2.1 Graded objects

In applications it is often convenient to consider gradings by objects more general than the integers, or even arbitrary abelian groups. This is because abstract stable homotopy theories (by which, following [Reference MathewMat16, 2.1], we mean presentable stable symmetric monoidal $\infty$-categories in which the tensor product commutes with colimits in each variable) are naturally ‘graded’ by their subcategories of invertible objects, their so-called Picard $\infty$-groupoids. Equivalence classes of objects in the Picard $\infty$-groupoid is the Picard groupoid, an object which naturally grades the homotopy category of the homotopy theory (that is, the latter is the $\infty$-category, and the former is its homotopy category). For instance, while the Picard groupoid of the stable homotopy category has an object $S^{n}$ for each integer $n\in \mathbb {Z}$, with automorphisms $\mathbb {Z}^{\times }$, the $K(n)$-local homotopy categories have much larger Picard groupoids, including families of ‘exotic spheres’.

We will abusively use the symbol $+$ to denote the symmetric monoidal structure on a Picard groupoid $\Gamma$, and write $0$ for the unit object.

Suppose $\mathcal {C}$ is cocomplete and symmetric monoidal under an operation $\otimes$ with unit $\mathbb {I}$. If $\otimes$ preserves colimits in each variable separately, then $\mathcal {C}_\Gamma$ has a symmetric monoidal closed structure given by the Day convolution product. Specifically, its values are given by

\[ (M \otimes N)_\gamma = {\operatorname{colim}}_{\alpha + \beta \to \gamma} M_\alpha \otimes N_\beta, \]

and the unit is given by the functor $\gamma \mapsto \coprod _{\operatorname {Hom}(\gamma , 0)} \mathbb {I}$. Making choices of representatives for all isomorphism classes $[\gamma ] \in \pi _0 \Gamma$ gives rise to a noncanonical isomorphism

\[ (M \otimes N)_{\gamma} \cong \coprod_{\{([\alpha], [\beta])\, | \, \alpha + \beta \cong \gamma\}} M_\alpha \otimes_{\operatorname{Aut}_\Gamma(0)} N_\beta. \]

For $\gamma \in \Gamma$, write $\mathbb {Z}^{\gamma }$ for the $\Gamma$-graded abelian group obtained from the $\Gamma$-graded set $\operatorname {Hom}_\Gamma (-,\gamma )$ by taking the free group levelwise. We have natural isomorphisms $\mathbb {Z}^{\alpha } \otimes \mathbb {Z}^{\beta } \to \mathbb {Z}^{\alpha + \beta }$ that determine a functor $\Gamma \to \operatorname {Pic}(\operatorname {Ab}_\Gamma )$. Let the suspension operator $\Sigma ^{\gamma }$ be the tensor product with $\mathbb {Z}^{\gamma }$, an automorphism of the category of ${{R_\star }}$-modules. There is an isomorphism $M_\delta \cong (\Sigma ^{\gamma } M)_{\gamma + \delta }$, and this extends to isomorphisms

\[ M_\gamma \cong \operatorname{Hom}_{{{R_\star}}}(\Sigma^{\gamma} {{R_\star}}, M_\star). \]

Definition 2.8 A finite $\Gamma$-graded set is a functor $I:\Gamma ^{\mathrm {op}}\to \mathrm {Set}$ such that

\[ I\cong\coprod_{i=1}^{n} \operatorname{Hom}_\Gamma(-,\gamma_i) \]

is isomorphic to a finite coproduct of representable functors $\operatorname {Hom}_\Gamma (-,\gamma _i)$, $1\leq i\leq n$. We write $|I|\cong \{1,\ldots ,n\}$ for the underlying finite set of $I$.

For example, ${{R_\star }}$ is always a graded generator of $\operatorname {Mod}_{{{R_\star }}}$. It is unlikely to be a generator of $\operatorname {Mod}_{{{R_\star }}}$ in the ordinary sense unless the $\Gamma$-graded ring ${{R_\star }}$ contains units in $R_\gamma$ for each $\gamma \in \Gamma$.

Let $\theta \colon \thinspace \Gamma \to \Gamma '$ be a homomorphism of Picard groupoids and let ${{R_\star }}$ be a $\Gamma$-graded commutative ring. The pullback functor $\theta ^{*}$ from $\Gamma '$-graded modules to $\Gamma$-graded modules has a left adjoint $\theta _!$, given by left Kan extension along $\theta$.

Proof. This is a special case of left Kan extension being symmetric monoidal for the Day convolution product, but we give a brief indication of the proof below. For $M$, $N$ objects of $\mathcal {C}_\Gamma$, we consider the following square.

The object $\theta _! (M \otimes N)$ is obtained by starting with $M \otimes N\colon \thinspace \Gamma ^\mathrm {op} \times \Gamma ^\mathrm {op} \to \mathcal {C}$ and taking Kan extension along the two functors in the upper-right portion of the square. Because the tensor product preserves colimits in each variable, the composite Kan extension of $M \otimes N$ along the lower-left portion of the square is canonically isomorphic to $(\theta _! M) \otimes (\theta _! N)$. The natural isomorphism making the square commute determines a natural isomorphism between these two composites. Similar diagrams show that when $\theta$ preserves the unit and is compatible with the associativity, symmetry, and unit isomorphisms, $\theta _!$ does the same.

In particular, the ring ${{R_\star }}$ gives rise to a $\Gamma '$-graded ring $(\theta _! R)_\star$ defined by the formula

\[ (\theta_! R)_{\gamma'}=\operatorname{colim}_{\gamma' \to \theta (\gamma)} R_\gamma. \]

Moreover, an ${{R_\star }}$-module $M_\star$ determines an $(\theta _! R)_\star$-module $(\theta _! M)_\star$.

We also have the notion of a $\theta$-graded ring map ${{R_\star }}\to {{R'_\star }}$, which is just a $\Gamma '$-graded ring map $\theta _! {{R_\star }}\to {{R'_\star }}$. Given a $\theta$-graded ring map ${{R_\star }}\to {{R'_\star }}$, we obtain a functor

\[ (-)\otimes_{{{R_\star}}} {{R'_\star}}\colon\thinspace \operatorname{Mod}_{{R_\star}}\longrightarrow\operatorname{Mod}_{{{R'_\star}}} \]

which sends the ${{R_\star }}$-module $M_\star$ to the ${{R'_\star }}$-module $M'_\star :=M_\star \otimes _{{{R_\star }}} {{R'_\star }}$ defined by

\[ M'=(\theta_! M)_\star\otimes_{(\theta_! R)_\star} {{R'_\star}}. \]

Here the tensor product on the right is the usual base-change along a $\Gamma '$-graded ring map.

As in Definition 2.1, if $\mathcal {A}$ is a symmetric monoidal category, we write $\operatorname {Pic}(\mathcal {A})$ for the maximal subgroupoid of $\mathcal {A}$ and refer to $\operatorname {Pic}(\mathcal {A})$ as the Picard groupoid of $\mathcal {A}$.

Proposition 2.13 Suppose $\mathcal {A}$ is an additive symmetric monoidal category with unit $\mathbb {I}$ such that the monoidal product is additive in each variable, and that we have a symmetric monoidal functor $\Gamma \to \operatorname {Pic}(\mathcal {A})$ given by $\gamma \mapsto A^{\gamma }$. Then there is a canonical additive, lax symmetric monoidal functor $\phi \colon \thinspace \mathcal {A} \to \operatorname {Ab}_\Gamma$, sending $M$ to the object $M_\star$ with

\[ M_\gamma = \operatorname{Hom}(A^{\gamma}, M). \]

In particular, $\mathbb {I}_\star$ is a $\Gamma$-graded commutative ring, and $\phi$ lifts to the category of $\mathbb {I}_\star$-modules.

Proof. Since $\mathcal {A}$ is additive, the set $\operatorname {Hom}(M,N)$ of maps from $M$ to $N$ admits an abelian group structure such that composition is bilinear. This determines the functor $\phi$. It remains to show that $\phi$ is lax symmetric monoidal.

The lax monoidal structure map sends a pair $(A^{\alpha } \to M)$ in $M_\alpha$ and $(A^{\beta } \to N)$ in $N_\beta$ to the composite determined by

\[ A^{\alpha + \beta} \xleftarrow{\sim} A^{\alpha} \otimes A^{\beta} \to M \otimes N, \]

an element in $(M \otimes N)_{\alpha + \beta }$. The natural associativity and commutativity diagrams

(together with a similar unitality diagram) reduce the proof that $\phi$ is a lax symmetric monoidal functor to the fact that $\Gamma \to \operatorname {Pic}(\mathcal {A})$ is symmetric monoidal.

