The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

15.67 Hom complexes

Let $R$ be a ring. Let $L^\bullet $ and $M^\bullet $ be two complexes of $R$-modules. We construct a complex $\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )$. Namely, for each $n$ we set

\[ \mathop{\mathrm{Hom}}\nolimits ^ n(L^\bullet , M^\bullet ) = \prod \nolimits _{n = p + q} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, M^ p) \]

It is a good idea to think of $\mathop{\mathrm{Hom}}\nolimits ^ n$ as the $R$-module of all $R$-linear maps from $L^\bullet $ to $M^\bullet $ (viewed as graded modules) which are homogenous of degree $n$. In this terminology, we define the differential by the rule

\[ \text{d}(f) = \text{d}_ M \circ f - (-1)^ n f \circ \text{d}_ L \]

for $f \in \mathop{\mathrm{Hom}}\nolimits ^ n(L^\bullet , M^\bullet )$. We omit the verification that $\text{d}^2 = 0$. This construction is a special case of Differential Graded Algebra, Example 22.19.6. It follows immediately from the construction that we have

15.67.0.1
\begin{equation} \label{more-algebra-equation-cohomology-hom-complex} H^ n(\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{K(R)}(L^\bullet , M^\bullet [n]) \end{equation}

for all $n \in \mathbf{Z}$.

Lemma 15.67.1. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet , M^\bullet $ of $R$-modules there is a canonical isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (\text{Tot}(K^\bullet \otimes _ R L^\bullet ), M^\bullet ) \]

of complexes of $R$-modules.

Proof. Let $\alpha $ be an element of degree $n$ on the left hand side. Thus

\[ \alpha = (\alpha ^{p, q}) \in \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q}, \mathop{\mathrm{Hom}}\nolimits ^ p(L^\bullet , M^\bullet )) \]

Each $\alpha ^{p, q}$ is an element

\[ \alpha ^{p, q} = (\alpha ^{r, s, q}) \in \prod \nolimits _{r + s + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q}, \mathop{\mathrm{Hom}}\nolimits _ R(L^{-s}, M^ r)) \]

If we make the identifications

15.67.1.1
\begin{equation} \label{more-algebra-equation-identification} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q}, \mathop{\mathrm{Hom}}\nolimits _ R(L^{-s}, M^ r)) = \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q} \otimes _ R L^{-s}, M^ r) \end{equation}

then by our sign rules we get

\begin{align*} \text{d}(\alpha ^{r, s, q}) & = \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )} \circ \alpha ^{r, s, q} - (-1)^ n \alpha ^{r, s, q} \circ \text{d}_ K \\ & = \text{d}_ M \circ \alpha ^{r, s, q} - (-1)^{r + s} \alpha ^{r, s, q} \circ \text{d}_ L - (-1)^{r + s + q} \alpha ^{r, s, q} \circ \text{d}_ K \end{align*}

On the other hand, if $\beta $ is an element of degree $n$ of the right hand side, then

\[ \beta = (\beta ^{r, s, q}) \in \prod \nolimits _{r + s + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q} \otimes _ R L^{-s}, M^ r) \]

and by our sign rule (Homology, Definition 12.22.3) we get

\begin{align*} \text{d}(\beta ^{r, s, q}) & = \text{d}_ M \circ \beta ^{r, s, q} - (-1)^ n \beta ^{r, s, q} \circ \text{d}_{\text{Tot}(K^\bullet \otimes L^\bullet )} \\ & = \text{d}_ M \circ \beta ^{r, s, q} - (-1)^{r + s + q} \left( \beta ^{r, s, q} \circ \text{d}_ K + (-1)^{-q} \beta ^{r, s, q} \circ \text{d}_ L \right) \end{align*}

Thus we see that the map induced by the identifications (15.67.1.1) indeed is a morphism of complexes. $\square$

Lemma 15.67.2. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet , M^\bullet $ of $R$-modules there is a canonical morphism

\[ \text{Tot}\left( \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet ) \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet ) \right) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , M^\bullet ) \]

of complexes of $R$-modules.

