The Stacks project

36.9 Koszul complexes

Let $A$ be a ring and let $f_1, \ldots , f_ r$ be a sequence of elements of $A$. We have defined the Koszul complex $K_\bullet (f_1, \ldots , f_ r)$ in More on Algebra, Definition 15.28.2. It is a chain complex sitting in degrees $r, \ldots , 0$. We turn this into a cochain complex $K^\bullet (f_1, \ldots , f_ r)$ by setting $K^{-n}(f_1, \ldots , f_ r) = K_ n(f_1, \ldots , f_ r)$ and using the same differentials. In the rest of this section all the complexes will be cochain complexes.

We define a complex $I^\bullet (f_1, \ldots , f_ r)$ such that we have a distinguished triangle

\[ I^\bullet (f_1, \ldots , f_ r) \to A \to K^\bullet (f_1, \ldots , f_ r) \to I^\bullet (f_1, \ldots , f_ r)[1] \]

in $K(A)$. In other words, we set

\[ I^ i(f_1, \ldots , f_ r) = \left\{ \begin{matrix} K^{i - 1}(f_1, \ldots , f_ r) & \text{if } i \leq 0 \\ 0 & \text{else} \end{matrix} \right. \]

and we use the negative of the differential on $K^\bullet (f_1, \ldots , f_ r)$. The maps in the distinguished triangle are the obvious ones. Note that $I^0(f_1, \ldots , f_ r) = A^{\oplus r} \to A$ is given by multiplication by $f_ i$ on the $i$th factor. Hence $I^\bullet (f_1, \ldots , f_ r) \to A$ factors as

\[ I^\bullet (f_1, \ldots , f_ r) \to I \to A \]

where $I = (f_1, \ldots , f_ r)$. In fact, there is a short exact sequence

\[ 0 \to H^{-1}(K^\bullet (f_1, \ldots , f_ s)) \to H^0(I^\bullet (f_1, \ldots , f_ s)) \to I \to 0 \]

and for every $i < 0$ we have $H^ i(I^\bullet (f_1, \ldots , f_ r)) = H^{i - 1}(K^\bullet (f_1, \ldots , f_ r)$. Observe that given a second sequence $g_1, \ldots , g_ r$ of elements of $A$ there are canonical maps

\[ I^\bullet (f_1g_1, \ldots , f_ rg_ r) \to I^\bullet (f_1, \ldots , f_ r) \quad \text{and}\quad K^\bullet (f_1g_1, \ldots , f_ rg_ r) \to K^\bullet (f_1, \ldots , f_ r) \]

compatible with the maps described above. The first of these maps is given by multiplication by $g_ i$ on the $i$th summand of $I^0(f_1g_1, \ldots , f_ rg_ r) = A^{\oplus r}$. In particular, given $f_1, \ldots , f_ r$ we obtain an inverse system of complexes

36.9.0.1
\begin{equation} \label{perfect-equation-system} I^\bullet (f_1, \ldots , f_ r) \leftarrow I^\bullet (f_1^2, \ldots , f_ r^2) \leftarrow I^\bullet (f_1^3, \ldots , f_ r^3) \leftarrow \ldots \end{equation}

which will play an important role in that which is to follow. To easily formulate the following lemmas we fix some notation.

Situation 36.9.1. Here $A$ is a ring and $f_1, \ldots , f_ r$ is a sequence of elements of $A$. We set $X = \mathop{\mathrm{Spec}}(A)$ and $U = D(f_1) \cup \ldots \cup D(f_ r) \subset X$. We denote $\mathcal{U} : U = \bigcup _{i = 1, \ldots , r} D(f_ i)$ the given open covering of $U$.

Our first lemma is that the complexes above can be used to compute the cohomology of quasi-coherent sheaves on $U$. Suppose given a complex $I^\bullet $ of $A$-modules and an $A$-module $M$. Then we define $\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet , M)$ to be the complex with $n$th term $\mathop{\mathrm{Hom}}\nolimits _ A(I^{-n}, M)$ and differentials given as the contragredients of the differentials on $I^\bullet $.

Lemma 36.9.2. In Situation 36.9.1. Let $M$ be an $A$-module and denote $\mathcal{F}$ the associated $\mathcal{O}_ X$-module. Then there is a canonical isomorphism of complexes

\[ \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M) \longrightarrow \check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}) \]

functorial in $M$.