Definition 2.14 Suppose $\mathcal {A}$ is an additive symmetric monoidal category such that the monoidal product is additive in each variable, and that we have a symmetric monoidal functor $\Gamma \to \operatorname {Pic}(\mathcal {A})$ given by $\gamma \mapsto A^{\gamma }$. The shift operator $\Sigma ^{\gamma }\colon \thinspace \mathcal {A} \to \mathcal {A}$ is defined by

\[ \Sigma^{\gamma} M = A^{\gamma} \otimes M. \]

We then define

\[ \operatorname{Hom}(M,N)_\gamma:=\operatorname{Hom}(\Sigma^{\gamma} M,N). \]

The notation is compatible with the shift notation for $\Gamma$-graded abelian groups, because there is a natural isomorphism $(\Sigma ^{\gamma } M)_\star \cong \Sigma ^{\gamma } (M_\star )$.

Proof. There are canonical isomorphisms $\Sigma ^{\alpha +\beta } L \cong \Sigma ^{\alpha } \Sigma ^{\beta } L$. Using this, we may define composition of graded maps by

\begin{align*} \operatorname{Hom}(\Sigma^{\alpha} M,N) \otimes \operatorname{Hom}(\Sigma^{\beta} L,M) &\longrightarrow \operatorname{Hom}(\Sigma^{\alpha} M,N) \otimes \operatorname{Hom}(\Sigma^{\alpha} \Sigma^{\beta} L,\Sigma^{\alpha} M)\\ &\longrightarrow\operatorname{Hom}(\Sigma^{\alpha + \beta} L,N). \end{align*}

This composition is associative, and the unit $\mathbb {I}_\star \to \operatorname {Hom}(M,M)_\star$ sends $f\colon \thinspace A^{\gamma } \to \mathbb {I}$ to $f \otimes \text {id}_M$.

Remark 2.16 In [Reference Hovey and StricklandHS99, § 14], a group cohomology element in $H^{3}(\pi _0 \Gamma ; \pi _1\Gamma )$ is described which obstructs our ability to make $\Gamma$-grading monoidal, in the sense of the functor $\otimes$ inducing an associative exterior product $\otimes \colon \thinspace \pi _\alpha (X) \otimes \pi _\beta (Y) \to \pi _{\alpha + \beta }(X \otimes Y)$. This group cohomology element is the unique $k$-invariant of the classifying space $B\Gamma$.

Since $\Gamma$ is assumed symmetric monoidal, $B\Gamma$ admits an infinite delooping and one can calculate that this $k$-invariant must vanish. This removes the obstruction to $\otimes$ inducing a monoidal pairing. However, this becomes replaced by a spectrum $k$-invariant

\[ \varepsilon \in H^{2}(H\pi_0 \Gamma, \pi_1\Gamma) \cong \operatorname{Hom}(\pi_0 \Gamma, \pi_1 \Gamma)[2] \]

which classifies the ‘sign rule’.

More specifically, the sign rule is equivalent to a bilinear pairing $\pi _0 \Gamma \times \pi _0 \Gamma \to \pi _1 \Gamma$ sending $\alpha , \beta \in \pi _0 \Gamma$ to the element $\varepsilon _{\alpha ,\beta } \in \pi _1\Gamma$. For $X$ and $Y$ with twist isomorphism $\tau \colon \thinspace X \otimes Y \to Y \otimes X$, $x \in \pi _\alpha X$, and $y \in \pi _\beta Y$, $\tau (x \otimes y) = \varepsilon _{\alpha ,\beta } (y \otimes x)$. (The elements $\varepsilon _{\alpha ,\beta }$ are not invariant under equivalence; the isomorphism with the group of $2$-torsion homomorphisms indicates that such Picard groupoids are determined completely by the $\varepsilon _{\alpha ,\alpha }$, together describing a $2$-torsion homomorphism $\pi _0 \Gamma \to \pi _1 \Gamma$ [Reference Johnson and OsornoJO12].)

Remark 2.17 One needs to be extremely cautious with isomorphisms between $\Gamma$-graded objects due to the sign rule. For example, a casual expression like

\[ F_{{{R_\star}}}(\Sigma^{\alpha} M_\star, \Sigma^{\beta} N_\star) = \Sigma^{\beta-\alpha} F_{{{R_\star}}}(M_\star, N_\star) \]

hides several implicit isomorphisms [Reference AdamsAda84].

2.2 Graded Azumaya algebras

We continue to fix a Picard groupoid $\Gamma$ and let ${{R_\star }}$ be a $\Gamma$-graded commutative ring with module category $\operatorname {Mod}_{{R_\star }}$.

Definition 2.21 Let $\operatorname {Cat}_{{R_\star }}$ be the $2$-category of Grothendieck abelian categories which are left-tensored over the monoidal category $\operatorname {Mod}_{{R_\star }}$: abelian categories $\mathcal {A}$ with a functor $\otimes \colon \thinspace \operatorname {Mod}_{{R_\star }} \times \mathcal {A} \to \mathcal {A}$ which preserves colimits in each variable, together with a natural isomorphism

\[ \mathbb{I} \otimes A \xrightarrow{\sim} A \]

and

\[ (M \otimes_{{{R_\star}}} N) \otimes A \xrightarrow{\sim} M \otimes (N \otimes A) \]

that respects the unit and pentagon axioms.

Morphisms in $\operatorname {Cat}_{{{R_\star }}}$ are $\operatorname {Mod}_{{R_\star }}$-linear: colimit-preserving functors $F\colon \thinspace \mathcal {A} \to \mathcal {A}'$, together with natural isomorphisms $M \otimes F(A) \to F(M \otimes A)$ that respect associativity and the unit isomorphisms. The $2$-morphisms in $\operatorname {Cat}_{{{R_\star }}}$ are natural isomorphisms of functors which commute with the tensor structure.

Remark 2.22 In particular, a left-tensored category $\mathcal {A}$ inherits suspension operators by defining $\Sigma ^{\gamma } M = (\Sigma ^{\gamma } R) \otimes M$ via the left action. This allows us to define graded function objects by

\[ F_\mathcal{A}(M, N)_\gamma = \operatorname{Hom}_\mathcal{A}(\Sigma^{\gamma} M, N). \]

This definition makes a $\operatorname {Mod}_{{{R_\star }}}$-linear category into a category enriched in ${{R_\star }}$-modules in such a way that $\operatorname {Mod}_{{{R_\star }}}$-linear functors preserve this enrichment.

Definition 2.23 The functor

\[ \operatorname{Mod}\colon\thinspace \operatorname{Alg}_{{R_\star}} \to \operatorname{Cat}_{{R_\star}} \]

sends an ${{R_\star }}$-algebra ${A_\star }$ to the category $\operatorname {Mod}_{{A_\star }}$ of right ${A_\star }$-modules in $\operatorname {Mod}_{{R_\star }}$, viewed as left-tensored over ${{R_\star }}$ via the tensor product in the underlying category $\operatorname {Mod}_{{R_\star }}$. A map ${A_\star } \to {{B_\star }}$ is sent to the functor $\operatorname {Mod}_{{A_\star }} \to \operatorname {Mod}_{{{B_\star }}}$ given by extension of scalars. (Composite ring maps have natural isomorphisms of composite functors which satisfy a coherence condition: $\operatorname {Mod}$ is a pseudofunctor.)

The following theorems have proofs which are essentially identical to their classical counterparts; for example, see [Reference IkaiIka99]. We will sketch the main points below.

Theorem 2.24 (Graded Eilenberg–Watts) The map sending an ${A_\star }$-${{B_\star }}$-bimodule $L_\star$ to the functor

\[ N_\star \mapsto N_\star \otimes_{{A_\star}} L_\star \]

determines a canonical equivalence of categories from the category ${}_{{A_\star }}\operatorname {Mod}_{{{B_\star }}}$ of ${A_\star }$-${{B_\star }}$-bimodules to the category of $\operatorname {Mod}_{{R_\star }}$-linear functors $\operatorname {Mod}_{{A_\star }} \to \operatorname {Mod}_{{{B_\star }}}$.

