Derived versus spectral#
There are mainly three models people use when doing “higher” commutative algebra over a discrete commutative ring $R$:
- topological spaces/simplicial sets with an $R$-algebra structure (sometimes called simplicial $R$-algebras),
- commutative differential graded $R$-algebras,
- $\mathbb{E}_ \infty$-ring spectra over $R$.
When $R$ does not have characteristic zero, the category of commutative differential graded $R$-algebras apparently badly behaved. On the other hand, if it has characteristic zero, it is equivalent to the third notion. So we are left with mainly two options, which are genuinely different objects.
Definition 1. For $R$ a discrete commutative ring, consider the category $\mathsf{Poly}_ R$ of $R$ algebras that are isomorphic to $R[x_1, \dotsc, x_n]$, with morphisms being $R$-algebra homomorphisms. (The objects don’t have a preferred choice of generators.) We define $$ \mathsf{CAlg}_ R^\Delta \subseteq \mathsf{Fun}(\mathsf{Poly}_ R^\mathrm{op}, \mathcal{S}) $$ to be the full subcategory of functors that preserve finite products.
Remark 2. If you’re used to working with actual simplicial rings, it turns out that there is a simplicial model structure on the category of simplicial $R$-algebras, and the simplicial nerve of the full subcategory of fibrant-cofibrant objects is equivalent to $\mathsf{CAlg}_ R^\Delta$.
Recall that this is the kind of thing that showed up when we were discussing nonabelian derived functors. The intuition is that by this “preserving products” condition, the values are determined by the value of $R[x]$, and there are a bunch of structure maps that look like an $R$-algebra structure on this space.
Theorem 3 (SAG 25.1.2.2). Using the natural inclusion functor $\mathsf{Poly}_ R \to \mathsf{CAlg}_ R^\mathrm{cn}$ (which preserves finite coproducts) and using universal property of nonabelian derived functors, we can uniquely extend it to a functor $$ \Theta \colon \mathsf{CAlg}_ R^\Delta \to \mathsf{CAlg}_ R^\mathrm{cn} $$ that preserves all colimits.
- The functor $\Theta$ preserves all limits and detects isomorphisms.
- If $R$ contains $\mathbb{Q}$, then $\Theta$ is an equivalence.
Here is an intuitive reason why this is not necessarily an equivalence. For any $A \in \mathsf{CAlg}_ R^\Delta$ and a point $a \in A$, there is one canonical way of square this element: pull back via the map $R[x] \to R[x]$ sending $x$ to $x^2$. On the other hand, if you think about the $\mathbb{E}_ \infty$-operad, there is a $B\Sigma_2$ worth of ways of squaring an element in a symmetric ring spectrum. Roughly speaking, the functor $\Theta$ is forgetting about this stricter homotopy coherence relations.
This difference gives rise to two different flavors of higher algebraic geometry. The one using $\mathsf{CAlg}^\Delta$ as models as rings is called derived algebraic geometry; the other using $\mathsf{CAlg}^\mathrm{cn}$ (or just $\mathsf{CAlg}$) is called spectral algebraic geometry. The two agree over $\mathbb{Q}$ but not in general.
Spectral schemes#
Recall that a scheme is a locally ringed space that locally looks like $\Spec A$. We will use the same definition. Denote by $\mathsf{CAlg}$ the category of $\mathbb{E}_ \infty$-ring spectra.
Definition 4 (SAG 1.1.2.5). A spectrally ringed space is a topological space $X$ together with a sheaf $\mathscr{O}_ X$ valued in $\mathsf{CAlg}$.
We haven’t really talked about sheaves valued in categories other than $\mathcal{S}$, but these are still presheaves satisfying the sheaf condition.
Example 5. Let $A \in \mathsf{CAlg}$. Then $\pi_0 A$ is a commutative ring, so we can consider the affine scheme $\Spec (\pi_0 A)$. We now put a structure sheaf on this topological space by $$ \mathscr{O}_ X(\Spec ((\pi_0 A)[a^{-1}])) = A[a^{-1}] $$ for all $a \in \pi_0 A$. (Recall we defined $A[a^{-1}]$ last time.)
To show that this is indeed a sheaf, we need the following lemma.
Lemma 6. Let $A \in \mathsf{CAlg}$ and let $a_1, \dotsc, a_n \in \pi_0 A$ generate the unit ideal. Then the map $$ A \to \varprojlim_{\emptyset \neq S \subseteq \lbrace 1, \dotsc, n \rbrace} A[a_S^{-1}] $$ is an equivalence.
Proof.
Here is a general fact: a map $N \to M$ of $A$-modules being an isomorphism can be checked after a tensoring with a flat $A$-module $F$ with $\pi_0 F$ faithfully flat over $\pi_0 A$. The reason is that $$ \pi_n (N \otimes_A F) = \pi_n N \otimes_{\pi_0 A} \pi_0 F $$ just by degeneration of the spectral sequence on the $E_2$-page.