Proof. An element $\alpha $ of degree $n$ of the left hand side is

\[ \alpha = (\alpha ^{p, q}) \in \bigoplus \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits ^ p(L^\bullet , M^\bullet ) \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^ q(K^\bullet , L^\bullet ) \]

The element $\alpha ^{p, q}$ is a finite sum $\alpha ^{p, q} = \sum \beta ^ p_ i \otimes \gamma ^ q_ i$ with

\[ \beta ^ p_ i = (\beta ^{r, s}_ i) \in \prod \nolimits _{r + s = p} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-s}, M^ r) \]

and

\[ \gamma ^ q_ i = (\gamma ^{u, v}_ i) \in \prod \nolimits _{u + v = q} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-v}, L^ u) \]

The map is given by sending $\alpha $ to $\delta = (\delta ^{r, v})$ with

\[ \delta ^{r, v} = \sum \nolimits _{i, s} \beta ^{r, s}_ i \circ \gamma ^{-s, v}_ i \in \mathop{\mathrm{Hom}}\nolimits _ R(K^{-v}, M^ r) \]

For given $r + v = n$ this sum is finite as there are only finitely many nonzero $\alpha ^{p, q}$, hence only finitely many nonzero $\beta ^ p_ i$ and $\gamma ^ q_ i$. By our sign rules we have

\begin{align*} \text{d}(\alpha ^{p, q}) & = \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )}(\alpha ^{p, q}) + (-1)^ p \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet )}(\alpha ^{p, q}) \\ & = \sum \Big( \text{d}_ M \circ \beta ^ p_ i \circ \gamma ^ q_ i - (-1)^ p \beta ^ p_ i \circ \text{d}_ L \circ \gamma ^ q_ i \Big) \\ & \quad + (-1)^ p \sum \Big( \beta ^ p_ i \circ \text{d}_ L \circ \gamma ^ q_ i - (-1)^ q \beta ^ p_ i \circ \gamma ^ q_ i \circ \text{d}_ K \Big) \\ & = \sum \Big( \text{d}_ M \circ \beta ^ p_ i \circ \gamma ^ q_ i -(-1)^ n \beta ^ p_ i \circ \gamma ^ q_ i \circ \text{d}_ K \Big) \end{align*}

It follows that the rules $\alpha \mapsto \delta $ is compatible with differentials and the lemma is proved. $\square$

Lemma 15.67.3. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet , M^\bullet $ of $R$-modules there is a canonical morphism

\[ \text{Tot}(\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet ) \otimes _ R K^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet ), M^\bullet ) \]

of complexes of $R$-modules functorial in all three complexes.

Proof. Consider an element $\beta $ of degree $n$ of the right hand side. Then

\[ \beta = (\beta ^{p, q}) \in \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits ^{-q}(K^\bullet , L^\bullet ), M^ p) \]

Each $\beta ^{p, q}$ is an element

\[ \beta ^{p, q} = (\beta ^{p, r, s}) \in \prod \nolimits _{p + r + s = n} \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(K^ s, L^{-r}), M^ p) \]

We can apply the differentials $\text{d}_ M$ and $\text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet )}$ to the element $\beta ^{p, q}$ and we can apply the differentials $\text{d}_ K$, $\text{d}_ L$, $\text{d}_ M$ to the element $\beta ^{p, r, s}$. We omit the precise definitions. The our sign rules tell us that

\begin{align*} \text{d}(\beta ^{p, r, s}) & = \text{d}_ M(\beta ^{p, r, s}) - (-1)^ n \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet )}(\beta ^{p, r, s}) \\ & = \text{d}_ M(\beta ^{p, r, s}) - (-1)^ n \left( \text{d}_ L(\beta ^{p, r, s}) - (-1)^{r + s} \text{d}_ K(\beta ^{p, r, s}) \right) \\ & = \text{d}_ M(\beta ^{p, r, s}) - (-1)^ n \text{d}_ L(\beta ^{p, r, s}) + (-1)^ p \text{d}_ K(\beta ^{p, r, s}) \end{align*}

On the other hand, an element $\alpha $ of degree $n$ of the left hand side looks like

\[ \alpha = (\alpha ^{t, s}) \in \bigoplus \nolimits _{t + s = n} \mathop{\mathrm{Hom}}\nolimits ^ t(L^\bullet , M^\bullet ) \otimes K^ s \]

Each $\alpha ^{t, s}$ maps to an element

\[ \alpha ^{t, s} \mapsto (\alpha ^{p, r, s}) \in \prod \nolimits _{p + r + s = n} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-r}, M^ p) \otimes _ R K^ s \]