Proof. Recall that the alternating Čech complex is the subcomplex of the usual Čech complex given by alternating cochains, see Cohomology, Section 20.23. As usual we view a $p$-cochain in $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ as an alternating function $s$ on $\{ 1, \ldots , r\} ^{p + 1}$ whose value $s_{i_0\ldots i_ p}$ at $(i_0, \ldots , i_ p)$ lies in $M_{f_{i_0}\ldots f_{i_ p}} = \mathcal{F}(U_{i_0\ldots i_ p})$. On the other hand, a $p$-cochain $t$ in $\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M)$ is given by a map $t : \wedge ^{p + 1}(A^{\oplus r}) \to M$. Write $[i] \in A^{\oplus r}$ for the $i$th basis element and write

\[ [i_0, \ldots , i_ p] = [i_0] \wedge \ldots \wedge [i_ p] \in \wedge ^{p + 1}(A^{\oplus r}) \]

Then we send $t$ as above to $s$ with

\[ s_{i_0\ldots i_ p} = \frac{t([i_0, \ldots , i_ p])}{f_{i_0}^ e\ldots f_{i_ p}^ e} \]

It is clear that $s$ so defined is an alternating cochain. The construction of this map is compatible with the transition maps of the system as the transition map

\[ I^\bullet (f_1^ e, \ldots , f_ r^ e) \leftarrow I^\bullet (f_1^{e + 1}, \ldots , f_ r^{e + 1}), \]

of the (36.9.0.1) sends $[i_0, \ldots , i_ p]$ to $f_{i_0}\ldots f_{i_ p}[i_0, \ldots , i_ p]$. It is clear from the description of the localizations $M_{f_{i_0}\ldots f_{i_ p}}$ in Algebra, Lemma 10.9.9 that these maps define an isomorphism of cochain modules in degree $p$ in the limit. To finish the proof we have to show that the map is compatible with differentials. To see this recall that

\begin{align*} d(s)_{i_0\ldots i_{p + 1}} & = \sum \nolimits _{j = 0}^{p + 1} (-1)^ j s_{i_0\ldots \hat i_ j \ldots i_ p} \\ & = \sum \nolimits _{j = 0}^{p + 1} (-1)^ j \frac{t([i_0, \ldots , \hat i_ j, \ldots i_{p + 1}])}{f_{i_0}^ e\ldots \hat f_{i_ j}^ e \ldots f_{i_{p + 1}}^ e} \end{align*}

On the other hand, we have

\begin{align*} \frac{d(t)([i_0, \ldots , i_{p + 1}])}{f_{i_0}^ e\ldots f_{i_{p + 1}}^ e} & = \frac{t(d[i_0, \ldots , i_{p + 1}])}{f_{i_0}^ e\ldots f_{i_{p + 1}}^ e} \\ & = \frac{\sum _ j (-1)^ j f_{i_ j}^ e t([i_0, \ldots , \hat i_ j, \ldots i_{p + 1}])}{f_{i_0}^ e \ldots f_{i_{p + 1}}^ e} \end{align*}

The two formulas agree by inspection. $\square$

Suppose given a finite complex $I^\bullet $ of $A$-modules and a complex of $A$-modules $M^\bullet $. We obtain a double complex $H^{\bullet , \bullet } = \mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet , M^\bullet )$ where $H^{p, q} = \mathop{\mathrm{Hom}}\nolimits _ A(I^ p, M^ q)$. The first differential comes from the differential on $\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet , M^ q)$ and the second from the differential on $M^\bullet $. Associated to this double complex is the total complex with degree $n$ term given by

\[ \bigoplus \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ A(I^ p, M^ q) \]

and differential as in Homology, Definition 12.18.3. As our complex $I^\bullet $ has only finitely many nonzero terms, the direct sum displayed above is finite. The conventions for taking the total complex associated to a Čech complex of a complex are as in Cohomology, Section 20.25.

Lemma 36.9.3. In Situation 36.9.1. Let $M^\bullet $ be a complex of $A$-modules and denote $\mathcal{F}^\bullet $ the associated complex of $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism of complexes

\[ \mathop{\mathrm{colim}}\nolimits _ e \text{Tot}(\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet )) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \]

functorial in $M^\bullet $.

Proof. Immediate from Lemma 36.9.2 and our conventions for taking associated total complexes. $\square$

Lemma 36.9.4. In Situation 36.9.1. Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism

\[ \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (U, \mathcal{F}^\bullet ) \]

in $D(A)$ functorial in $\mathcal{F}^\bullet $.