So we can check this is an isomorphism after base changing to the faithfully flat $A$-algebra $\prod_{i}^{} A[a_i^{-1}]$, i.e., to each $A[a_i^{-1}]$ separately. Once we invert $a_i$, we have the map $$ A[a_i^{-1}] \to \varprojlim_{\emptyset \neq S} A[a_{S \cup \lbrace 1 \rbrace}^{-1}]. $$ But if you look at this diagram, it is just a right Kan extension of diagram restricted the part where $i \in S$. So we can compute the limit on that part, but now this is clearly just $A[a_i^{-1}]$.
Once we have this fact, we use the fact that subsets of this form (i.e., basic opens) form a basis for the Zariski topology, and then glue these things to define the value on all opens.
Remark 7. Even if $A = \pi_0 A$ is a classical ring, there is some “derivedness” in the structure sheaf of $X = \Spec A$. For any open subset $U \subseteq X$, we will have $$ \pi_n \mathscr{O} _X(U) = H^{-n}(U, \mathscr{O}_X \vert_U). $$
Now that we know what an affine scheme is, we can make the following definition.
Definition 8. A non-connective spectral scheme is a spectrally ringed space $(X, \mathscr{O}_ X)$ that is locally isomorphic to $\Spec A$ for $A \in \mathsf{CAlg}$.
Given this structure sheaf $\mathscr{O}_ X$, we can look at $\pi_n \mathscr{O}_ X$, which is the sheafification of the presheaf obtained from taking $\pi_n$. Then $\pi_0 \mathscr{O}_ X$ will be a sheaf of rings, and then $\pi_n \mathscr{O}_ X$ will be a sheaf of $\pi_0 \mathscr{O}_ X$-modules.
Proposition 9. Let $X$ be a non-connected spectral scheme.
- The ringed space $(X, \pi_0 \mathscr{O}_ X)$ is a scheme. In particular, if $X = \Spec A$ then $(X, \pi_0 \mathscr{O}_ X)$ is $\Spec \pi_0 A$.
- Each sheaf of modules $\pi_n X$ is a quasi-coherent sheaf on $(X, \pi_0 \mathscr{O}_ X)$. In particular, if $X = \Spec A$ then $\pi_n X$ is the quasi-coherent sheaf corresponding to $\pi_n A$ over $\pi_0 A$.
- The structure sheaf $\mathscr{O}_ X$ is hypercomplete.
Theorem 10 (SAG 1.1.6.2). Let $(X, \mathscr{O}_ X)$ be a spectrally ringed space. Suppose
- the ring space $(X, \pi_0 \mathscr{O}_ X)$ is a scheme,
- the sheaves of modules $\pi_n \mathscr{O}_ X$ are quasi-coherent,
- for every open subset $U \subseteq X$ that corresponds to an affine open subscheme of $(X, \pi_0 \mathscr{O}_ X)$ and every integer $n$, the natural map $\pi_n (\mathscr{O}_ X(U)) \to (\pi_n \mathscr{O}_ X)(U)$ is an isomorphism.
Then $(X, \mathscr{O}_ X)$ is a non-connective spectral scheme.
Proof.
We may reduce to when $(X, \pi_0 \mathscr{O}_ X)$ is affine. The claim is that $(X, \mathscr{O}_ X)$ is affine. Let $R = \mathscr{O}_ X(X)$ so that $\pi_0 R = (\pi_0 \mathscr{O}_ X)(X)$. The first step is building a map $$ (X, \mathscr{O}_ X) \to \Spec R $$ of spectrally ringed spaces. On topological spaces, just use the identification $X \cong \Spec \pi_0 R \cong \Spec R$. Then on the level of sheaves, use the universal property of localization: for $f \in \pi_0 R$, since $\mathscr{O}_ X(\Spec R_f)$ is an $R$-algebra with $f$ invertible, it is an $R_f$-algebra.
The claim is that this map has to be an isomorphism. On each $\Spec R_f$, $$ \mathscr{O}_X (\Spec R_f) \to R_f $$ induces an isomorphism on each $\pi_n$, and hence is an isomorphism. Since this is a map of sheaves that is an isomorphism on all basic opens, it is an isomorphism. (Note that we don’t need hypercompleteness because intersections of basic opens are basic opens!)
Definition 11. A spectrally ringed space $(X, \mathscr{O}_ X)$ is a locally spectrally ringed space if the underlying ringed space $(X, \pi_0 \mathscr{O}_ X)$ is a locally ringed space. A morphism between non-connective spectral schemes $(X, \mathscr{O}_ X) \to (Y, \mathscr{O}_ Y)$ is a continuous map $f \colon X \to Y$ of topological spaces together with a map of sheaves $\mathscr{O}_ Y \to f_\ast \mathscr{O}_ X$ inducing a local homomorphism $(\pi_0 \mathscr{O}_ Y)_ {f(x)} \to (\pi_0 \mathscr{O}_ X)_ x$ for all $x \in X$. We denote by $\mathsf{SpSch}^\mathrm{nc}$ the category of non-connective spectral schemes.
