The Stacks project

38.40 Blowing up complexes

This section finds normal forms for perfect objects of the derived category after blowups.

Lemma 38.40.1. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent. For every $p, k \in \mathbf{Z}$ there is an finite type quasi-coherent sheaf of ideals $\text{Fit}_{p, k}(E) \subset \mathcal{O}_ X$ with the following property: for $U \subset X$ open such that $E|_ U$ is isomorphic to

\[ \ldots \to \mathcal{O}_ U^{\oplus n_{b - 2}} \xrightarrow {d_{b - 2}} \mathcal{O}_ U^{\oplus n_{b - 1}} \xrightarrow {d_{b - 1}} \mathcal{O}_ U^{\oplus n_ b} \to 0 \to \ldots \]

the restriction $\text{Fit}_{p, k}(E)|_ U$ is generated by the minors of the matrix of $d_ p$ of size

\[ - k + n_{p + 1} - n_{p + 2} + \ldots + (-1)^{b - p + 1} n_ b \]

Convention: the ideal generated by $r \times r$-minors is $\mathcal{O}_ U$ if $r \leq 0$ and the ideal generated by $r \times r$-minors where $r > \min (n_ p, n_{p + 1})$ is zero.

Proof. Observe that $E$ locally on $X$ has the shape as stated in the lemma, see More on Algebra, Section 15.63, Cohomology, Section 20.44, and Derived Categories of Schemes, Section 36.10. Thus it suffices to prove that the ideal of minors is independent of the chosen representative. To do this, it suffices to check in local rings. Over a local ring $(R, \mathfrak m, \kappa )$ consider a bounded above complex

\[ F^\bullet : \ldots \to R^{\oplus n_{b - 2}} \xrightarrow {d_{b - 2}} R^{\oplus n_{b - 1}} \xrightarrow {d_{b - 1}} R^{\oplus n_ b} \to 0 \to \ldots \]

Denote $\text{Fit}_{k, p}(F^\bullet ) \subset R$ the ideal generated by the minors of size $k - n_{p + 1} + n_{p + 2} - \ldots + (-1)^{b - p} n_ b$ in the matrix of $d_ p$. Suppose some matrix coefficient of some differential of $F^\bullet $ is invertible. Then we pick a largest integer $i$ such that $d_ i$ has an invertible matrix coefficient. By Algebra, Lemma 10.102.2 the complex $F^\bullet $ is isomorphic to a direct sum of a trivial complex $\ldots \to 0 \to R \to R \to 0 \to \ldots $ with nonzero terms in degrees $i$ and $i + 1$ and a complex $(F')^\bullet $. We leave it to the reader to see that $\text{Fit}_{p, k}(F^\bullet ) = \text{Fit}_{p, k}((F')^\bullet )$; this is where the formula for the size of the minors is used. If $(F')^\bullet $ has another differential with an invertible matrix coefficient, we do it again, etc. Continuing in this manner, we eventually reach a complex $(F^\infty )^\bullet $ all of whose differentials have matrices with coefficients in $\mathfrak m$. Here you may have to do an infinite number of steps, but for any cutoff only a finite number of these steps affect the complex in degrees $\geq $ the cutoff. Thus the “limit” $(F^\infty )^\bullet $ is a well defined bounded above complex of finite free modules, comes equipped with a quasi-isomorphism $(F^\infty )^\bullet \to F^\bullet $ into the complex we started with, and $\text{Fit}_{p, k}(F^\bullet ) = \text{Fit}_{p, k}((F^\infty )^\bullet )$. Since the complex $(F^\infty )^\bullet $ is unique up to isomorphism by More on Algebra, Lemma 15.74.5 the proof is complete. $\square$

Lemma 38.40.2. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. Let $U \subset X$ be a scheme theoretically dense open subscheme such that $H^ i(E|_ U)$ is finite locally free of constant rank $r_ i$ for all $i \in \mathbf{Z}$. Then there exists a $U$-admissible blowup $b : X' \to X$ such that $H^ i(Lb^*E)$ is a perfect $\mathcal{O}_{X'}$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$.

Proof. We will construct and study the blowup affine locally. Namely, suppose that $V \subset X$ is an affine open subscheme such that $E|_ V$ can be represented by the complex

\[ \mathcal{O}_ V^{\oplus n_ a} \xrightarrow {d_ a} \ldots \xrightarrow {d_{b - 1}} \mathcal{O}_ V^{\oplus n_ b} \]

Set $k_ i = r_{i + 1} - r_{i + 2} + \ldots + (-1)^{b - i + 1}r_ b$. A computation which we omit show that over $U \cap V$ the rank of $d_ i$ is

\[ \rho _ i = - k_ i + n_{i + 1} - n_{i + 2} + \ldots + (-1)^{b - i + 1}n_ b \]

in the sense that the cokernel of $d_ i$ is finite locally free of rank $n_{i + 1} - \rho _ i$. Let $\mathcal{I}_ i \subset \mathcal{O}_ V$ be the ideal generated by the minors of size $\rho _ i \times \rho _ i$ in the matrix of $d_ i$.

On the one hand, comparing with Lemma 38.40.1 we see the ideal $\mathcal{I}_ i$ corresponds to the global ideal $\text{Fit}_{i, k_ i}(E)$ which was shown to be independent of the choice of the complex representing $E|_ V$. On the other hand, $\mathcal{I}_ i$ is the $(n_{i + 1} - \rho _ i)$th Fitting ideal of $\mathop{\mathrm{Coker}}(d_ i)$. Please keep this in mind.

