The Stacks project

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 15.64, Cohomology, Section 20.47, 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 15.75.6 the proof is complete. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ESP. Beware of the difference between the letter 'O' and the digit '0'.