Proof. Functors of the form $(-) \otimes _{{A_\star }} L_\star$ are colimit-preserving and come with a natural associativity isomorphism

\[ M_\star{\otimes} _{{{R_\star}}} (N_\star{\otimes} _{{A_\star}} L_\star) \to (M_\star{\otimes} _{{{R_\star}}} N_\star) \otimes _{{A_\star}} L_\star, \]

making them maps $\operatorname {Mod}_{{A_\star }} \to \operatorname {Mod}_{{{B_\star }}}$ in $\operatorname {Cat}_{{R_\star }}$. This produces the desired functor. Conversely, any $\operatorname {Mod}_{{R_\star }}$-linear functor $G\colon \thinspace \operatorname {Mod}_{{A_\star }} \to \operatorname {Mod}_{{{B_\star }}}$ preserves the shift operators $\Sigma ^{\gamma }$ and extends to a $\Gamma$-graded functor. In particular, the action map

\[ {A_\star} \otimes_{{R_\star}}G(A_\star) \to G({A_\star} \otimes_{{R_\star}} {A_\star}) \to G({A_\star}) \]

induced by the multiplication is adjoint to a ring map ${A_\star } \to F_{{{B_\star }}}(G({A_\star }),G({A_\star }))$ making $G({A_\star })$ into an ${A_\star }$-${{B_\star }}$-bimodule. Given the canonical presentation

\[ N_\star{\cong} \operatorname{colim} \bigg( \bigoplus_{\Sigma^{\delta} {A_\star} \to \Sigma^{\gamma} {A_\star} \to N_\star} \Sigma^{\delta} {A_\star} \rightrightarrows \bigoplus_{\Sigma^{\gamma} {A_\star} \to N_\star} \Sigma^{\gamma} A_\star \bigg), \]

the two colimit-preserving functors $G$ and $(-)\otimes _{{A_\star }}G({A_\star })$ both give us naturally isomorphic presentations

\[ G(N_\star) \cong \operatorname{colim} \bigg( \bigoplus_{\Sigma^{\delta} {A_\star} \to \Sigma^{\gamma} {A_\star} \to N_\star} \Sigma^{\delta} G({A_\star}) \rightrightarrows \bigoplus_{{A_\star} \to N_\star} \Sigma^{\gamma} G(A_\star) \bigg). \]

Therefore, these two functors are canonically equivalent.

Theorem 2.25 (Graded Morita theory) Let ${A_\star }$ be an ${{R_\star }}$-algebra, and $\operatorname {Mod}_{{A_\star }}^{\mathrm {cg}}$ be the full subcategory of $\operatorname {Mod}_{{A_\star }}$ spanned by the graded generators $P_\star$. Then there are canonical pullback diagrams of categories

in which $\{{A_\star }\}$ and $\{\operatorname {Mod}_{{A_\star }}\}$ denote categories with a single object and (identity) arrow, viewed as subcategories of $\operatorname {Alg}_{{{R_\star }}}$ and $\operatorname {Cat}_{{{R_\star }}}$, respectively, and the middle vertical arrow is the functor which sends the graded generator $P_\star$ to the ${{R_\star }}$-algebra $\operatorname {End}_{{A_\star }}(P_\star )$. More generally, the fiber of $(\operatorname {Mod}^{\mathrm {\mathrm {cg}}}_{{A_\star }})^{\simeq } \to (\operatorname {Alg}_{{R_\star }})^{\simeq }$ over an algebra $B_\star$ is either empty or a principal torsor for the Picard groupoid $\operatorname {Pic}({}_{{A_\star }}\!\!\operatorname {Mod}_{{A_\star }})$ of the category of bimodules.

Proof. We will first identify $\operatorname {Mod}^{\mathrm {cg}}_{{A_\star }}$ with the right-hand fiber product. The pullback of the diagram $\operatorname {Alg}_{{{R_\star }}} \to \operatorname {Cat}_{{{R_\star }}} \leftarrow \{\operatorname {Mod}_{{A_\star }}\}$ is the category of pairs $({{B_\star }},\phi )$, where ${{B_\star }}$ is an ${{R_\star }}$-algebra and $\phi$ is an equivalence $\operatorname {Mod}_{{{B_\star }}} \to \operatorname {Mod}_{{A_\star }}$ in $\operatorname {Cat}_{{{R_\star }}}$. Such a functor is colimit-preserving, so by the graded Eilenberg–Watts theorem such a functor is represented by a certain type of pair $({{B_\star }}, P_\star )$. For this functor to be an equivalence, the graded generator ${{B_\star }}$ must map to a graded generator $P_\star$, and we must have ${{B_\star }} = \operatorname {End}_{{A_\star }}(P_\star )$. It remains to show that any such $P_\star$ determines an equivalence of categories.

Given a right ${A_\star }$-module $P_\star$ as in the statement, we obtain an ${{R_\star }}$-algebra ${{B_\star }} = F_{{A_\star }}(P_\star ,P_\star )$ and a functor $(-) \otimes _{{{B_\star }}} P_\star \colon \thinspace \operatorname {Mod}_{{{B_\star }}} \to \operatorname {Mod}_{{A_\star }}$. This functor is colimit-preserving. It also has a colimit-preserving right adjoint $F_{{A_\star }}(P_\star ,-)$ because $P_\star$ is finitely generated projective.

The unit map

\[ M_\star \to F_{{A_\star}}(P_\star, P_\star \otimes_{{{B_\star}}} M_\star) \]

is an isomorphism when $M_\star = \Sigma ^{\gamma } B_\star$. Both sides preserve colimits, and so applying this unit to a resolution $F_1 \to F_0 \to M_\star \to 0$ where $F_i$ are (graded) free modules shows that the unit is always an isomorphism.

The counit map

\[ F_{{A_\star}}(P_\star, N_\star) \otimes_{{{B_\star}}} P_\star \to N_\star \]

is an isomorphism when $N_\star = P_\star$. Because the set of objects $\Sigma ^{\gamma } P_\star$ is a set of generators there always exists a resolution $F_1 \to F_0 \to N_\star \to 0$ where $F_i$ are direct sums of shifts of $P_\star$. Again, as the functors in question preserve colimits, the counit is always an isomorphism.

We now consider the left-hand square. As pullbacks can be calculated iteratively, the pullback of a diagram ${{B_\star }} \to \operatorname {Alg}_{{{R_\star }}} \leftarrow (\operatorname {Mod}^{\mathrm {cg}}_{{A_\star }})^{\simeq }$ is equivalent to the pullback of the diagram $\{\operatorname {Mod}_{{{B_\star }}}\} \to \operatorname {Cat}_{{{R_\star }}} \leftarrow \{\operatorname {Mod}_{{A_\star }}\}$. If these categories are inequivalent as ${{R_\star }}$-linear categories, this is empty. If these categories are equivalent, then composition with any chosen equivalence makes the groupoid of ${{R_\star }}$-linear equivalences $\{\operatorname {Mod}_{{{B_\star }}}\} \to \{\operatorname {Mod}_{{A_\star }}\}$ isomorphic to the groupoid of self-equivalences of $\operatorname {Mod}_{{A_\star }}$: without making such a choice, it is a principal torsor for the groupoid of self-equivalences of $\operatorname {Mod}_{{A_\star }}$.

Equivalences of $\operatorname {Mod}_{{A_\star }}$ are given up to unique isomorphism by tensoring with an ${A_\star }$-bimodule $P_\star$, and there must exist an inverse given by tensoring with an ${A_\star }$-bimodule $Q_\star$. For these to be inverse to each other, we must have isomorphisms of ${A_\star }$-bimodules

\[ P_\star \otimes_{{A_\star}} Q_\star\, {\cong}\, Q_\star \otimes_{{A_\star}} P_\star\,{\cong}\, {A_\star}. \]

Such a $Q_\star$ exists if and only if $P_\star$ is an invertible element in the category of bimodules.

The following is a graded analogue of results of [Reference Rosenberg and ZelinskyRZ61], relating outer automorphisms to the Picard group. In the classical case of ungraded algebras over a field $k$, the Picard group of $k$-modules is trivial and so it reduces to the Noether–Skolem theorem: all automorphisms of an algebra are inner.

Corollary 2.26 (Graded Rosenberg–Zelinsky exact sequence) For an Azumaya ${{R_\star }}$-algebra ${A_\star }$, there is an exact sequence of groups

\[ 1 \to (R_0)^{{\times}} \to (A_0)^{{\times}} \to \operatorname{Aut}_{\operatorname{Alg}_{{{R_\star}}}}({A_\star}) \to \pi_0 \operatorname{Pic}(\operatorname{Mod}_{{R_\star}}). \]

The group $\operatorname {Pic}(\operatorname {Mod}_{{R_\star }})$ acts on the set of isomorphism classes of compact generators of $\operatorname {Mod}_{{A_\star }}$ with quotient the set of isomorphism classes of Azumaya ${{R_\star }}$-algebras ${{B_\star }}$ such that $\operatorname {Mod}_{{A_\star }} \simeq \operatorname {Mod}_{{{B_\star }}}$. The stabilizer of ${A_\star }$, viewed as a right ${A_\star }$-module, is the image of the outer automorphism group in $\operatorname {Pic}(\operatorname {Mod}_{{R_\star }})$.