We let $b : X' \to X$ be the blowing up in the product of the ideals $\text{Fit}_{i, k_ i}(E)$; this makes sense as locally on $X$ almost all of these ideals are equal to the unit ideal (see above). This blowup dominates the blowups $b_ i : X'_ i \to X$ in the ideals $\text{Fit}_{i, k_ i}(E)$, see Divisors, Lemma 31.32.12. By Divisors, Lemma 31.35.3 each $b_ i$ is a $U$-admissible blowup. It follows that $b$ is a $U$-admissible blowup (tiny detail omitted; compare with the proof of Divisors, Lemma 31.34.4). Finally, $U$ is still a scheme theoretically dense open subscheme of $X'$. Thus after replacing $X$ by $X'$ we end up in the situation discussed in the next paragraph.

Assume $\text{Fit}_{i, k_ i}(E)$ is an invertible ideal for all $i$. Choose an affine open $V$ and a complex of finite free modules representing $E|_ V$ as above. It follows from Divisors, Lemma 31.35.3 that $\mathop{\mathrm{Coker}}(d_ i)$ has tor dimension $\leq 1$. Thus $\mathop{\mathrm{Im}}(d_ i)$ is finite locally free as the kernel of a map from a finite locally free module to a finitely presented module of tor dimension $\leq 1$. Hence $\mathop{\mathrm{Ker}}(d_ i)$ is finite locally free as well (same argument). Thus the short exact sequence

\[ 0 \to \mathop{\mathrm{Im}}(d_{i - 1}) \to \mathop{\mathrm{Ker}}(d_ i) \to H^ i(E)|_ V \to 0 \]

shows what we want and the proof is complete. $\square$

Lemma 38.40.3. Let $X$ be an integral scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then there exists a nonempty open $U \subset X$ such that $H^ i(E|_ U)$ is finite locally free of constant rank $r_ i$ for all $i \in \mathbf{Z}$ and there exists a $U$-admissible blowup $b : X' \to X$ such that $H^ i(Lb^*E)$ is a perfect $\mathcal{O}_{X'}$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$.

Proof. We strongly urge the reader to find their own proof of the existence of $U$. Let $\eta \in X$ be the generic point. The restriction of $E$ to $\eta $ is isomorphic in $D(\kappa (\eta ))$ to a finite complex $V^\bullet $ of finite dimensional vector spaces with zero differentials. Set $r_ i = \dim _{\kappa (\eta )} V^ i$. Then the perfect object $E'$ in $D(\mathcal{O}_ X)$ represented by the complex with terms $\mathcal{O}_ X^{\oplus r_ i}$ and zero differentials becomes isomorphic to $E$ after pulling back to $\eta $. Hence by Derived Categories of Schemes, Lemma 36.35.9 there is an open neighbourhood $U$ of $\eta $ such that $E|_ U$ and $E'|_ U$ are isomorphic. This proves the first assertion. The second follows from the first and Lemma 38.40.2 as any nonempty open is scheme theoretically dense in the integral scheme $X$. $\square$

Remark 38.40.4. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be a perfect object such that $H^ i(E)$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. This property sometimes allows one to reduce questions about $E$ to questions about $H^ i(E)$. For example, suppose

\[ \mathcal{E}^ a \xrightarrow {d^ a} \ldots \xrightarrow {d^{b - 2}} \mathcal{E}^{b - 1} \xrightarrow {d^{b - 1}} \mathcal{E}^ b \]

is a bounded complex of finite locally free $\mathcal{O}_ X$-modules representing $E$. Then $\mathop{\mathrm{Im}}(d^ i)$ and $\mathop{\mathrm{Ker}}(d^ i)$ are finite locally free $\mathcal{O}_ X$-modules for all $i$. Namely, suppose by induction we know this for all indices bigger than $i$. Then we can first use the short exact sequence

\[ 0 \to \mathop{\mathrm{Im}}(d^ i) \to \mathop{\mathrm{Ker}}(d^{i + 1}) \to H^{i + 1}(E) \to 0 \]

and the assumption that $H^{i + 1}(E)$ is perfect of tor dimension $\leq 1$ to conclude that $\mathop{\mathrm{Im}}(d^ i)$ is finite locally free. The same argument used again for the short exact sequence

\[ 0 \to \mathop{\mathrm{Ker}}(d^ i) \to \mathcal{E}^ i \to \mathop{\mathrm{Im}}(d^ i) \to 0 \]

then gives that $\mathop{\mathrm{Ker}}(d^ i)$ is finite locally free. It follows that the distinguished triangles

\[ \tau _{\leq k - 1}E \to \tau _{\leq k}E \to H^ k(E)[-k] \to (\tau _{\leq k - 1}E)[1] \]

are represented by the following short exact sequences of bounded complexes of finite locally free modules

\[ \begin{matrix} & & & & & & 0 \\ & & & & & & \downarrow \\ \mathcal{E}^ a & \to & \ldots & \to & \mathcal{E}^{k - 2} & \to & \mathop{\mathrm{Ker}}(d^{k - 1}) \\ \downarrow & & & & \downarrow & & \downarrow \\ \mathcal{E}^ a & \to & \ldots & \to & \mathcal{E}^{k - 2} & \to & \mathcal{E}^{k - 1} & \to & \mathop{\mathrm{Ker}}(d^ k) \\ & & & & & & \downarrow & & \downarrow \\ & & & & & & \mathop{\mathrm{Im}}(d^{k - 1}) & \to & \mathop{\mathrm{Ker}}(d^ k) \\ & & & & & & \downarrow \\ & & & & & & 0 \end{matrix} \]

Here the complexes are the rows and the “obvious” zeros are omitted from the display.


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