The Stacks project

73.26 Perfect complexes

We first talk about jumping loci for betti numbers of perfect complexes. First we have to define betti numbers.

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $x \in |X|$. We want to define $\beta _ i(x) \in \{ 0, 1, 2, \ldots \} \cup \{ \infty \} $. To do this, choose a morphism $f : \mathop{\mathrm{Spec}}(k) \to X$ in the equivalence class of $x$. Then $Lf^*E$ is an object of $D(\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}, \mathcal{O})$. By Étale Cohomology, Lemma 58.58.4 and Theorem 58.17.4 we find that $D(\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}, \mathcal{O}) = D(k)$ is the derived category of $k$-vector spaces. Hence $Lf^*E$ is a complex of $k$-vector spaces and we can take $\beta _ i(x) = \dim _ k H^ i(Lf^*E)$. It is easy to see that this does not depend on the choice of the representative in $x$. Moreover, if $X$ is a scheme, this is the same as the notion used in Derived Categories of Schemes, Section 36.31.

Lemma 73.26.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent (for example perfect). For any $i \in \mathbf{Z}$ consider the function

\[ \beta _ i : |X| \longrightarrow \{ 0, 1, 2, \ldots \} \]

defined above. Then we have

  1. formation of $\beta _ i$ commutes with arbitrary base change,

  2. the functions $\beta _ i$ are upper semi-continuous, and

  3. the level sets of $\beta _ i$ are étale locally constructible.

Proof. Choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. Then $L\varphi ^*E$ is a pseudo-coherent complex on the scheme $U$ (use Lemma 73.13.2) and we can apply the result for schemes, see Derived Categories of Schemes, Lemma 36.31.1. The meaning of part (3) is that the inverse image of the level sets to $U$ are locally constructible, see Properties of Spaces, Definition 64.8.2. $\square$

Lemma 73.26.2. Let $Y$ be a scheme and let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is flat, proper, and of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $Y$. For fixed $i \in \mathbf{Z}$ consider the function

\[ \beta _ i : |Y| \to \{ 0, 1, 2, \ldots \} ,\quad y \longmapsto \dim _{\kappa (y)} H^ i(X_ y, \mathcal{F}_ y) \]

Then we have

  1. formation of $\beta _ i$ commutes with arbitrary base change,

  2. the functions $\beta _ i$ are upper semi-continuous, and

  3. the level sets of $\beta _ i$ are locally constructible in $Y$.

Proof. By cohomology and base change (more precisely by Lemma 73.25.4) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $Y$ whose formation commutes with arbitrary base change. In particular we have

\[ H^ i(X_ y, \mathcal{F}_ y) = H^ i(K \otimes _{\mathcal{O}_ Y}^\mathbf {L} \kappa (y)) \]

Thus the lemma follows from Lemma 73.26.1. $\square$

Lemma 73.26.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect. The function

\[ \chi _ E : |X| \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^ i \beta _ i(x) \]

is locally constant on $X$.

Proof. Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 36.31.2. $\square$

Lemma 73.26.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect. Given $i, r \in \mathbf{Z}$, there exists an open subspace $U \subset X$ characterized by the following

  1. $E|_ U \cong H^ i(E|_ U)[-i]$ and $H^ i(E|_ U)$ is a locally free $\mathcal{O}_ U$-module of rank $r$,

  2. a morphism $f : Y \to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$.

Proof. Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 36.31.3. $\square$

Lemma 73.26.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is proper, flat, and of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $Y$. Fix $i, r \in \mathbf{Z}$. Then there exists an open subspace $V \subset Y$ with the following property: A morphism $T \to Y$ factors through $V$ if and only if $Rf_{T, *}\mathcal{F}_ T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$.

Proof. By cohomology and base change ( Lemma 73.25.4) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $Y$ whose formation commutes with arbitrary base change. Thus this lemma follows immediately from Lemma 73.26.4. $\square$

Lemma 73.26.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $r \geq 0$. Then there exists a locally closed subspace $j : Z \to X$ characterized by the following

  1. $H^ a(Lj^*E)$ is a locally free $\mathcal{O}_ Z$-module of rank $r$, and

  2. a morphism $f : Y \to X$ factors through $Z$ if and only if for all morphisms $g : Y' \to Y$ the $\mathcal{O}_{Y'}$-module $H^ a(L(f \circ g)^*E)$ is locally free of rank $r$.