Proof. We consider the pullback diagram of categories

obtained from graded Morita theory. This is a homotopy pullback diagram of groupoids, and so we may take the nerve and obtain a long exact sequence in homotopy groups at the basepoint ${A_\star }$ of $\operatorname {Pic}$. Put together, this gives an exact sequence

\[ \!\!\! 1 \to \operatorname{Aut}_{\operatorname{Pic}({}_{{A_\star}}\!\!\operatorname{Mod}_{{A_\star}})}({A_\star}) \to \operatorname{Aut}_{\operatorname{Mod}_{{A_\star}}}({A_\star}) \to \operatorname{Aut}_{\operatorname{Alg}_{{{R_\star}}}}({A_\star}) \to \pi_0 \operatorname{Pic}({}_{{A_\star}}\!\!\operatorname{Mod}_{{A_\star}}). \]

Moreover, the category of ${A_\star }$-bimodules is equivalent to the category of modules over ${A_\star } \otimes _{{{R_\star }}} {A_\star}^\mathrm{\!\!\!op}$, which is Morita equivalent to ${{R_\star }}$. This gives us an equivalence of categories $\operatorname {Pic}({}_{{A_\star }}\operatorname {Mod}_{{A_\star }}) \simeq \operatorname {Pic}({{{R_\star }}})$ that carries ${A_\star }$ to ${{R_\star }}$. The desired description of this exact sequence follows by identifying $\operatorname {Aut}_{\operatorname {Mod}_{{A_\star }}}({A_\star })$ with $A_0^{\times }$ and $\operatorname {Aut}_{\operatorname {Pic}({{R_\star }})}({{R_\star }})$ with $R_0^{\times }$.

Similarly, the description of the action of $\operatorname {Pic}$ follows by identifying this fiber square with the principal fibration associated to the map $(\operatorname {Alg}_{{{R_\star }}})^{\simeq } \to (\operatorname {Cat}_{{{R_\star }}})^{\simeq }$.

2.3 Matrix algebras over graded commutative rings

Definition 2.28 Let ${{R_\star }}$ be a $\Gamma$-graded commutative ring. An ${{R_\star }}$-algebra is a matrix ${{R_\star }}$-algebra if it is isomorphic to the endomorphism ${{R_\star }}$-algebra

\[ \operatorname{End}_{{{R_\star}}}(M_\star) = F_{{{R_\star}}}(M_\star,M_\star) \]

of an ${{R_\star }}$-module of the form $M_\star \cong {{R^{I}_\star }}$ for some $\Gamma$-graded set $I$ (Definition 2.3).

In general, we write $\operatorname {Mat}_{I}({{R_\star }})$ for the $\Gamma$-graded matrix algebra $\operatorname {End}_{{{R_\star }}}({{R_\star }}^{I})$ and $\operatorname {GL}_{I}({{R_\star }})$ for the group $[\operatorname {Aut}_{{{R_\star }}}({{R_\star }}^{I})]_0^{\times }$ of automorphisms of the graded ${{R_\star }}$-module ${{R_\star }}^{I}$.

Proposition 2.29 If $R_\gamma =0$ for $\gamma \neq 0$ then there is an isomorphism of groups

\[ \operatorname{GL}_{I}(R)\cong\prod_{\gamma\in\Gamma}\operatorname{GL}_{I_\gamma}(R_0), \]

where the groups on the right are the usual general linear groups of the commutative ring $R_0$.

Proposition 2.30 If $I$ is a finite $\Gamma$-graded set with underlying finite set $|I|$, meaning that $I\cong \coprod _{i\in |I|}\operatorname {Hom}(-,\gamma _i)$ for some $|I|$-indexed collection of objects $\gamma _i$ of $\Gamma$, then there is a canonical isomorphism of ${{R_\star }}$-modules

\[ \operatorname{End}_{{{R_\star}}}({{R_\star}}^{\!\!\!I}) \cong \bigoplus_{i,j \in | I |} \Sigma^{\gamma_i - \gamma_j} {{R_\star}}. \]

In particular, there is a natural $\Gamma$-graded set $\partial I$ such that it is of the form $R^{\partial I}_\star$.

Proposition 2.31 The formation of matrix algebras is compatible with base-change. That is, for any homomorphism $\theta \colon \thinspace \Gamma \to \Gamma '$ of abelian groups, any $\theta$-graded ring map ${{R_\star }}\to {{R'_\star }}$, and any finite $\Gamma$-graded set $I$, the canonical ${{R'_\star }}$-algebra map

\[ \operatorname{Mat}_{I}({{R_\star}})\otimes_{{{R_\star}}} {{R'_\star}}\longrightarrow\operatorname{Mat}_{\theta_! I}({{R'_\star}}) \]

is an isomorphism.

Proof. Write $\partial I$ for the $\Gamma$-graded set as in the previous proposition. First, let us assume that $\theta$ is the identity of $\Gamma$, so that ${{R_\star }}\to {{R'_\star }}$ is just a $\Gamma$-graded ring map. Then $\theta _! \partial I=\partial I$ and the map $R^{\partial I}_\star \otimes _{{{R_\star }}} {{R'_\star }}\to ({{R'_\star }})^{\partial I}$ is an equivalence between free ${{R'_\star }}$-modules on the same $\Gamma$-graded set.

Now suppose instead that $\theta$ is arbitrary and ${{R'_\star }}=\theta _! {{R_\star }}$. Then the desired map is a composite

\[ R{}^{\partial I}_\star \otimes_{{{R_\star}}} {{R'_\star}} \cong\theta_!(R{}^{\partial I}_\star)\cong(\theta_! {{R_\star}})^{\partial\theta_! I}. \]

Finally, an arbitrary $\theta$-graded ring map $R\to R'$ is a composite of ring maps of the type treated above, so the result follows.

2.4 Derivations and Hochschild cohomology

The following recalls some of Quillen's work on cohomology for associative rings [Reference QuillenQui70]. We suppose for simplicity that we are working in a setting in which the relevant derived functors exist, such as the case in which there are enough projectives and the tensor product of projective objects is again projective.

In a symmetric monoidal abelian category in which the symmetric monoidal ‘tensor product’ operation $\otimes$ preserves colimits (separately in each variable), any algebra $A$ sits in a short exact sequence

\[ 0 \to \Omega_A \to A \otimes A^\mathrm{op} \to A \to 0 \]

of $A$-bimodules, split (as left modules) by the unit. If $A$ is the tensor algebra on a projective object $P$, then $\Omega _A$ can be identified with the projective bimodule $A \otimes P \otimes A^\mathrm {op}$. Moreover, for any $A$-bimodule $M$ with associated square-zero extension $M \rtimes A \to A$ in $\operatorname {Alg}_A$, there are canonical isomorphisms

\[ \operatorname{Der}(A,M) = \operatorname{Hom}_{\operatorname{Alg}_A / A}(A, M \rtimes A) \cong \operatorname{Hom}_{{}_A\operatorname{Mod}_A}(\Omega_A,M). \]

This allows us to relate the derived functors of derivations, in the sense of [Reference QuillenQui70], to Hochschild cohomology in this category. The André–Quillen cohomology groups of $A$ with coefficients in $M$ may be identified with the nonabelian derived functors $\operatorname {Der}^{s}(A,M)$. Applying the right derived functors of $\operatorname {Hom}_{{}_A\operatorname {Mod}_A}(-,M)$ to the exact sequence defining $\Omega _A$ gives us isomorphisms

\[ \operatorname{Der}^{s}(A,M) \to HH^{s+1}(A,M) \]

for $s > 0$ and an exact sequence

\[ 0 \to HH^{0}(A,M) \to M \to \operatorname{Der}(A,M) \to HH^{1}(A,M) \to 0. \]

Proposition 2.32 Suppose ${A_\star }$ is an Azumaya ${{R_\star }}$-algebra. For any ${A_\star }$-bimodule $M_\star$ in the category of $\Gamma$-graded ${{R_\star }}$-modules, we have a short exact sequence

\[ 0 \to HH^{0}({A_\star}, M_\star) \to M_\star \to \operatorname{Der}({A_\star},M_\star) \to 0. \]

Both the Hochschild cohomology groups $HH^{s}_{{{R_\star }}}({A_\star }, M_\star )$ and the derived functors $\operatorname {Der}^{s}_{{{R_\star }}}({A_\star }, M_\star )$ vanish for $s > 0$.

Proof. Consider the short exact sequence

\[ 0 \to \Omega_{{A_\star}} \to {A_\star} \otimes_{{{R_\star}}} {A_\star}^\mathrm{\!\!\!op} \to {A_\star} \to 0 \]

of bimodules. The center bimodule is free, hence projective. Moreover, under the chain of Morita equivalences

\[ \operatorname{Mod}_{{{R_\star}}} \simeq \operatorname{Mod}_{\operatorname{End}_{{{R_\star}}}({A_\star})} \simeq \operatorname{Mod}_{{A_\star} \otimes_{{{R_\star}}} {A_\star}^\mathrm{\!\!\!op}}, \]

the image of the projective ${{R_\star }}$-module ${{R_\star }}$ is ${A_\star }$, and hence ${A_\star }$ is also projective. Therefore, the sequence splits and $\Omega _{{A_\star }}$ is projective too.

