ is full, so that $\Psi$ must also be full. Let $M, N$ be $G$-modules and consider a map from $M[0]$ to $N[0]$ in $D^b(\mathit {Mod}(G))$ represented by a ‘roof’ $M[0] \leftarrow C^\bullet \to N[0]$ in $\mathcal {K}^b(\mathit {Mod}(G))$. This roof factors through the quotient map $C^\bullet \twoheadrightarrow \tau _{\geq 0}(C^\bullet )$ (which is a quasi-isomorphism). On the other hand, $M \simeq ker\{ d^0\}/im\{d^{-1}\} \hookrightarrow \tau _{\geq 0}(C^\bullet )^0$. Thus, there is a natural map of roofs

\[ (M[0] \stackrel={\leftarrow} M[0] \to N[0]) \to (M[0] \leftarrow \tau_{\geq 0}(C^\bullet) \to N[0]), \]

thereby establishing the fullness of $\mathit {Mod}(G) \to D^b(\mathit {Mod}(G))$.

We next show that $\mathit {Hom}_{D^b(\mathit {Mod}(G))}(M,N) \to \mathit {Hom}_{D^b(\mathit {Mod}(G))/\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))}(M,N)$ is surjective. Consider a roof of maps $M[0] \leftarrow C^\bullet \to N[0]$ in $D^b(\mathit {Mod}(G))$, with associated exact triangle $J^\bullet \to C^\bullet \to M[0]$ in $D^b(\mathit {Mod}(G))$ for some bounded complex $J^\bullet$ of injective $G$-modules. Replacing $C^\bullet$ by the mapping cylinder of $J^\bullet \to C^\bullet$ if necessary, we may assume that $J^\bullet \hookrightarrow C^\bullet$ so that there is an isomorphism $C^\bullet \to ((C^\bullet /J^\bullet ) \oplus J^\bullet )$ and $C^\bullet \to M[0]$ factors as $C^\bullet \to ((C^\bullet /J^\bullet ) \oplus J^\bullet ) \to M[0]$ with $C^\bullet /J^\bullet \to M[0]$ an isomorphism in $D^b(\mathit {Mod}(G))$. Thus, the given roof is equal as a map in $D^b(\mathit {Mod}(G))/\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))$ to the roof $M[0] \leftarrow C^\bullet /J^\bullet \to N[0]$ which is the type considered in the previous paragraph and, thus, is the image under $\Psi$ of a map $M \to N$ in $\mathit {Mod}(G)$.

As shown in the proof of [Reference RickardRic89, Theorem 2.1], exactness and fullness of the exact functor $\Psi$ imply that $\Psi$ is also faithful (because $\Psi (M) = 0$ for a $G$-module $M$ implies that $M = 0$ in $\mathit {StMod}(G)$). As any summand of an injective $G$-module is itself injective, we conclude that $\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))$ is a thick subcategory of $D^b(\mathit {Mod}(G))$. To check that every object $C^\bullet \in D^b(\mathit {Mod}(G))/\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))$ lies in the image of $\Psi$ (up to isomorphism), we argue by induction on the length $d$ of $C^\bullet$, recognizing that any complex of length $0$ is of the form $M[n]$ and, thus, is in the image of $\Psi$. We construct $\tilde \Phi : D^b(\mathit {Mod}(G)) \to \mathit {StMod}(G)$ (which induces $\Phi$ of (4.2.1)) with the property that $\Psi \circ \tilde \Phi$ sends $X^\bullet \in D^b(\mathit {Mod}(G))$ to its image in $D^b(\mathit {Mod}(G))/\mathcal {I}\mathit {nj}^b(\mathit {Mod}(G))$ under the quotient map. We proceed by induction, assuming that $\tilde \Phi$ has been defined on the full subcategory of $D^b(\mathit {Mod}(G))$ whose objects are quasi-isomorphic to complexes of length $\leq d$.

Let $X^\bullet \in D^b(\mathit {Mod}(G))$ be a complex of length $d+1$ with $X^{n+1} \not = 0$ and $X^i = 0$ for $i > n+1$. We consider the exact triangle in $\mathcal {K}^b(\mathit {Mod}(G))$

(4.2.2)\begin{equation} H^{n+1}(X^\bullet)[n] \to \tau_{\leq n}(X^\bullet) \to X^\bullet \to H^{n+1}(X^\bullet)[n+1], \end{equation}

and recognize that $\tau _{\leq n}(X^\bullet )$ is a complex of length $\leq d$. We extend $\tilde \Phi (H^{n+1}(X^\bullet )[n]) \to \tilde \Phi ( \tau _{\leq n}(X^\bullet ))$ to an exact triangle in $\mathit {StMod}(G)$

(4.2.3)\begin{equation} \tilde \Phi(H^{n+1}(X^\bullet)[n]) \to \tilde\Phi(\tau_{\leq n}(X^\bullet)) \to M \to \tilde\Phi(H^{n+1}(X^\bullet)[n+1]). \end{equation}

Then $\Psi$ defines an isomorphism of exact triangles in $D^b(\mathit {Mod}(G))$ from the result of applying $\Psi$ to the exact triangle (4.2.3) to the exact triangle (4.2.2). We define $\tilde \Phi (X^\bullet ) \equiv M$. In particular, $X^\bullet = \Psi (M)$ is in the image of $\Psi$.

Assume we have defined $\tilde \Phi (X^\bullet )$ for all complexes of length $d+1$ (with arbitrary $n$ as top non-vanishing degree). Let $Y^\bullet$ be another bounded complex of length $d+1$. For any map $f: X^\bullet \to Y^\bullet \in D^b(\mathit {Mod}(G))$, we define $\tilde \Phi (f): \tilde \Phi (X^\bullet ) \to \tilde \Phi (Y^\bullet )$ to be the unique map satisfying the condition that $\Psi (\tilde \Phi (f))$ equals the image of $f$ in $D^b(\mathit {Mod}(G))/\mathit {Inj}^b(\mathit {Mod}(G))$. One readily verifies that $\tilde \Phi$ so defined is a functor which preserves exact triangles and induces $\Phi$ inverse to $\Psi$.