Moreover, $j : Z \to X$ is of finite presentation and we have

  1. if $f : Y \to X$ factors as $Y \xrightarrow {g} Z \to X$, then $H^ a(Lf^*E) = g^*H^ a(Lj^*E)$,

  2. if $\beta _ a(x) \leq r$ for all $x \in |X|$, then $j$ is a closed immersion and given $f : Y \to X$ the following are equivalent

    1. $f : Y \to X$ factors through $Z$,

    2. $H^0(Lf^*E)$ is a locally free $\mathcal{O}_ Y$-module of rank $r$,

    and if $r = 1$ these are also equivalent to

    1. $\mathcal{O}_ Y \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective.

Proof. Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 36.31.4. $\square$

Lemma 73.26.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume

  1. $f$ is proper, flat, and of finite presentation, and

  2. for a morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field, we have $k = H^0(X_ k, \mathcal{O}_{X_ k})$.

Then we have

  1. $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds after any base change,

  2. étale locally on $Y$ we have

    \[ Rf_*\mathcal{O}_ X = \mathcal{O}_ Y \oplus P \]

    in $D(\mathcal{O}_ Y)$ where $P$ is perfect of tor amplitude in $[1, \infty )$.

Proof. It suffices to prove (a) and (b) étale locally on $Y$, thus we may and do assume $Y$ is an affine scheme. By cohomology and base change (Lemma 73.25.4) the complex $E = Rf_*\mathcal{O}_ X$ is perfect and its formation commutes with arbitrary base change. In particular, for $y \in Y$ we see that $H^0(E \otimes ^\mathbf {L} \kappa (y)) = H^0(X_ y, \mathcal{O}_{X_ y}) = \kappa (y)$. Thus $\beta _0(y) \leq 1$ for all $y \in Y$ with notation as in Lemma 73.26.1. Apply Lemma 73.26.6 with $a = 0$ and $r = 1$. We obtain a universal closed subscheme $j : Z \to Y$ with $H^0(Lj^*E)$ invertible characterized by the equivalence of (4)(a), (b), and (c) of the lemma. Since formation of $E$ commutes with base change, we have

\[ Lf^*E = R\text{pr}_{1, *}\mathcal{O}_{X \times _ Y X} \]

The morphism $\text{pr}_1 : X \times _ Y X$ has a section namely the diagonal morphism $\Delta $ for $X$ over $Y$. We obtain maps

\[ \mathcal{O}_ X \longrightarrow R\text{pr}_{1, *}\mathcal{O}_{X \times _ Y X} \longrightarrow \mathcal{O}_ X \]

in $D(\mathcal{O}_ X)$ whose composition is the identity. Thus $R\text{pr}_{1, *}\mathcal{O}_{X \times _ Y X} = \mathcal{O}_ X \oplus E'$ in $D(\mathcal{O}_ X)$. Thus $\mathcal{O}_ X$ is a direct summand of $H^0(Lf^*E)$ and we conclude that $X \to Y$ factors through $Z$ by the equivalence of (4)(c) and (4)(a) of the lemma cited above. Since $\{ X \to Y\} $ is an fppf covering, we have $Z = Y$. Thus $f_*\mathcal{O}_ X$ is an invertible $\mathcal{O}_ Y$-module. We conclude $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is an isomorphism because a ring map $A \to B$ such that $B$ is invertible as an $A$-module is an isomorphism. Since the assumptions are preserved under base change, we see that (a) is true.

Proof of (b). Above we have seen that for every $y \in Y$ the map $\mathcal{O}_ Y \to H^0(E \otimes ^\mathbf {L} \kappa (y))$ is surjective. Thus we may apply More on Algebra, Lemma 15.75.2 to see that in an open neighbourhood of $y$ we have a decomposition $Rf_*\mathcal{O}_ X = \mathcal{O}_ Y \oplus P$ $\square$

Lemma 73.26.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume

  1. $f$ is proper, flat, and of finite presentation, and

  2. the geometric fibres of $f$ are reduced and connected.

Then $f_*\mathcal{O}_ X = \mathcal{O}_ Y$ and this holds after any base change.

Proof. By Lemma 73.26.7 it suffices to show that $k = H^0(X_ k, \mathcal{O}_{X_ k})$ for all morphisms $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field. This follows from Spaces over Fields, Lemma 70.14.3 and the fact that $X_ k$ is geometrically connected and geometrically reduced. $\square$


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