3.1 Gradings for ring spectra

Definition 3.1 Let $R$ be an $\mathbb {E}_\infty$-ring spectrum, with $\Gamma _R$ the algebraic Picard groupoid of invertible $R$-modules and homotopy classes of equivalences; similarly, let $\Gamma _\mathbb {S}$ be the Picard groupoid of the sphere spectrum. A grading for $R$ is a Picard groupoid $\Gamma$ together with a commutative diagram

of Picard groupoids.

The period of the grading is the minimum of the set

\[ \{n > 0 \mid \nu[S^{n}] = 0\text{ in }\pi_0 \Gamma\} \cup \{\infty\}, \]

where $[S^{n}] \in \pi _0 \Gamma _{\mathbb {S}}$ is the equivalence class of the $n$-sphere.

A grading provides a chosen lift of the suspension $\Sigma R$ to $\Gamma$ such that the twist on $\Sigma R\otimes \Sigma R$ lifts the automorphism $-1 \in (\pi _0 R)^{\times }$; it also provides an action $\Gamma _\mathbb {S} \times \Gamma \to \Gamma$, $(n,\gamma ) \mapsto n + \gamma$, compatible with that on $\Gamma _R$. The minimal and maximal options are $\Gamma _\mathbb {S}$-grading (usually referred to as ‘$\mathbb {Z}$-grading’) and $\Gamma _R$-grading (usually referred to as ‘Picard grading’). If $R$ is connective (and nontrivial) then $\pi _0 \Gamma _\mathbb {S} \to \pi _0 \Gamma _R$ is a monomorphism, and so $R$ has period $\infty$ (usually referred to as ‘not being periodic’).

Throughout this section we will assume that we have chosen a grading for $R$. This produces elements $R^{\gamma } \in \operatorname {Mod}_R$ for $\gamma \in \Gamma$ and gives the category of $R$-modules $\Gamma$-graded homotopy groups $\pi _\star M$ as in § 5.1. These homotopy groups preserve coproducts and filtered colimits, as well as take cofiber sequences to long exact sequences. The fact that weak equivalences are detected on $\mathbb {Z}$-graded homotopy groups implies the following propositions.

3.2 Picard-graded model structures

In this section we describe model structures on categories of $R$-modules and $R$-algebras based on using elements of $\operatorname {Pic}(R)$ as basic cells. The structure of this section is based on Goerss and Hopkins’ work on obstruction theory for algebras over an operad [Reference Goerss and HopkinsGH04], which in turn is based on Bousfield's work [Reference BousfieldBou03]. We carry this out under the simplifying assumptions that we are not using an auxiliary homology theory, and that the operad in question is the associative operad. However, we will remove the assumption that the base category is the stable homotopy category, and allow ourselves the use of homotopy groups graded by a Picard groupoid $\Gamma$ rather than integer-graded homotopy groups.

In this section we work in the flat stable model category structure on symmetric spectra (the $S$-model structure of [Reference ShipleyShi04]). Fix a commutative model for our $\mathbb {E}_\infty$-ring spectrum $R$, and let $\operatorname {Mod}_R^{\Delta }$ denote the (ordinary) category of $R$-module objects in symmetric spectra (which should not be confused with its underlying $\infty$-category $\operatorname {Mod}_R$). We also fix a grading $\Gamma$ for $R$ as in the previous section, giving any $R$-module $M$ natural $\Gamma$-graded homotopy groups $\pi _\star M$.

According to [Reference ShipleyShi04, 2.6–2.7], the category $\operatorname {Mod}_R^{\Delta }$ is a cofibrantly generated, proper, stable model category with generating sets of cofibrations and acyclic cofibrations with cofibrant source; it is also, compatibly, a simplicial model category (see, for example, [Reference Davis and LawsonDL14] for references in this direction). The smash product $\wedge _R$ and function object $F_R(-,-)$ give $\operatorname {Mod}_R^{\Delta }$ a symmetric monoidal closed structure under which $\operatorname {Mod}_R^{\Delta }$ is a monoidal model category, and the category $\operatorname {Alg}_R^{\Delta }$ of associative $R$-algebras is a cofibrantly generated simplicial model category with fibrations and weak equivalences detected in $\operatorname {Mod}_R^{\Delta }$ [Reference Schwede and ShipleySS00]. We let $\mathbb {T}$ denote the monad taking $M$ to the free $R$-algebra $\mathbb {T}(M) = \bigvee M^{\wedge _R n}$; algebras over $\mathbb {T}$ are associative $R$-algebras.

The following definitions are dual to those in Bousfield [Reference BousfieldBou03], taking the category $\Gamma$ as generating a class $\mathcal {P}$ of cogroup objects.

Any object $P \in \operatorname {Pic}(R)$ with a lift to an element $\gamma \in \Gamma$ is automatically $\operatorname {Pic}$-projective, and the class of projective cofibrations is closed under coproducts, suspensions, and desuspensions. There are enough $\operatorname {Pic}$-projective objects: to construct a $\operatorname {Pic}$-projective $P$ and a map $P \to X$ inducing a surjection $\pi _\star P \to \pi _\star X$, we can choose generators $\{x_\alpha \in \pi _{\gamma _\alpha } X\}$ of $\pi _\star X$ which are represented by a map $\bigvee _\alpha R^{\gamma _\alpha } \to X$. We can then describe a model structure on the category $s\operatorname {Mod}_R^{\Delta }$ of simplicial $R$-modules.

As in [Reference Goerss and HopkinsGH04, § 3], for a simplicial $R$-module $X$ and $\gamma \in \Gamma$ we have ‘natural’ homotopy groups $\pi _n^{\natural }(X;\gamma )$. On geometric realization there is a homotopy spectral sequence with $E_2$-term

\[ \pi_p \pi_\gamma(X) \Rightarrow \pi_{p+\gamma} |X|\,. \]

The $E_2$-term of this spectral sequence comes from an exact couple, the spiral exact sequence [Reference Goerss and HopkinsGH04, Lemma 3.9]:

\[ \cdots \to \pi_{n-1}^{\natural}(X;\gamma) \to \pi_n^{\natural}(X;\gamma) \to \pi_n \pi_\gamma(X) \to \pi_{n-2}^{\natural}(X;\gamma) \to \cdots \]

As applications of the $\operatorname {Pic}$-resolution model structure, we obtain $\operatorname {Pic}$-graded Künneth and universal coefficient spectral sequences.

Theorem 3.8 For $X, Y \in \mathcal {D}_R$, there are spectral sequences of $\Gamma$-graded ${{R_\star }}$-modules:

\begin{align*} \operatorname{Tor}^{{{R_\star}}}_{p,\gamma}(\pi_\star X, \pi_\star Y) &\Rightarrow \pi_{p+\gamma}(X \wedge_R Y),\\ \operatorname{Ext}^{s,\tau}_{{{R_\star}}}(\pi_\star X, \pi_\star Y) &\Rightarrow \pi_{{-}s-\tau} F_R(X,Y). \end{align*}

Proof. This follows by first observing that the Künneth formula degenerates to isomorphisms

\[ \pi_\star (P \wedge_R \cdots \wedge_R P) \cong \pi_\star P \otimes_{\pi_\star R} \cdots \otimes_{\pi_\star R} \pi_\star P, \]

and then applying $\pi _\star$ to the identification

\[ \mathbb{T}(P) \cong \bigvee_{k \geq 0} P^{\wedge_R k} \]

of $R$-modules.

The $\operatorname {Pic}$-resolution model structure on simplicial $R$-modules now lifts to $R$-algebras. The following results are originally due to Bousfield (cf. [Reference BousfieldBou03, Reference BousfieldBou89]) and Goerss and Hopkins (cf. [Reference Goerss and HopkinsGH04, Reference Goerss and HopkinsGH]), respectively; see also [Reference Pstragowski and VanKoughnettPV19] for a more recent treatment.

Given this structure, we can use Goerss and Hopkins’ moduli tower of Postnikov approximations to produce an obstruction theory. This both classifies objects and constructs a Bousfield–Kan spectral sequence for spaces of maps between $R$-algebras using $\Gamma$-graded homotopy groups. In order to describe the resulting obstruction theories, let $\operatorname {Der}^{s}_{\operatorname {Alg}_{\pi _\star R}}$ denote the derived functors of derivations in the category of $\Gamma$-graded $\pi _\star R$-algebras as in § 2.4.

Theorem 3.11 1. There are successively defined obstructions to realizing an algebra ${A_\star } \in \operatorname {Alg}_{\pi _\star R}$ by an $R$-algebra $A$ in the groups

\[ \operatorname{Der}^{s+2}_{\operatorname{Alg}_{\pi_\star R}}({A_\star}, \Omega^{s} {A_\star}), \]

and obstructions to uniqueness in the groups

\[ \operatorname{Der}^{s+1}_{\operatorname{Alg}_{\pi_\star R}}({A_\star}, \Omega^{s} {A_\star}), \]

for $s \geq 1$.