Proof. Let $\mathcal{B}$ be the set of affine opens of $U$. Since the higher cohomology groups of a quasi-coherent module on an affine scheme are zero (Cohomology of Schemes, Lemma 30.2.2) this is a special case of Cohomology, Lemma 20.37.2. $\square$

In Situation 36.9.1 denote $I_ e$ the object of $D(\mathcal{O}_ X)$ corresponding to the complex of $A$-modules $I^\bullet (f_1^ e, \ldots , f_ r^ e)$ via the equivalence of Lemma 36.3.5. The maps (36.9.0.1) give a system

\[ I_1 \leftarrow I_2 \leftarrow I_3 \leftarrow \ldots \]

Moreover, there is a compatible system of maps $I_ e \to \mathcal{O}_ X$ which become isomorphisms when restricted to $U$. Thus we see that for every object $E$ of $D(\mathcal{O}_ X)$ there is a canonical map

36.9.4.1
\begin{equation} \label{perfect-equation-comparison} \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(I_ e, E) \longrightarrow H^0(U, E) \end{equation}

constructed by sending a map $I_ e \to E$ to its restriction to $U$ and using that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{O}_ U, E|_ U) = H^0(U, E)$.

Proof. By Lemma 36.3.5 we may assume that $E$ is given by a complex of quasi-coherent sheaves $\mathcal{F}^\bullet $. Let $M^\bullet = \Gamma (X, \mathcal{F}^\bullet )$ be the corresponding complex of $A$-modules. By Lemmas 36.9.3 and 36.9.4 we have quasi-isomorphisms

\[ \mathop{\mathrm{colim}}\nolimits _ e \text{Tot}(\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet )) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (U, \mathcal{F}^\bullet ) \]

Taking $H^0$ on both sides we obtain

\[ \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _{D(A)}(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet ) = H^0(U, E) \]

Since $\mathop{\mathrm{Hom}}\nolimits _{D(A)}(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(I_ e, E)$ by Lemma 36.3.5 the lemma follows. $\square$

In Situation 36.9.1 denote $K_ e$ the object of $D(\mathcal{O}_ X)$ corresponding to the complex of $A$-modules $K^\bullet (f_1^ e, \ldots , f_ r^ e)$ via the equivalence of Lemma 36.3.5. Thus we have distinguished triangles

\[ I_ e \to \mathcal{O}_ X \to K_ e \to I_ e[1] \]

and a system

\[ K_1 \leftarrow K_2 \leftarrow K_3 \leftarrow \ldots \]

compatible with the system $(I_ e)$. Moreover, there is a compatible system of maps

\[ K_ e \to H^0(K_ e) = \mathcal{O}_ X/(f_1^ e, \ldots , f_ r^ e) \]

Lemma 36.9.6. In Situation 36.9.1. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Assume that $H^ i(E)|_ U = 0$ for $i = - r + 1, \ldots , 0$. Then given $s \in H^0(X, E)$ there exists an $e \geq 0$ and a morphism $K_ e \to E$ such that $s$ is in the image of $H^0(X, K_ e) \to H^0(X, E)$.

Proof. Since $U$ is covered by $r$ affine opens we have $H^ j(U, \mathcal{F}) = 0$ for $j \geq r$ and any quasi-coherent module (Cohomology of Schemes, Lemma 30.4.2). By Lemma 36.3.4 we see that $H^0(U, E)$ is equal to $H^0(U, \tau _{\geq -r + 1}E)$. There is a spectral sequence

\[ H^ j(U, H^ i(\tau _{\geq -r + 1}E)) \Rightarrow H^{i + j}(U, \tau _{\geq -N}E) \]

see Derived Categories, Lemma 13.21.3. Hence $H^0(U, E) = 0$ by our assumed vanishing of cohomology sheaves of $E$. We conclude that $s|_ U = 0$. Think of $s$ as a morphism $\mathcal{O}_ X \to E$ in $D(\mathcal{O}_ X)$. By Proposition 36.9.5 the composition $I_ e \to \mathcal{O}_ X \to E$ is zero for some $e$. By the distinguished triangle $I_ e \to \mathcal{O}_ X \to K_ e \to I_ e[1]$ we obtain a morphism $K_ e \to E$ such that $s$ is the composition $\mathcal{O}_ X \to K_ e \to E$. $\square$


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