Fully Faithful Functors and Dimension

Noah Olander

Abstract

We define the countable Rouquier dimension of a triangulated category and use this notion together with Theorem 2 of [Ola21] to prove that if there is a fully faithful embedding $D^{b}_{coh}(X)\subset D^{b}_{coh}(Y)$ with $X,Y$ smooth proper varieties, then $\mathrm{dim}(X)\leq\mathrm{dim}(Y)$ .

We will show how Theorem 2 follows from [Ola21, Theorem 2]. Theorem 2 was expected to be true by Conjecture 10 of [Orl09], but to the best of the author’s knowledge, it was unknown until now. The author is very grateful to Dmitrii Pirozhkov for suggesting this proof almost immediately after the paper [Ola21] was posted. The author also thanks Dmitri Orlov for a helpful email. The key is the following definition due to Pirozhkov:

Definition 1.

Let $\mathcal{T}$ be a triangulated category. The countable Rouquier dimension of $\mathcal{T}$ , denoted $\mathrm{CRdim}(\mathcal{T})$ , is the smallest $n$ such that there exists a countable set $\{E_{i}\}_{i\in I}$ of objects of $\mathcal{T}$ such that $\mathcal{T}=\langle\{E_{i}\}_{i\in I}\rangle_{n+1}$ .

We allow the countable Rouquier dimension to be infinity. We have an easy lemma:

Lemma 1.

Let $\mathcal{A}\subset\mathcal{T}$ be an admissible subcategory in a triangulated category. Then $\mathrm{CRdim}(\mathcal{A})\leq\mathrm{CRdim}(\mathcal{T})$ .

Proof.

Let $R$ be the right adjoint of the inclusion. If $\mathcal{T}=\langle\{E_{i}\}_{i\in I}\rangle_{n+1}$ then since $R$ is essentially surjective, $\mathcal{A}=\langle\{R(E_{i})\}_{i\in I}\rangle_{n+1}$ . ∎

The remark following the proof of [Ola21] directly implies:

Theorem 1.

Let $X$ be a Noetherian regular scheme with affine diagonal. Then $\mathrm{CRdim}(D^{b}_{coh}(X))\leq\mathrm{dim}(X)$ .

In fact Theorem 1 is true without the assumption on the diagonal of $X$ but we don’t need it here. The reverse inequality is not true in general. For example, if $X$ is a variety over a countable field $k$ , then $D^{b}_{coh}(X)$ has only countably many objects up to isomorphism, hence in fact $\mathrm{CRdim}(D^{b}_{coh}(X))=0$ . However for varieties over $\mathbf{C}$ we do get the reverse inequality:

Proposition 1.

Let $k$ be an uncountable field. Let $X$ be a reduced scheme of finite type over $k$ . Then $\mathrm{CRdim}(D^{b}_{coh}(X))\geq\mathrm{dim}(X)$ .

Proof.

Compare to the proof of [Rou08, Proposition 7.17]. Let $n=\mathrm{CRdim}(D^{b}_{coh}(X))$ and let $\{E_{i}\}_{i\in I}$ be a countable family of objects such that $D^{b}_{coh}(X)=\langle\{E_{i}\}_{i\in I}\rangle_{n+1}$ . Consider the set of closed points $x\in X$ such that for every $i\in I$ , the cohomology modules of $(E_{i})_{x}$ are free $\mathcal{O}_{X,x}$ -modules. Since a variety over $k$ is not a countable union of proper closed subsets ([Liu02, Exercise 2.5.10]), the set contains a closed point $x$ such that $\mathrm{dim}(\mathcal{O}_{X,x})=\mathrm{dim}(X)$ . We have $(E_{i})_{x}\in\langle\mathcal{O}_{X,x}\rangle_{1}$ for each $i$ since a complex with projective cohomology modules is decomposable, hence

\kappa(x)\in\langle\{(E_{i})_{x}\}_{i\in I}\rangle_{n+1}\subset\langle\mathcal{O}_{X,x}\rangle_{n+1},

hence $n\geq\mathrm{dim}(X)$ by [Rou08, Proposition 7.14]. ∎

Remark.

Since the countable Rouquier dimension only gives the expected answer for varieties over a sufficiently large field, Orlov suggests an alternative notion of countable Rouquier dimension which makes sense for a $k$ -linear dg-category $\mathcal{A}$ with $k$ a field: Take the smallest $n$ such that there exists a countable family $\{E_{i}\}_{i}$ of objects of $\mathcal{A}$ such that $\mathcal{A}_{K}=\langle\{(E_{i})_{K}\}_{i}\rangle_{n+1}$ for every field extension $K/k$ .

Finally, we prove our main result:

Theorem 2.

Let $k$ be a field. Let $X,Y$ be smooth proper varieties over $k$ . Assume there exists a fully faithful, exact, $k$ -linear functor $F:D^{b}_{coh}(X)\to D^{b}_{coh}(Y)$ . Then $\mathrm{dim}(X)\leq\mathrm{dim}(Y)$ .

Proof.

Choose any uncountable extension field $K/k$ . By [Ola20, Theorem 1], $F$ is the Fourier–Mukai transform with respect to a kernel $E\in D^{b}_{coh}(X\times_{k}Y)$ . Then $E_{K}$ gives rise to a functor $F_{K}:D^{b}_{coh}(X_{K})\to D^{b}_{coh}(Y_{K})$ which remains fully faithful: The fact that $F$ is fully faithful may be rephrased as $R\circ F\cong\mathrm{id}$ where $R$ is the right adjoint of $F$ . By the calculus of kernels, this may be rephrased as the existence of an isomorphism

\mathbf{R}pr_{13*}(\mathbf{L}pr_{12}^{*}(E)\otimes_{\mathcal{O}_{X\times Y\times X}}^{\mathbf{L}}\mathbf{L}pr_{23}^{*}(E^{\prime}))\cong\mathcal{O}_{\Delta}

(1)

where $E^{\prime}=\mathbf{R}\mathcal{H}om_{\mathcal{O}_{X\times Y}}(E,\mathbf{L}pr_{1}^{*}(\omega_{X}))[\mathrm{dim}(X)]$ , viewed as an object of $D^{b}_{coh}(Y\times_{k}X)$ , is the kernel of $R$ , and (1) remains valid upon base change to $K$ .

Now $F_{K}$ is the inclusion of an admissible subcategory since $X,Y$ are smooth and proper ([Sta18, Tag 0FYN]). We have $\mathrm{CRdim}(D^{b}_{coh}(X_{K}))=\mathrm{dim}(X_{K})=\mathrm{dim}(X)$ by Theorem 1 and Proposition 1 and similarly for $Y$ . Thus by Lemma 1, we have

\mathrm{dim}(X)=\mathrm{CRdim}(D^{b}_{coh}(X_{K}))\leq\mathrm{CRdim}(D^{b}_{coh}(Y_{K}))=\mathrm{dim}(Y),

as needed. ∎

References

[Liu02] Qing Liu. Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné, Oxford Science Publications.
[Ola20] Noah Olander. Orlov’s theorem in the smooth proper case, 2020. arXiv:2006.15173.
[Ola21] Noah Olander. The rouquier dimension of quasi-affine schemes, 2021. arXiv:2108.12005.
[Orl09] Dmitri Orlov. Remarks on generators and dimensions of triangulated categories. Moscow Mathematical Journal, 9:143–149, 2009.
[Rou08] Raphaël Rouquier. Dimensions of triangulated categories. Journal of K-Theory, 1(2):193–256, 2008.
[Sta18] The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018.

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