2. For $R$-algebras $X$ and $Y$, there are successively defined obstructions to realizing a map $f \in \operatorname {Hom}_{\operatorname {Alg}_{\pi _\star R}}(\pi _\star X, \pi _\star Y)$ in the groups

\[ \operatorname{Der}^{s+1}_{\operatorname{Alg}_{\pi_\star R}}(\pi_\star X, \Omega^{s} \pi_\star Y), \]

and obstructions to uniqueness in the groups

\[ \operatorname{Der}^{s}_{\operatorname{Alg}_{\pi_\star R}}(\pi_\star X, \Omega^{s} \pi_\star Y), \]

for $s \geq 1$.

3. Let $\phi \in \operatorname {Map}_{\operatorname {Alg}_R^{\Delta }}(X,Y)$ be a map of $R$-algebras. Then there is a fringed, second quadrant spectral sequence abutting to

\[ \pi_{t-s} (\operatorname{Map}_{\operatorname{Alg}_R^{\Delta}}(X,Y),\phi), \]

with $E_2$-term given by

\[ E_2^{0,0} = \operatorname{Hom}_{\operatorname{Alg}_{\pi_\star R}}(\pi_\star X, \pi_\star Y) \]

and

\[ E_2^{s,t} = \operatorname{Der}^{s}_{\operatorname{Alg}_{\pi_\star R}}(\pi_\star X, \Omega^{t} \pi_\star Y)\quad\text{for }t > 0. \]

This theorem is obtained using simplicial resolutions. Given an $R$-algebra $A$, we form a simplicial resolution of $A$ by free $R$-algebras, which becomes a resolution of $\pi _\star A$ by free $\pi _\star R$-algebras by Corollary 3.9. We get the spectral sequences for mapping spaces from the associated homotopy spectral sequence (see [Reference BousfieldBou03]). The obstruction theory for the construction of such $A$, instead, relies on constructing partial resolutions $P_n A$ as simplicial free $R$-algebras whose homotopy spectral sequence degenerates in a specific way, and then identifying the obstruction to extending the construction of $P_n(A)$ to $P_{n+1}(A)$ as lying in an André–Quillen cohomology group.

3.3 Algebraic Azumaya algebras

We now apply the obstruction theory of the previous section to the algebraic case. We continue to let $R$ be an $\mathbb {E}_\infty$-ring spectrum with a grading by $\Gamma$, and $\mathcal {D}_R$ the homotopy category of left $R$-modules.

We recall that an algebra $A$ is an Azumaya $R$-algebra if $A$ is a compact generator of $\mathcal {D}_R$, and the left-right action map $A \wedge _R A^\mathrm {op} \to \operatorname {End}_R(A)$ is an equivalence in $\mathcal {D}_R$ [Reference Baker, Richter and SzymikBRS12].

We have a similar result about Morita triviality.

We may apply the Goerss–Hopkins obstruction theory to algebraic Azumaya $R$-algebras. Much of the following is originally due to Baker, Richter, and Szymik [Reference Baker, Richter and SzymikBRS12, 6.1].

Theorem 3.15 1. Any Azumaya $\pi _\star R$-algebra is isomorphic to $\pi _\star A$ for some $\Gamma$-graded algebraic Azumaya $R$-algebra $A$.

2. Suppose $A$ is a $\Gamma$-graded algebraic Azumaya $R$-algebra. For any $R$-algebra $S$ (not necessarily Azumaya), the natural map

\[ [A, S]_{\operatorname{Alg}_R^{\Delta}} \xrightarrow{\pi_\star} \operatorname{Hom}_{\operatorname{Alg}_{\pi_\star R}} (\pi_\star A, \pi_\star S) \]

is an isomorphism. For any map $\phi \colon \thinspace A \to S$ of $R$-algebras (making $\pi _\star S$ into a $\pi _\star A$-bimodule) and any $t > 0$, we have an isomorphism

\[ \pi_t (\operatorname{Map}_{\operatorname{Alg}_R^{\Delta}} (A, S), \phi) \cong (\pi_t S)/HH^{0}(\pi_\star A, \Omega^{t} \pi_\star S). \]

3. If $A$ is a $\Gamma$-graded algebraic Azumaya $R$-algebra, the homotopy groups of the space $\operatorname {Aut}_{\operatorname {Alg}_R^{\Delta }}(A)$ satisfy

\[ \pi_t(\operatorname{Aut}_{\operatorname{Alg}_R^{\Delta}}(A), \mathrm{id}) \cong \begin{cases} \operatorname{Aut}_{\operatorname{Alg}_{\pi_\star R}}(\pi_\star A) & \text{if }t = 0,\\ \pi_t A/ \pi_t R & \text{if }t > 0. \end{cases} \]

4. If $A$ is a $\Gamma$-graded algebraic Azumaya $R$-algebra, then for $t > 0$ the sequence

\[ 0 \to \pi_t \operatorname{GL}_1(R) \to \pi_t \operatorname{GL}_1(A) \to \pi_t \operatorname{Aut}_{\operatorname{Alg}_R^{\Delta}}(A) \to 0 \]

is exact, and there is an exact sequence of potentially nonabelian groups

\[ 1 \to \pi_0 \operatorname{GL}_1(R) \to \pi_0 \operatorname{GL}_1(A) \to \pi_0 \operatorname{Aut}_{\operatorname{Alg}_R^{\Delta}}(A) \to \pi_0 \operatorname{Pic}(R). \]

The image in $\pi _0 \operatorname {Pic}(R)$ of the last map is the group of outer automorphisms of $\pi _\star A$ as a $\pi _\star R$-algebra.

Proof. The Goerss–Hopkins obstruction groups $\operatorname {Der}^{s}_{\operatorname {Alg}_{\pi _\star R}}(\pi _\star A, M)$ appearing in Theorem 3.11 vanish identically for $s > 0$ by Proposition 2.32. In particular, the obstructions to existence and uniqueness vanish, so every Azumaya $\pi _\star R$-algebra lifts to an Azumaya $R$-algebra. Moreover, the obstructions to existence and uniqueness for lifting maps also vanish, and so every map of Azumaya $\pi _\star R$-algebras lifts uniquely to a map of $R$-algebras.

We then apply the vanishing and exact sequence of Proposition 2.32 to the spectral sequence calculating the homotopy groups of $\operatorname {Map}_{\operatorname {Alg}_R^{\Delta }}(A,S)$. We find that there are short exact sequences

\[ 0 \to HH^{0}(\pi_\star A, \pi_\star S) \to \pi_\star S \to \pi_t \operatorname{Map}_{\operatorname{Alg}_R^{\Delta}}(A,S) \to 0 \]

for $t > 1$, and thus obtain the stated results on $\pi _1$ and $\pi _0$, once due caution is exercised regarding basepoints.

4.1 Closed symmetric monoidal $\infty$-categories

Recall [Reference LurieLur17, 4.8] that the $\infty$-category of $\mathcal {P}r^{\mathrm {L}}$ of presentable $\infty$-categories and colimit-preserving functors [Reference LurieLur09, 5.5.3.1] admits a symmetric monoidal structure with unit the $\infty$-category $\mathcal {S}$ of spaces. We refer to (commutative) algebra objects in this $\infty$-category as presentable (symmetric) monoidal $\infty$-categories.

Note that this implies that (the underlying $\infty$-category of) a presentable symmetric monoidal $\infty$-category $\mathcal {C}^{\otimes }$ is canonically enriched, tensored and cotensored over itself. If $\mathcal {C}$ is stable, then $\mathcal {C}$ is enriched, tensored and cotensored over $\mathrm {Sp}$, the $\infty$-category of spectra. We will not normally notationally distinguish between the internal mapping object and the mapping spectrum, which should always be clear from the context.

There is also the following multiplicative version of Morita theory.

Proposition 4.5 ([Reference LurieLur17, 7.1.2.7], [Reference Antieau and GepnerAG14, 3.1])

The functor

\[ \operatorname{Mod}\colon\thinspace\operatorname{CAlg}(\mathrm{Sp})\longrightarrow\operatorname{CAlg}(\mathcal{P}r^{\mathrm{L}}), \]

sending $R$ to the (symmetric monoidal, presentable, stable) $\infty$-category of $R$-modules, is a fully faithful embedding.

4.2 Structured fibrations

We will write $\operatorname {Cat}^{\land }_\infty$ for the very large $\infty$-category of large $\infty$-categories.

We have a cocartesian fibration $\operatorname {Mod} \to \operatorname {Ring}$ [Reference LurieLur17, 4.5.3.6] whose fiber over the $\mathbb {E}_\infty$-ring spectrum $R$ is the (large) $\infty$-category $\operatorname {Mod}_R$ of $R$-modules.

