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

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 $U$.

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

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

functorial in $M$ where the differentials on the $\mathop{\mathrm{Hom}}\nolimits$-complex are the contragredients of the differentials on $I^\bullet (f_1^ e, \ldots , f_ r^ e)$.

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 $\mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet (f_1^ e, \ldots , f_ r^ e), M)$ is 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

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

For $t$ as above we set

$\Psi (t)_{i_0 \ldots i_ p} = (-1)^ p \frac{t([i_0, \ldots , i_ p])}{f_{i_0}^ e\ldots f_{i_ p}^ e}$

It is clear that $\Psi (t)$ is an alternating cochain. The rule above 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 (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 the rule $\Psi$ defines an isomorphism of cochain modules in degree $p$ in the colimit. To finish the proof we have to show that the map is compatible with differentials. To see this, for $t$ as above we compute

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

Recall that the differentials on $I^\bullet (f_1^ e, \ldots , f_ r^ e)$ are the negative of the differentials on $K^\bullet (f_1, \ldots , f_ r)$. Thus

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

The two formulas agree concluding the proof. $\square$

Suppose given a finite complex $I^\bullet$ of $A$-modules and a complex of $A$-modules $M^\bullet$. Then we have the corresponding Hom complex $\mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet , M^\bullet )$. This is a complex with degree $n$ term given by

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

and differential as described in More on Algebra, Section 15.71. 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

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

functorial in $M^\bullet$.

Proof. Consider the double complex $F^{\bullet , \bullet }$ with terms $F^{p, q} = \mathcal{C}_{alt}^ p(\mathcal{U}, \mathcal{F}^ q)$ discussed in Cohomology, Section 20.25. Consider the double complex $G^{\bullet , \bullet }$ with terms $G^{p, q} = \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _ A(I^{-p}(f_1^ e, \ldots , f_ r^ e), M^ q)$ and differentials given by functoriality (without the intervention of signs). The maps $\psi ^{p, q} : G^{p, q} \to F^{p, q}$ constructed in the proof of Lemma 36.9.2 are isomorphisms and compatible with the differentials $d_1$ (by the lemma) and $d_2$ (this is clear). However, the differentials $d$ on the complexes on the left and right hand side of the arrow in the lemma have different signs. Namely, for $g \in G^{p, q}$ is given by

$d(g) = d_2(g) - (-1)^{p + q} d_1(g)$

(see More on Algebra, Section 15.71) and the differential for $f \in F^{p, q}$ is given by

$d(f) = d_1(f) + (-1)^ p d_2(f)$

Thus we can fix the signs by multiplying $\psi ^{p, q}$ by $(-1)^{pq + p(p - 1)/2}$. $\square$

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 20.40.2. $\square$

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

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

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

By More on Algebra, Lemma 15.73.2 and Equation (15.73.0.2) taking $H^0$ of the complex $\mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet )$ computes $\mathop{\mathrm{Hom}}\nolimits$ in $D(A)$. Thus 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 $\mathcal{O}_ X \to K_ e \to E$. $\square$

Comment #8620 by nkym on

5 lines above 36.9.0.1, you need an additional ) to $H^{i-1}$.

Comment #8626 by nkym on

In the definition of Hom complexes after Lemma 36.9.2, in the equation $\oplus_{p+q=n}Hom_A(I^p, M^q)$, one of $p$ should be negated.

Comment #9416 by on

Thanks for your comments. I also tried to match the hom complexes with the definitions of them elsewhere and unfortunately this made things worse. Anyway, see changes here.

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