Finite approximations as a tool for studying triangulated categories

I have often been asked where the name “compact” came from. In the preprint version of [26] these objects went by a different name, but the (anonymous) referee didn’t like it. I was given a choice: I was allowed to baptize them either “compact” or “small”.

Who was I to argue with a referee?

The reader should note that an example would not be illuminating, the categories ${\langle G\rangle}_{n}$, $\overline{\langle G\rangle}_{n}$, $\overline{\langle G\rangle}^{(-\infty,n]}$ and $\overline{\langle G\rangle}^{[-n,n]}_{n}$ are not usually overly computable. For example: let $R$ be an associative ring, and let ${\mathscr{T}}=\mathbf{D}(R)$ be the unbounded derived category of complexes of left $R$-modules. The object $R\in{\mathscr{T}}$, that is the complex which is $R$ in degree zero and vanishes in all other degrees, is a compact generator for ${\mathscr{T}}=\mathbf{D}(R)$.

But if we wonder what the categories ${\langle R\rangle}_{n}$, $\overline{\langle R\rangle}_{n}$, $\overline{\langle R\rangle}^{(-\infty,n]}$ and $\overline{\langle R\rangle}^{[-n,n]}_{n}$ might turn out to be, only the category $\overline{\langle R\rangle}^{(-\infty,n]}$ is straightforward: it is the category of all cochain complexes whose cohomology vanishes in degrees $>n$. The three categories ${\langle R\rangle}_{n}$, $\overline{\langle R\rangle}_{n}$ and $\overline{\langle R\rangle}^{[-n,n]}_{n}$ are mysterious in general. In fact, the computation of ${\langle G\rangle}_{n}$ is the subject of conjectures that have attracted much interest. We will say a tiny bit about theorems in this direction in Section 4, and will mention one of the active, open conjectures in the discussion between Definition 7.7 and Problem 7.8.

In the definition of approximable triangulated categories, which is about to come, the category $\overline{\langle G\rangle}^{[-n,n]}_{n}$ will play the role of the replacement for the vector space of trigonometric Laurent polynomials of degree $\leq n$, which came up in the desiderata of Discussion 2.1(3). The older categories ${\langle G\rangle}_{n}$, $\overline{\langle G\rangle}_{n}$ and $\overline{\langle G\rangle}^{(-\infty,n]}$ will be needed later in the article.

Let us return to the heuristics of Discussion 2.1. Assume we have chosen the t-structure $\left({\mathscr{T}}^{\leq 0},{\mathscr{T}}^{\geq 0}\right)$ as in Discussion 2.1(2), which we think of as our replacement for the $L^{p}$–norm on $M(\mathbb{S}^{1})$. And we have also chosen a compact generator $G\in{\mathscr{T}}$ as in Discussion 2.1(1), which we think of as the analog of the exponential function $g(x)=e^{2\pi ix}$. We have declared that the subcategories $\overline{\langle G\rangle}^{[-n,n]}_{n}$ will be our replacement for the vector space of trigonometric Laurent polynomials of degree $\leq n$, as in Discussion 2.1(3). It’s now time to start approximating functions by trigonometric Laurent polynomials.

Let us therefore assume we start with some object $F\in{\mathscr{T}}$, and find a good approximation of it by the object $E\in\overline{\langle G\rangle}^{[-m,m]}_{m}$. Recall: this means that we find a morphism $E\longrightarrow F$ such that, in the triangle $E\longrightarrow F\longrightarrow D$, the object $D$ belongs to ${\mathscr{T}}^{\leq-M}$ for some suitably large $M$.

Now we can try to iterate, and find a good approximation for $D$. Thus we can look for a morphism $E^{\prime\prime}\longrightarrow D$, with $E^{\prime\prime}\in\overline{\langle G\rangle}^{[-n,n]}_{n}$, and such that in the triangle $E^{\prime\prime}\longrightarrow D\longrightarrow D^{\prime}$ the object $D^{\prime}$ belongs to ${\mathscr{T}}^{\leq-N}$, with $N>M$ even more enormous than $M$. Can we combine these to improve our initial approximation of $F$?

To do this, let’s build up the octahedron on the composable morphisms $F\longrightarrow D\longrightarrow D^{\prime}$. We end up with a diagram where the rows and columns are triangles

and in particular the triangle $E^{\prime}\longrightarrow F\longrightarrow D^{\prime}$ gives that $E^{\prime}$ is an even better approximation of $F$ than $E$ was. We are therefore interested in knowing if the triangle $E\longrightarrow E^{\prime}\longrightarrow E^{\prime\prime}$, coupled with the fact that $E\in\overline{\langle G\rangle}^{[-m,m]}_{m}$ and $E^{\prime\prime}\in\overline{\langle G\rangle}^{[-n,n]}_{n}$, gives any information about where $E^{\prime}$ might lie with respect to the construction of Black Box 2.2(3). Hence it is useful to know the following.

