The Stacks project

Lemma 15.76.2. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $K^\bullet $ be a pseudo-coherent complex of $R$-modules. Assume that for some $i \in \mathbf{Z}$ the map

\[ H^ i(K^\bullet ) \otimes _ R \kappa (\mathfrak p) \longrightarrow H^ i(K^\bullet \otimes _ R^{\mathbf{L}} \kappa (\mathfrak p)) \]

is surjective. Then there exists an $f \in R$, $f \not\in \mathfrak p$ such that $\tau _{\geq i + 1}(K^\bullet \otimes _ R R_ f)$ is a perfect object of $D(R_ f)$ with tor amplitude in $[i + 1, \infty ]$ and a canonical isomorphism

\[ K^\bullet \otimes _ R R_ f \cong \tau _{\leq i}(K^\bullet \otimes _ R R_ f) \oplus \tau _{\geq i + 1}(K^\bullet \otimes _ R R_ f) \]

in $D(R_ f)$.

Proof. In this proof all tensor products are over $R$ and we write $\kappa = \kappa (\mathfrak p)$. We may assume that $K^\bullet $ is a bounded above complex of finite free $R$-modules. Let us inspect what is happening in degree $i$:

\[ \ldots \to K^{i - 1} \xrightarrow {d^{i - 1}} K^ i \xrightarrow {d^ i} K^{i + 1} \to \ldots \]

Let $0 \subset V \subset W \subset K^ i \otimes \kappa $ be defined by the formulas

\[ V = \mathop{\mathrm{Im}}\left( K^{i - 1} \otimes \kappa \to K^ i \otimes \kappa \right) \quad \text{and}\quad W = \mathop{\mathrm{Ker}}\left( K^ i \otimes \kappa \to K^{i + 1} \otimes \kappa \right) \]

Set $\dim (V) = r$, $\dim (W/V) = s$, and $\dim (K^ i \otimes \kappa /W) = t$. We can pick $x_1, \ldots , x_ r \in K^{i - 1}$ which map by $d^{i - 1}$ to a basis of $V$. By our assumption we can pick $y_1, \ldots , y_ s \in \mathop{\mathrm{Ker}}(d^ i)$ mapping to a basis of $W/V$. Finally, choose $z_1, \ldots , z_ t \in K^ i$ mapping to a basis of $K^ i \otimes \kappa /W$. Then we see that the elements $d^ i(z_1), \ldots , d^ i(z_ t) \in K^{i + 1}$ are linearly independent in $K^{i + 1} \otimes _ R \kappa $. By Algebra, Lemma 10.79.4 we may after replacing $R$ by $R_ f$ for some $f \in R$, $f \not\in \mathfrak p$ assume that

  1. $d^ i(x_ a), y_ b, z_ c$ is an $R$-basis of $K^ i$,

  2. $d^ i(z_1), \ldots , d^ i(z_ t)$ are $R$-linearly independent in $K^{i + 1}$, and

  3. the quotient $E^{i + 1} = K^{i + 1}/\sum Rd^ i(z_ c)$ is finite projective.

Since $d^ i$ annihilates $d^{i - 1}(x_ a)$ and $y_ b$, we deduce from condition (2) that $E^{i + 1} = \mathop{\mathrm{Coker}}(d^ i : K^ i \to K^{i + 1})$. Thus we see that

\[ \tau _{\geq i + 1}K^\bullet = (\ldots \to 0 \to E^{i + 1} \to K^{i + 2} \to \ldots ) \]

is a bounded complex of finite projective modules sitting in degrees $[i + 1, b]$ for some $b$. Thus $\tau _{\geq i + 1}K^\bullet $ is perfect of amplitude $[i + 1, b]$. Since $\tau _{\leq i}K^\bullet $ has no cohomology in degrees $> i$, we may apply Lemma 15.76.1 to the distinguished triangle

\[ \tau _{\leq i}K^\bullet \to K^\bullet \to \tau _{\geq i + 1}K^\bullet \to (\tau _{\leq i}K^\bullet )[1] \]

(Derived Categories, Remark 13.12.4) to conclude. $\square$

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