Proposition 4.7 [Reference LurieLur17, 4.5.3.1, 4.5.3.2]

The cocartesian fibration $\operatorname {Mod}\to \operatorname {Ring}$ admits a canonical symmetric monoidal structure: there is a cocartesian family of $\infty$-operads

\[ \operatorname{Mod}^{{\otimes}} \to \operatorname{Ring} \times \operatorname{Comm}^{{\otimes}} \]

classifying a functor $R \mapsto \operatorname {Mod}_R\colon \thinspace \operatorname {Ring} \to \operatorname {CAlg}(\mathcal {P}r^{\mathrm {L}}_{\mathrm {st}})$ from $\mathbb {E}_\infty$-ring spectra to presentable stable symmetric monoidal $\infty$-categories.

We next consider algebra objects. By applying [Reference LurieLur17, 4.8.3.13], we similarly find that we have a cocartesian fibration $\operatorname {Alg}\to \operatorname {Ring}$ whose fiber over the ring $R$ is the (large) $\infty$-category $\operatorname {Alg}_R$ of $R$-algebras.

Proposition 4.8 The cocartesian fibration $\operatorname {Alg}\to \operatorname {Ring}$ admits a canonical symmetric monoidal structure such that the forgetful functor from algebras to modules induces a morphism of symmetric monoidal cocartesian fibrations

over $\operatorname {Ring}$.

Proof. As in [Reference LurieLur17, 5.3.1.20], the cocartesian family of $\infty$-operads $\operatorname {Mod}^{\otimes } \to \operatorname {Ring} \times \operatorname {Comm}^{\otimes }$ classifies a functor $\operatorname {Ring} \to (\operatorname {Op}_\infty )^{/\operatorname {Comm}^{\otimes }}$, taking $R$ to the cocartesian fibration $\operatorname {Mod}_R^{\otimes } \to \operatorname {Comm}^{\otimes }$. Applying [Reference LurieLur17, 3.4.2.1], we obtain a functor $\operatorname {Alg}\colon \thinspace \operatorname {Ring} \to (\operatorname {Op}_\infty )^{/\operatorname {Comm}^{\otimes }}$, taking $R$ to a cocartesian fibration $\operatorname {Alg}_R^{\otimes } \to \operatorname {Comm}^{\otimes }$ with a forgetful map

that preserves cocartesian arrows [Reference LurieLur17, 3.2.4.3]. Converting this back, we obtain a diagram

of cocartesian $\operatorname {Ring}$-families of symmetric monoidal $\infty$-operads, lifting the underlying map $\operatorname {Alg} \to \operatorname {Mod}$ to one compatible with the symmetric monoidal structure.

Restricting the cocartesian fibration $\operatorname {Mod}\to \operatorname {Ring}$ to the subcategory of cocartesian arrows between compact modules, we obtain a left fibration

\[ \operatorname{Mod}^{\omega,\mathrm{cocart}}\longrightarrow\operatorname{Ring} \]

whose fiber over $R$ is the $\infty$-groupoid $\operatorname {Mod}^{\omega ,\simeq }_R$ of compact (or perfect [Reference LurieLur17, 7.2.5.2]) $R$-modules. More precisely, an arrow $(R,M)\to (R',M')$ of $\operatorname {Mod}^{\omega ,\mathrm {cocart}}$ is an arrow $(R,M)\to (R',M')$ of $\operatorname {Mod}$ such that $M$ is compact (as an $R$-module) and the map $M\otimes _R R'\to M'$ is an equivalence.

Lastly, let $\operatorname {Alg}^\mathrm {prop}\to \operatorname {Ring}$ denote the left fibration whose source $\infty$-category is the subcategory $\operatorname {Alg}^\mathrm {prop}$ of proper algebras defined by the following pullback.

This time, however, $\operatorname {Alg}^\mathrm {prop}_R$ is not the full subgroupoid of $\operatorname {Alg}_R$ on the compact $R$-algebras, but rather the full subgroupoid of $\operatorname {Alg}_R$ consisting of the $R$-algebras $A$ whose underlying $R$-module is compact.

4.3 Functoriality of endomorphisms

In order to construct the endomorphism algebra as a functor, we need to extend the results of [Reference LurieLur17, 4.7.2]. In this, Lurie considers the category of tuples $(A,M,\phi \colon \thinspace A \otimes M \to M)$, which has a forgetful functor $p$ given by $p(A,M,\phi ) = M$. He extends it in such a way as to give this functor $p$ monoidal fibers; this gives the terminal object $\operatorname {End}(M)$ in the fiber over $M$ a canonical monoid structure. For the reader's convenience, we will first review some details of Lurie's construction.

Let $\mathcal {L}\mathcal {M}^{\otimes }$ denote the $\infty$-operad parametrizing pairs of an algebra and a left module [Reference LurieLur17, 4.2.1.7]. A cocartesian fibration $\mathcal {O}^{\otimes } \to \mathcal {L}\mathcal {M}^{\otimes }$ of $\infty$-operads determines a monoidal $\infty$-category $\mathcal {C}$ and an $\infty$-category $\mathcal {M}$ such that $\mathcal {M}$ is left-tensored over $\mathcal {C}$ [Reference LurieLur17, 4.2.1.19]; in particular, there exist objects $A \otimes M$ for $A \in \mathcal {C}$ and $M \in \mathcal {M}$. Associated to this there is a category $\operatorname {LMod}(\mathcal {M})$ of left module objects in $\mathcal {M}$ [Reference LurieLur17, 4.2.1.13]; such an object is determined by an algebra $A \in \mathcal {C}$ and a left $A$-module $M \in \mathcal {M}$. There is a forgetful map $\operatorname {LMod}(\mathcal {M}) \to \mathcal {M}$ which is a categorical fibration.

Definition 4.11 [Reference LurieLur17, 4.2.1.28]

Suppose that $\mathcal {M}$ is left-tensored over the monoidal $\infty$-category $\mathcal {C}$. A morphism object for $M$ and $N$ is an object $F_\mathcal {M}(M,N)$ of $\mathcal {C}$ equipped with a map $F_\mathcal {M}(M,N) \otimes M \to N$ such that the resulting natural homotopy class of map

\[ \operatorname{Map}_\mathcal{C}(C,F_\mathcal{M}(M,N)) \to \operatorname{Map}_\mathcal{M}(C \otimes M,N) \]

is a homotopy equivalence for all $C \in \mathcal {C}$. If morphism objects exist for all $M$ and $N$, we say that the left-tensor structure gives $\mathcal {M}$ a $\mathcal {C}$-enrichment.

Proof. By [Reference LurieLur09, 2.4.4.9], the full subcategory of $\operatorname {Act}(\mathcal {M})$ spanned by the final objects determines a trivial Kan fibration $\operatorname {End}(\mathcal {M}) \to \mathcal {M}^{\simeq }$. Choosing a section of this map, we obtain a composite functor

\[ \mathcal{M}^{{\simeq}} \to \operatorname{LMod}(\mathcal{M}) \to \operatorname{Alg}_\mathcal{C} \]

with the desired properties.

5.1 Picard spectra

In this section we recall the relevant notions and derive a useful long exact sequence (Corollary 5.20), related to the graded Rosenberg–Zelinsky sequence of Corollary 2.26, which generalizes the short exact sequences of Theorem 3.15.

If $\mathcal {C}$ is a small $\infty$-category, we write $\pi _0\mathcal {C}$ for the set of equivalence classes of objects of $\mathcal {C}$. By definition, $\pi _0\mathcal {C}$ is an invariant of the underlying $\infty$-groupoid $\mathcal {C}^{\simeq }$ of $\mathcal {C}$ (the $\infty$-groupoid obtained by discarding the noninvertible arrows).

A symmetric monoidal $\infty$-category $\mathcal {C}$ has a unique maximal grouplike symmetric monoidal subgroupoid $\mathcal {C}^{\times }$, the subcategory $\mathcal {C}^{\times }\subset \mathcal {C}$ consisting of the invertible objects and the equivalences thereof. That this is actually a symmetric monoidal subcategory in the $\infty$-categorical sense follows from the fact that invertibility and equivalence are both detected upon passage to the symmetric monoidal homotopy category; the grouplike condition is guaranteed by considering only the invertible objects.

Let $\mathcal {P}r^{\mathrm {L}}_{\mathrm {st}} \subset \mathcal {P}r^{\mathrm {L}}$ denote the $\infty$-category of stable presentable $\infty$-categories and colimit-preserving functors; by [Reference LurieLur17, 4.8.2.18] this is the category $\operatorname {Mod}_\mathrm {Sp}$ of left modules over the $\infty$-category of spectra. We have the $\infty$-category $\operatorname {CAlg}(\operatorname {Mod}_\mathrm {Sp})$ of commutative ring objects in $\mathcal {P}r^{\mathrm {L}}_{\mathrm {st}}$; these are the same as commutative $\mathrm {Sp}$-algebras or presentable symmetric monoidal stable $\infty$-categories.