While part (2) of Definition 2.7 comes motivated by the analogy with Fourier analysis, part (1) of the definition seems random. It requires the t-structure, which is our replacement for the $L^{p}$–norm, to be compatible with the compact generator, which is the analog of $g(x)=e^{2\pi ix}$. As the reader will see in Proposition 5.5, this has the effect of uniquely specifying the t-structure (up to equivalence). So maybe a better parallel would be to fix our norm to be a particularly nice one, for example the $L^{2}$–norm on $M(\mathbb{S}^{1})$.

Let me repeat myself: as with all new mathematics, Definition 2.7 should be viewed as provisional. In the remainder of this survey we will discuss the applications as they now stand, to highlight the power of the methods. But I wouldn’t be surprised in the slightest if future applications turn out to require modifications, and/or generalizations, of the definitions and of the theorems that have worked so far.

If $X$ is noetherian and separated, then Theorem 4.2 specializes to saying that $\mathbf{D}^{\mathbf{perf}}(X)$ is regular if and only if $X$ is regular and finite dimensional. Hence the terminology.

The reader is referred to [33] and to Aoki [4] for the proofs of Theorems 4.2 and 4.4. More precisely: for Theorem 4.2 see [33, Theorem 0.5]. About Theorem 4.4: if we add the assumption that every closed subvariety of $X$ admits a regular alteration then the result may be found in [33, Theorem 0.15], but Aoki [4] found a lovely argument that allowed him to extend the statement to all quasiexcellent $X$.

There is a rich literature on strong generation, with beautiful papers by many authors. In the introduction to [33], as well as in [29] and [31, Section 7], the reader can find an extensive discussion of (some of) this fascinating work and of the way Theorems 4.2 and 4.4 compare to the older literature. For a survey taking an entirely different tack see Minami [22], which places in historical perspective a couple of the key steps in the proofs that [33] gives for Theorems 4.2 and 4.4.

Since all of this is now well documented in the published literature, let us focus the remainder of the current survey on the other applications of approximability. Those are all still in preprint form, see [25, 24, 23], although there are (published) surveys in [31, Sections 8 and 9] and in [30]. Those surveys are fuller and more complete than the sketchy one we are about to embark on. As we present the material, we will feel free to refer the reader to the more extensive surveys whenever we deem it appropriate.

Strangely enough, the value of Proposition 5.5 can be that it allows us to find an explicit t-structure in the preferred equivalence class.

Consider the case where $X$ is a quasicompact, separated scheme. By Bondal and Van den Bergh [6, Theorem 3.1.1(ii)] we know that the category ${\mathbf{D}_{\mathbf{qc}}}(X)$ has a compact generator, but in Example 3.2 we mentioned that the existence proof isn’t overly constructive, it doesn’t give us a handle on any explicit compact generator. Let $G$ be some compact generator. From Alonso, Jeremías and Souto [1, Theorem A.1] we know that the subcategory $\overline{\langle G\rangle}^{(-\infty,0]}$ of Black Box 2.2(2) is equal to ${\mathscr{T}}_{G}^{\leq 0}$ for a unique t-structure $\left({\mathscr{T}}_{G}^{\leq 0},{\mathscr{T}}_{G}^{\geq 0}\right)$ in the preferred equivalence class. But this doesn’t leave us a whole lot wiser—the compact generator $G$ isn’t explicit, hence neither is the t-structure.

However: the combination of [33, Theorem 5.8] and [25, Lemma 3.5] tells us that the category ${\mathbf{D}_{\mathbf{qc}}}(X)$ is approximable, and it so happens that the t-structure used in the proof, that is the t-structure for which a compact generator $H$ and an integer $n>0$ satisfying (1) and (2) of Proposition 5.5 are shown to exist, happens to be the standard t-structure. From Proposition 5.5 we now deduce that the standard t-structure is in the preferred equivalence class.

Let ${\mathscr{T}}$ be an approximable triangulated category. Aside from the classical, full subcategory ${\mathscr{T}}^{c}$ of compact objects, which we encountered back in Definition 1.1, we have in this section concocted five more intrinsic, full subcategories of ${\mathscr{T}}$: they are ${\mathscr{T}}^{-}$, ${\mathscr{T}}^{+}$, ${\mathscr{T}}^{b}$, ${\mathscr{T}}^{-}_{c}$ and ${\mathscr{T}}^{b}_{c}$. It can be proved that all six subcategories, that is the old ${\mathscr{T}}^{c}$ and the five new ones, are thick subcategories of ${\mathscr{T}}$. In particular each of them is a triangulated category.

In this survey we spent some effort introducing the notion of approximable triangulated categories. In Example 3.2 we told the reader that it is a theorem (and not a trivial one) that, as long as a scheme $X$ is quasicompact and separated, the derived category ${\mathbf{D}_{\mathbf{qc}}}(X)$ is approximable. In this section we showed that every approximable triangulated category comes with canonically defined, intrinsic subcategories ${\mathscr{T}}^{-}$, ${\mathscr{T}}^{+}$, ${\mathscr{T}}^{b}$, ${\mathscr{T}}^{c}$, ${\mathscr{T}}^{-}_{c}$ and ${\mathscr{T}}^{b}_{c}$, and in Example 6.3 we informed the reader that, in the special case where ${\mathscr{T}}={\mathbf{D}_{\mathbf{qc}}}(X)$, these turn out to be, respectively, ${\mathbf{D}_{\mathbf{qc}}^{-}}(X)$, ${\mathbf{D}_{\mathbf{qc}}^{+}}(X)$, ${\mathbf{D}_{\mathbf{qc}}^{b}}(X)$, $\mathbf{D}^{\mathbf{perf}}(X)$, $\mathbf{D}^{-}_{\mathbf{coh}}(X)$ and $\mathbf{D}^{b}_{\mathbf{coh}}(X)$.