By our sign rules and with conventions as above we get

\begin{align*} \text{d}(\alpha ^{p, r, s}) & = \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )}(\alpha ^{p, r, s}) + (-1)^{p + r} \text{d}_ K(\alpha ^{p, r, s}) \\ & = \text{d}_ M(\alpha ^{p, r, s}) - (-1)^{p + r} \text{d}_ L(\alpha ^{p, r, s}) + (-1)^{p + r} \text{d}_ K(\alpha ^{p, r, s}) \end{align*}

To define our map we will use the canonical maps

\[ c_{p, r, s} : \mathop{\mathrm{Hom}}\nolimits _ R(L^{-r}, M^ p) \otimes _ R K^ s \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(K^ s, L^{-r}), M^ p) \]

which sends $\varphi \otimes k$ to the map $\psi \mapsto \varphi (\psi (k))$. This is functorial in all three variables. However, since the signs above do not match we need to use instead some map

\[ \epsilon _{p, r, s} c_{p, r, s} \]

for some sign $\epsilon _{p, r, s}$. Looking at the signs above we find that we need to find a solution for the equations

\[ \epsilon _{p, r, s} = \epsilon _{p + 1, r, s}, \quad \epsilon _{p, r, s} (-1)^ s = \epsilon _{p, r + 1, s}, \quad \epsilon _{p, r, s} (-1)^ r = \epsilon _{p, r, s + 1} \]

A good solution is to take $\epsilon _{p, r, s} = (-1)^{rs}$. The choice of this sign is explained in the remark following the proof. $\square$

Remark 15.67.4. In the yoga of super vector spaces the sign used in the proof of Lemma 15.67.3 above can be explained as follows. A super vector space is just a vector space $V$ which comes with a direct sum decomposition $V = V^+ \oplus V^-$. Here we think of the elements of $V^+$ as the even elements and the elements of $V^-$ as the odd ones. Given two super vector spaces $V$ and $W$ we set

\[ (V \otimes W)^+ = (V^+ \otimes W^+) \oplus (V^- \otimes W^-) \]

and similarly for the odd part. In the category of super vector spaces the isomorphism

\[ V \otimes W \longrightarrow W \otimes V \]

is defined to be the usual one, except that on the summand $V^- \otimes W^-$ we use the negative of the usual identification. In this way we obtain a tensor category (where $\otimes $ is symmetric and associative with $1$). The category of super vector spaces has an internal hom which we denote $V^\vee $. One checks that the canonical isomorphisms $\mathop{\mathrm{Hom}}\nolimits (V, W) = W \otimes V^\vee $ and $\mathop{\mathrm{Hom}}\nolimits (V, W)^\vee = V \otimes W^\vee $ do not involve signs. Finally, given three super vector spaces $U$, $V$, $W$ we can consider the analogue

\[ c : \mathop{\mathrm{Hom}}\nolimits (V, W) \otimes U \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\mathop{\mathrm{Hom}}\nolimits (U, V), W) \]

of the maps $c_{p, r, s}$ which occur in the lemma above. Using the formulae given above (which do not involve signs) this becomes a map

\[ W \otimes V^\vee \otimes U \longrightarrow W \otimes U \otimes V^\vee \]

which involves a $(-1)$ on elements $w \otimes v^\vee \otimes u$ if $v^\vee $ and $u$ are odd.

Lemma 15.67.5. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet , M^\bullet $ of $R$-modules there is a canonical morphism

\[ \text{Tot}(K^\bullet \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , L^\bullet )) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , \text{Tot}(K^\bullet \otimes _ R L^\bullet )) \]

of complexes of $R$-modules functorial in all three complexes.

Proof. Let $\alpha $ be an element of degree $n$ of the right hand side. Thus

\[ \alpha = (\alpha ^{p, q}) \in \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, \text{Tot}^ p(K^\bullet \otimes _ R L^\bullet )) \]

