Lemma 115.13.1.reference Let f : X' \to X be a proper birational morphism of integral Noetherian schemes with X regular. The map \mathcal{O}_ X \to Rf_*\mathcal{O}_{X'} canonically splits in D(\mathcal{O}_ X).
115.13 Representability in the regular proper case
This section is obsolete because we improved Derived Categories of Varieties, Theorem 57.6.3 to apply to all proper schemes over a field (whereas before we only proved it for projective schemes over a field).
Proof. Set E = Rf_*\mathcal{O}_{X'} in D(\mathcal{O}_ X). Observe that E is in D^ b_{\textit{Coh}}(\mathcal{O}_ X) by Derived Categories of Schemes, Lemma 36.11.3. By Derived Categories of Schemes, Lemma 36.11.8 we find that E is a perfect object of D(\mathcal{O}_ X). Since \mathcal{O}_{X'} is a sheaf of algebras, we have the relative cup product \mu : E \otimes _{\mathcal{O}_ X}^\mathbf {L} E \to E by Cohomology, Remark 20.28.7. Let \sigma : E \otimes E^\vee \to E^\vee \otimes E be the commutativity constraint on the symmetric monoidal category D(\mathcal{O}_ X) (Cohomology, Lemma 20.50.6). Denote \eta : \mathcal{O}_ X \to E \otimes E^\vee and \epsilon : E^\vee \otimes E \to \mathcal{O}_ X the maps constructed in Cohomology, Example 20.50.7. Then we can consider the map
E \xrightarrow {\eta \otimes 1} E \otimes E^\vee \otimes E \xrightarrow {\sigma \otimes 1} E^\vee \otimes E \otimes E \xrightarrow {1 \otimes \mu } E^\vee \otimes E \xrightarrow {\epsilon } \mathcal{O}_ X
We claim that this map is a one sided inverse to the map in the statement of the lemma. To see this it suffices to show that the composition \mathcal{O}_ X \to \mathcal{O}_ X is the identity map. This we may do in the generic point of X (or on an open subscheme of X over which f is an isomorphism). In this case E = \mathcal{O}_ X and \mu is the usual multiplication map and the result is clear. \square
Lemma 115.13.2. Let X be a proper scheme over a field k which is regular. Let K \in \mathop{\mathrm{Ob}}\nolimits (D_\mathit{QCoh}(\mathcal{O}_ X)). The following are equivalent
K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D_{perf}(\mathcal{O}_ X), and
\sum _{i \in \mathbf{Z}} \dim _ k \mathop{\mathrm{Ext}}\nolimits ^ i_ X(E, K) < \infty for all perfect E in D(\mathcal{O}_ X).
Proof. The equality in (1) holds by Derived Categories of Schemes, Lemma 36.11.8. The implication (1) \Rightarrow (2) follows from Derived Categories of Varieties, Lemma 57.5.3. The implication (2) \Rightarrow (1) follows from More on Morphisms, Lemma 37.69.6. \square
Lemma 115.13.3. Let X be a proper scheme over a field k which is regular.
Let F : D_{perf}(\mathcal{O}_ X)^{opp} \to \text{Vect}_ k be a k-linear cohomological functor such that
\sum \nolimits _{n \in \mathbf{Z}} \dim _ k F(E[n]) < \inftyfor all E \in D_{perf}(\mathcal{O}_ X). Then F is isomorphic to a functor of the form E \mapsto \mathop{\mathrm{Hom}}\nolimits _ X(E, K) for some K \in D_{perf}(\mathcal{O}_ X).
Let G : D_{perf}(\mathcal{O}_ X) \to \text{Vect}_ k be a k-linear homological functor such that
\sum \nolimits _{n \in \mathbf{Z}} \dim _ k G(E[n]) < \inftyfor all E \in D_{perf}(\mathcal{O}_ X). Then G is isomorphic to a functor of the form E \mapsto \mathop{\mathrm{Hom}}\nolimits _ X(K, E) for some K \in D_{perf}(\mathcal{O}_ X).
Proof. This follows from Derived Categories of Varieties, Theorem 57.6.3 and Lemma 57.6.4. We also give another proof below.
Proof of (1). The derived category D_\mathit{QCoh}(\mathcal{O}_ X) has direct sums, is compactly generated, and D_{perf}(\mathcal{O}_ X) is the full subcategory of compact objects, see Derived Categories of Schemes, Lemma 36.3.1, Theorem 36.15.3, and Proposition 36.17.1. By Derived Categories of Varieties, Lemma 57.6.2 we may assume F(E) = \mathop{\mathrm{Hom}}\nolimits _ X(E, K) for some K \in \mathop{\mathrm{Ob}}\nolimits (D_\mathit{QCoh}(\mathcal{O}_ X)). Then it follows that K is in D^ b_{\textit{Coh}}(\mathcal{O}_ X) by Lemma 115.13.2.
Proof of (2). Consider the contravariant functor E \mapsto E^\vee on D_{perf}(\mathcal{O}_ X), see Cohomology, Lemma 20.50.5. This functor is an exact anti-self-equivalence of D_{perf}(\mathcal{O}_ X). Hence we may apply part (1) to the functor F(E) = G(E^\vee ) to find K \in D_{perf}(\mathcal{O}_ X) such that G(E^\vee ) = \mathop{\mathrm{Hom}}\nolimits _ X(E, K). It follows that G(E) = \mathop{\mathrm{Hom}}\nolimits _ X(E^\vee , K) = \mathop{\mathrm{Hom}}\nolimits _ X(K^\vee , E) and we conclude that taking K^\vee works. \square
Comments (0)