Big deal. This teaches us that the traditional subcategories ${\mathbf{D}_{\mathbf{qc}}^{-}}(X)$, ${\mathbf{D}_{\mathbf{qc}}^{+}}(X)$, ${\mathbf{D}_{\mathbf{qc}}^{b}}(X)$, $\mathbf{D}^{\mathbf{perf}}(X)$, $\mathbf{D}^{-}_{\mathbf{coh}}(X)$ and $\mathbf{D}^{b}_{\mathbf{coh}}(X)$ of the category ${\mathbf{D}_{\mathbf{qc}}}(X)$ all have intrinsic descriptions. This might pass as a curiosity, unless we can actually use it to prove something we care about that we didn’t use to know.

The proof of part (1) of Theorem 6.6 may be found in [25], while the proof of part (2) of Theorem 6.6 occupies [24]. These aren’t easy theorems.

Let ${\mathscr{T}}={\mathbf{D}_{\mathbf{qc}}}(X)$, with $X$ a scheme proper over a noetherian ring $R$. Then the hypotheses of Theorem 6.6(1) are satisfied. We learn (among other things) that the natural functor, taking an object $B\in\mathbf{D}^{b}_{\mathbf{coh}}(X)$ to the $R$-linear functor $\mathrm{Hom}(-,B):{\mathbf{D}^{\mathbf{perf}}(X)}^{\mathrm{op}}\longrightarrow\text{Mod--}R$, is a fully faithful embedding

whose essential image is precisely the finite homological functors.

If we further assume that the scheme $X$ is finite-dimensional and quasiexcellent then the hypotheses of Theorem 6.6(2) are also satisfied. We learn that the functor, taking an object $A\in\mathbf{D}^{\mathbf{perf}}(X)$ to the $R$-linear functor $\mathrm{Hom}(A,-)$, is a fully faithful embedding

whose essential image is also the finite homological functors.

In [31, Historical Survey 8.2] the reader can find a discussion of the (algebro-geometric) precursors of Theorem 6.6. As for the applications: let us go through one of them.

Let $X$ be a scheme proper over the field $\mathbb{C}$ of complex numbers, and let $X^{\mathrm{an}}$ be the underlying complex analytic space. The analytification induces a functor we will call ${\mathscr{L}}:\mathbf{D}^{b}_{\mathbf{coh}}(X)\longrightarrow\mathbf{D}^{b}_{\mathbf{coh}}(X^{\mathrm{an}})$, it is the functor taking a bounded complex of coherent algebraic sheaves on $X$ to the analytification, which is a bounded complex of coherent analytic sheaves on $X^{\mathrm{an}}$. The pairing sending an object $A\in\mathbf{D}^{\mathbf{perf}}(X)$ and an object $B\in\mathbf{D}^{b}_{\mathbf{coh}}(X^{\mathrm{an}})$ to $\mathrm{Hom}\big{(}{\mathscr{L}}(A),B\big{)}$ delivers a map

Since the image lands in the finite homological functors, Theorem 6.6(1) allows us to factor this uniquely through the inclusion ${\mathscr{Y}}\circ i$, that is there exists (up to canonical natural isomorphism) a unique functor ${\mathscr{R}}$ rendering commutative the triangle

And proving Serre’s GAGA theorem reduces to the easy exercise of showing that ${\mathscr{L}}$ and ${\mathscr{R}}$ are inverse equivalences, the reader can find this in the (short) [25, Section 8 and Appendix A].

The brilliant inspiration underpinning the approach is due to Jack Hall [12], he is the person who came up with the idea of using the pairing above, coupled with representability theorems, to prove GAGA. The representability theorems available to Jack Hall at the time weren’t powerful enough, and Theorem 6.6 was motivated by trying to find a direct path from the ingenious, simple idea to a fullblown proof.

First of all: in Theorem 6.11(3) we assumed that the approximable triangulated ${\mathscr{T}}$ is noetherian as in [23, Definition 5.1]. The only observation we want to make here is that if $X$ is a noetherian, separated scheme then the approximable triangulated category ${\mathscr{T}}={\mathbf{D}_{\mathbf{qc}}}(X)$ is noetherian. Thus, for noetherian, separated schemes $X$, Theorem 6.11 gives exact equivalences of triangulated categories

The research paper [23] contains the proofs of the assertions in Theorem 6.11. The reader can find a skimpy survey in [31, Section 9] and a more extensive one in [30]. In [31, Historical Survey 9.1] there is a discussion of precursors of the results.