Each $\alpha ^{p, q}$ is an element

\[ \alpha ^{p, q} = (\alpha ^{r, s, q}) \in \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, \bigoplus \nolimits _{r + s + q = n} K^ r \otimes _ R L^ s) \]

where we think of $\alpha ^{r, s, q}$ as a family of maps such that for every $x \in M^{-q}$ only a finite number of $\alpha ^{r, s, q}(x)$ are nonzero. By our sign rules we get

\begin{align*} \text{d}(\alpha ^{r, s, q}) & = \text{d}_{\text{Tot}(K^\bullet \otimes _ R L^\bullet )} \circ \alpha ^{r, s, q} - (-1)^ n \alpha ^{r, s, q} \circ \text{d}_ M \\ & = \text{d}_ K \circ \alpha ^{r, s, q} + (-1)^ r \text{d}_ L \circ \alpha ^{r, s, q} - (-1)^ n \alpha ^{r, s, q} \circ \text{d}_ M \end{align*}

On the other hand, if $\beta $ is an element of degree $n$ of the left hand side, then

\[ \beta = (\beta ^{p, q}) \in \bigoplus \nolimits _{p + q = n} K^ p \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^ q(M^\bullet , L^\bullet ) \]

and we can write $\beta ^{p, q} = \sum \gamma _ i^ p \otimes \delta _ i^ q$ with $\gamma _ i^ p \in K^ p$ and

\[ \delta _ i^ q = (\delta _ i^{r, s}) \in \prod \nolimits _{r + s = q} \mathop{\mathrm{Hom}}\nolimits _ R(M^{-s}, L^ r) \]

By our sign rules we have

\begin{align*} \text{d}(\beta ^{p, q}) & = \text{d}_ K(\beta ^{p, q}) + (-1)^ p \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , L^\bullet )}(\beta ^{p, q}) \\ & = \sum \text{d}_ K(\gamma _ i^ p) \otimes \delta _ i^ q + (-1)^ p \sum \gamma _ i^ p \otimes (\text{d}_ L \circ \delta _ i^ q - (-1)^ q \delta _ i^ q \circ \text{d}_ M) \end{align*}

We send the element $\beta $ to $\alpha $ with

\[ \alpha ^{r, s, q} = c^{r, s, q}(\sum \gamma _ i^ r \otimes \delta _ i^{s, q}) \]

where $c^{r, s, q} : K^ r \otimes _ R \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, L^ s) \to \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, K^ r \otimes _ R L^ s)$ is the canonical map. For a given $\beta $ and $r$ there are only finitely many nonzero $\gamma _ i^ r$ hence only finitely many nonzero $\alpha ^{r, s, q}$ are nonzero (for a given $r$). Thus this family of maps satisfies the conditions above and the map is well defined. Comparing signs we see that this is compatible with differentials. $\square$

Lemma 15.67.6. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet $ of $R$-modules there is a canonical morphism

\[ K^\bullet \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , \text{Tot}(K^\bullet \otimes _ R L^\bullet )) \]

of complexes of $R$-modules functorial in both complexes.

Proof. This is a special case of Lemma 15.67.5 but we will also construct it directly here. Let $\alpha $ be an element of degree $n$ of the right hand side. Thus

\[ \alpha = (\alpha ^{p, q}) \in \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, \text{Tot}^ p(K^\bullet \otimes _ R L^\bullet )) \]

Each $\alpha ^{p, q}$ is an element

\[ \alpha ^{p, q} = (\alpha ^{r, s, q}) \in \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, \bigoplus \nolimits _{r + s + q = n} K^ r \otimes _ R L^ s) \]

where we think of $\alpha ^{r, s, q}$ as a family of maps such that for every $x \in L^{-q}$ only a finite number of $\alpha ^{r, s, q}(x)$ are nonzero. By our sign rules we get

\begin{align*} \text{d}(\alpha ^{r, s, q}) & = \text{d}_{\text{Tot}(K^\bullet \otimes _ R L^\bullet )} \circ \alpha ^{r, s, q} - (-1)^ n \alpha ^{r, s, q} \circ \text{d}_ L \\ & = \text{d}_ K \circ \alpha ^{r, s, q} + (-1)^ r \text{d}_ L \circ \alpha ^{r, s, q} - (-1)^ n \alpha ^{r, s, q} \circ \text{d}_ L \end{align*}

Now an element $\beta \in K^ n$ we send to $\alpha $ with $\alpha ^{n, -q, q} = \beta \otimes \text{id}_{L^{-q}}$ and $\alpha ^{r, s, q} = 0$ if $r \not= n$. This is indeed an element as above, as for fixed $q$ there is only one nonzero $\alpha ^{r, s, q}$. The description of the differential shows this is compatible with differentials. $\square$


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