Theorem 12 (SAG 1.1.5.7). There is a natural equivalence $$ \Hom(\Spec A, \Spec B) \simeq \Hom_\mathsf{CAlg}(B, A). $$
Sometimes one would want to work with connective spectra rather than all spectra. We need to be a bit careful here, because even for a classical scheme $X$, the sheaf $\mathscr{O}_ X$ is not necessarily valued in connective spectra.
Definition 13. We say that a non-connective spectral scheme $(X, \mathscr{O}_ X)$ is a spectral scheme if $\pi_n \mathscr{O}_ X = 0$ for all $n \lt 0$ (or equivalently, it locally looks like $\Spec A$ for $A \in \mathsf{CAlg}^\mathrm{cn}$). We denote by $\mathsf{SpSch}$ the full subcategory of $\mathsf{SpSch}^\mathrm{nc}$ spanned by spectral schemes.
Theorem 14 (SAG 1.1.7.6). The above inclusion admits a left adjoint $(X, \mathscr{O}_ X) \mapsto (X, \tau_{\ge 0} \mathscr{O}_ X)$. In particular, there is always a map of non-connected spectral schemes $$ (X, \mathscr{O}_ X) \to (X, \tau_{\ge 0} \mathscr{O}_ X). $$
Of course, locally we are just taking $\Spec A$ to $\Spec \tau_{\ge 0} A$.
Spectral Deligne–Mumford stacks#
Actually, Lurie’s SAG doesn’t work with spectral schemes. Somehow he wants to work with with étale topology rather than the Zariski topology, and so everything is written in terms of Deligne–Mumford stacks. This introduces an additional layer of technicality, because these are not spectrally ringed spaces. One might imagine using functor of points to define Deligne–Mumford stacks, but for some reason this is not the approach taken in SAG.
Instead, you generalize the notion of a topological space and works with topoi. Roughly speaking, an ∞-topos is a category that looks like the category of $\mathcal{S}$-valued on a topological space $X$. It turns out that this category contains enough information to recover the lattice of opens in $X$. But more general things are ∞-topoi, e.g., the category of $\mathcal{S}$-valued small étale sheaves on a scheme.
Definition 15. Given $\mathcal{X}$ an ∞-topos, we define a sheaf on $\mathcal{X}$ valued in a category $\mathcal{C}$ to be a functor $\mathcal{X}^\mathrm{op} \to \mathcal{C}$ that preserves all small limits.
Definition 16. A spectrally ringed ∞-topos is a pair $(\mathcal{X}, \mathscr{O}_ \mathcal{X})$ where $\mathcal{X}$ is an ∞-topos and $\mathscr{O}_ \mathcal{X}$ is a sheaf on $\mathcal{X}$ valued in $\mathsf{CAlg}$.
Again, given any $R \in \mathsf{CAlg}$ we can define an affine spectrally ringed ∞-topos, this time using the small étale site. But I need to first tell you what étale morphisms are.
Definition 17. An map $A \to B$ in $\mathsf{CAlg}$ is said to be étale when it is flat and $\pi_0 A \to \pi_0 B$ is étale.
Theorem 18 (HA 7.5.0.6). The full subcategory of $\mathsf{CAlg}_ {A/}$ spanned by étale maps is equivalent to the full subcategory of the ordinary category $(\mathsf{CAlg}^\heartsuit)_ {\pi_0 A/}$ spanned by étale maps.
In a sense, all the higher homotopy information behaves like a “nilpotent thickening” and is orthogonal to the étale direction.
Definition 19. For $R \in \mathsf{CAlg}$, consider $\mathcal{S}hv_R^\mathrm{et}$ the ∞-topos of sheaves on the small étale site $\mathsf{CAlg}_ R^\mathrm{et}$ of $R$. There is a forgetful functor $\mathscr{O} \colon \mathsf{CAlg}_ R^\mathrm{et} \to \mathsf{CAlg}$, which is an étale functor. We then define the étale spectrum of $R$ as $$ \operatorname{Spet} R = (\mathcal{S}hv_R^\mathrm{et}, \mathscr{O}). $$
Definition 20. A non-connective spectral Deligne–Mumford stack is a spectral ringed ∞-topos $(\mathcal{X}, \mathscr{O}_ \mathcal{X})$ such that there exist $U_\alpha \in \mathcal{X}$ covering the final object $1 \in \mathcal{X}$ satisfying the property that each $(\mathcal{X}_ {/U_\alpha}, \mathscr{O} \vert_{U_\alpha})$ is isomorphic to $\operatorname{Spet} R_\alpha$ for some $R_\alpha \in \mathsf{CAlg}$.