By [Reference Ando, Blumberg and GepnerABG18, 8.9] $\operatorname {Pic}(\mathcal {R})$ is equivalent to a small space, and by [Reference Ando, Blumberg and GepnerABG18, 8.10] the functor $\operatorname {Pic}$ commutes with limits.

We have a symmetric monoidal cocartesian fibration

\[ \operatorname{Mod}(\operatorname{Mod}_\mathrm{Sp})\longrightarrow\operatorname{CAlg}(\operatorname{Mod}_\mathrm{Sp}) \]

whose fiber over a commutative $\mathrm {Sp}$-algebra $\mathcal {R}$ is the symmetric monoidal $\infty$-category $\operatorname {Cat}_\mathcal {R}$ of $\mathcal {R}$-linear $\infty$-categories. Writing

\[ \operatorname{Mod}_{\mathrm{Sp},\omega}\subset\operatorname{Mod}_\mathrm{Sp} \]

for the symmetric monoidal subcategory consisting of the compactly generated $\mathrm {Sp}$-modules and compact-object-preserving functors, this restricts to a symmetric monoidal cocartesian fibration

\[ \operatorname{Mod}(\operatorname{Mod}_{\mathrm{Sp},\omega})\longrightarrow\operatorname{CAlg}(\operatorname{Mod}_{\mathrm{Sp},\omega}) \]

over the subcategory $\operatorname {CAlg}(\operatorname {Mod}_{\mathrm {Sp},\omega })\subset \operatorname {CAlg}(\operatorname {Mod}_\mathrm {Sp})$ of commutative algebra objects in $\operatorname {Mod}_{\mathrm {Sp},\omega }\subset \operatorname {Mod}_\mathrm {Sp}$. For a commutative algebra object $\mathcal {R}\in \operatorname {CAlg}(\operatorname {Mod}_{\mathrm {Sp},\omega })$, also known as a compactly generated commutative $\mathrm {Sp}$-algebra, we write $\operatorname {Cat}_{\mathcal {R},\omega }^{\simeq }$ for the full subgroupoid of the fiber $\operatorname {Cat}_{\mathcal {R},\omega }$ over $\mathcal {R}$, the symmetric monoidal $\infty$-category of compactly generated $\mathcal {R}$-linear $\infty$-categories in the sense of [Reference LurieLur09, 5.3.5], and note that the map $\mathcal {R} \mapsto \operatorname {Cat}_{\mathcal {R},\omega }^{\simeq }$ defines a left fibration $\operatorname {Mod}^{\simeq }(\operatorname {Mod}_{\mathrm {Sp},\omega })\to \operatorname {CAlg}(\operatorname {Mod}_{\mathrm {Sp},\omega })$.

Proof. Let $I$ be any invertible object of $\mathcal {R}$ and let $\{M_\alpha \}$ be a filtered system of objects of $\mathcal {R}$. Then there are natural equivalences

\begin{align*} \operatorname{Map}(I,\operatorname{colim} M_\alpha) &\simeq\operatorname{Map}(\textbf{1},\operatorname{colim} I^{{-}1}\otimes M_\alpha)\\ &\simeq\operatorname{colim}\operatorname{Map}(\textbf{1},I^{{-}1}\otimes M_\alpha)\\ &\simeq\operatorname{colim}\operatorname{Map}(I,M_\alpha). \end{align*}

The first equivalence follows because $\otimes$ commutes with colimits. The second follows because the monoidal unit $\textbf {1}$ (the image of the sphere spectrum under the map $\mathrm {Sp} \to \mathcal {R}$) is compact by definition.

Because $\operatorname {Pic}(\mathcal {R})$ is closed under the symmetric monoidal product on $\mathcal {R}$, it is a grouplike symmetric monoidal $\infty$-groupoid, so by the recognition principle for infinite loop spaces we may regard $\operatorname {Pic}(\mathcal {R})$ as having an associated (connective) spectrum $\operatorname {\mathfrak {pic}}(\mathcal {R}) = K(\operatorname {Pic}(\mathcal {R}))$. Let $\Gamma _\mathcal {R}$ be the algebraic Picard groupoid of $\mathcal {R}$: the homotopy category of $\operatorname {Pic}(\mathcal {R})$, which is the $1$-truncation of $\operatorname {Pic}(\mathcal {R})$. If $\mathcal {R}$ is unambiguous, we drop it and simply write $\Gamma$. We will notationally distinguish between an object $\gamma \in \Gamma$ and the associated invertible object $R^{\gamma } \in \mathcal {R}$.

Proof. Since $\mathcal {R}$ is stable, the set $\pi _0\operatorname {Map}(M,N)$ of homotopy classes of maps from $M$ to $N$ admits an abelian group structure which is natural in the variables $M$ and $N$ of $\mathcal {R}$, and composition is bilinear. The result then follows from Proposition 2.15, defining $\pi _\star \operatorname {Map}(M,N)$ by the rule

\[ \pi_\gamma\operatorname{Map}(M,N):=\pi_0\operatorname{Map}(\Sigma^{\gamma} M,N). \]

If $R$ is an $\mathbb {E}_\infty$-ring spectrum, then we will typically write $\operatorname {Pic}(R)$ in place of $\operatorname {Pic}(\operatorname {Mod}_R)$ and $\Gamma _R$ in place of $\Gamma _{\operatorname {Mod}_R}$.

5.2 Brauer spectra

The results in this subsection and the next are essentially a summary of some of the results of Toën [Reference ToënToë12], in the differential-graded context, and Antieau and Gepner [Reference Antieau and GepnerAG14], in the spectral context.

Because $\operatorname {Br}(\mathcal {R})$ is closed under the symmetric monoidal product on $\operatorname {Cat}_\mathcal {R}$, it is a grouplike symmetric monoidal $\infty$-groupoid, so we may associate to it a connective spectrum $\operatorname {\mathfrak {br}}(\mathcal {R})$.

Proof. We first observe that $\operatorname {Cat}_{\mathcal {R},\omega }$ is a symmetric monoidal $\infty$-category in which the tensor product is induced from the tensor product of (compactly generated) presentable stable $\infty$-categories [Reference LurieLur17, § 4.8.1]. Furthermore, this symmetric monoidal structure is compatible with the tensor product of associative algebra spectra; in particular, there is a canonical equivalence

\[ \operatorname{Mod}_A\otimes\operatorname{Mod}_B\simeq\operatorname{Mod}_{A\otimes B}. \]

In particular, $\operatorname {Br}(\mathcal {R})\simeq \operatorname {Pic}(\operatorname {Cat}_{\mathcal {R},\omega })$ (and also $\operatorname {Pic}(\mathcal {R})$) are grouplike symmetric monoidal $\mathbb {E}_\infty$-spaces.

Observe that if $X$ is a grouplike $\mathbb {E}_\infty$-space, regarded as an $\infty$-groupoid, then $\Omega X\simeq \operatorname {Aut}_X(*)$ is the space of automorphisms of the distinguished object $*$ of $X$. Moreover, $\Omega X$ is again a grouplike $\mathbb {E}_\infty$-space, as limits of grouplike $\mathbb {E}_\infty$-spaces are computed in the $\infty$-category of spaces. Hence $\Omega \operatorname {Pic}(\operatorname {Mod}_\mathcal {R}^{\omega })\simeq \operatorname {Aut}_{\mathcal {R}}(\mathcal {R})\simeq \operatorname {Pic}(\mathcal {R})$, where the last equivalence follows from the fact that invertible $\mathcal {R}$-module endomorphisms of $\mathcal {R}$ correspond to invertible objects of $\mathcal {R}$ under the equivalence $\operatorname {End}_{\mathcal {R}}(\mathcal {R})\simeq \mathcal {R}$ [Reference LurieLur17, 4.8.4]. The spectrum level equivalence $\operatorname {\mathfrak {pic}}(\mathcal {R})\to \Omega \operatorname {\mathfrak {br}}(\mathcal {R})$ now follows from the fact that $\operatorname {\mathfrak {pic}}(\mathcal {R})$ and $\operatorname {\mathfrak {br}}(\mathcal {R})$ are the connective spectra associated to the grouplike $\mathbb {E}_\infty$-spaces $\operatorname {Pic}(\mathcal {R})$ and $\operatorname {Br}(\mathcal {R})$, respectively.

If $R$ is an $\mathbb {E}_\infty$-ring spectrum, we will typically write $\operatorname {Br}_R$ for the $\infty$-groupoid $\operatorname {Br}(\operatorname {Mod}_R)$.

5.3 Azumaya algebras

We write $\operatorname {Az}$ for the full subcategory of $\operatorname {Alg}$ determined by pairs $(R,A)$ such that $A$ is an Azumaya $R$-algebra. Because Azumaya algebras are stable under base-change, we have a morphism of cocartesian fibrations

over $\operatorname {Ring}$.