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## Tag: 012X

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### 11.19. Spectral sequences: double complexes

Definition 11.19.1. Let $\mathcal{A}$ be an additive category. A double complex in $\mathcal{A}$ is given by a system $(\{A^{p, q}, d_1^{p, q}, d_2^{p, q}\}_{p, q\in \mathbf{Z}})$, where each $A^{p, q}$ is an object of $\mathcal{A}$ and $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ and $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ are morphisms of $\mathcal{A}$ such that the following rules hold:

1. $d_1^{p + 1, q} \circ d_1^{p, q} = 0$
2. $d_2^{p, q + 1} \circ d_2^{p, q} = 0$
3. $d_1^{p, q + 1} \circ d_2^{p, q} = d_2^{p + 1, q} \circ d_1^{p, q}$
for all $p, q \in \mathbf{Z}$.

This is just the cochain version of the definition. It says that each $A^{p, \bullet}$ is a cochain complex and that each $d_1^{p, \bullet}$ is a morphism of complexes $A^{p, \bullet} \to A^{p + 1, \bullet}$ such that $d_1^{p + 1, \bullet} \circ d_1^{p, \bullet} = 0$ as morphisms of complexes. In other words a double complex can be seen as a complex of complexes. So in the diagram $$\xymatrix{ \ldots & \ldots & \ldots & \ldots \\ \ldots \ar[r] & A^{p, q + 1} \ar[r]^{d_1^{p, q + 1}} \ar[u] & A^{p + 1, q + 1} \ar[r] \ar[u] & \ldots \\ \ldots \ar[r] & A^{p, q} \ar[r]^{d_1^{p, q}} \ar[u]^{d_2^{p, q}} & A^{p + 1, q} \ar[r] \ar[u]_{d_2^{p + 1, q}} & \ldots \\ \ldots & \ldots \ar[u] & \ldots \ar[u] & \ldots }$$ any square commutes. Warning: In the literature one encouters a different definition where a ''bicomplex'' or a ''double complex'' has the property that the squares in the diagram anti-commute.

It is customary to denote $H^p_I(K^{\bullet, \bullet})$ the complex with terms $\text{Ker}(d_1^{p, q})/\text{Im}(d_1^{p - 1, q})$ (varying $q$) and differential induced by $d_2$. Then $H^q_{II}(H^p_I(K^{\bullet, \bullet}))$ denotes its cohomology in degree $q$. It is also customary to denote $H^q_{II}(K^{\bullet, \bullet})$ the complex with terms $\text{Ker}(d_2^{p, q})/\text{Im}(d_2^{p, q - 1})$ (varying $p$) and differential induced by $d_1$. Then $H^p_I(H^q_{II}(K^{\bullet, \bullet}))$ denotes its cohomology in degree $q$.

Definition 11.19.2. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet}$ be a double complex. The {\it associated simple complex $sA^\bullet$}, also sometimes called the associated total complex is given by $$sA^n = \bigoplus\nolimits_{n = p + q} A^{p, q}$$ (if it exists) with differential $$d_{sA}^n = \sum\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q})$$ Alternatively, we sometimes write $\text{Tot}(A^{\bullet, \bullet})$ to denote this complex.

If countable direct sums exist in $\mathcal{A}$ or if for each $n$ at most finitely many $A^{p, n - p}$ are nonzero, then $sA^\bullet$ exists. Note that the definition is not symmetric in the indices $(p, q)$.

There are two natural filtrations on the simple complex $sA^\bullet$ associated to the double complex $A^{\bullet, \bullet}$. Namely, we define $$F_I^p(sA^n) = \bigoplus\nolimits_{i + j = n, i \geq p} A^{i, j} \quad \text{and} \quad F_{II}^p(sA^n) = \bigoplus\nolimits_{i + j = n, j \geq p} A^{i, j}.$$ It is immediately verified that $(sA^\bullet, F_I)$ and $(sA^\bullet, F_{II})$ are filtered complexes. By Section 11.18 we obtain two spectral sequences. It is customary to denote $({}'E_r, {}'d_r)_{r \geq 0}$ the spectral sequence associated to the filtration $F_I$ and to denote $({}''E_r, {}''d_r)_{r \geq 0}$ the spectral sequence associated to the filtration $F_{II}$. Here is a description of these spectral sequences.

Lemma 11.19.3. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet, \bullet}$ be a double complex. The spectral sequences associated to $K^{\bullet, \bullet}$ have the following terms:

1. ${}'E_0^{p, q} = K^{p, q}$ with ${}'d_0^{p, q} = (-1)^p d_2^{p, q} : K^{p, q} \to K^{p, q + 1}$,
2. ${}''E_0^{p, q} = K^{q, p}$ with ${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \to K^{q + 1, p}$,
3. ${}'E_1^{p, q} = H^q(K^{p, \bullet})$ with ${}'d_1^{p, q} = H^q(d_1^{p, \bullet})$,
4. ${}''E_1^{p, q} = H^q(K^{\bullet, p})$ with ${}''d_1^{p, q} = (-1)^q H^q(d_2^{\bullet, p})$,
5. ${}'E_2^{p, q} = H^p_I(H^q_{II}(K^{\bullet, \bullet}))$,
6. ${}''E_2^{p, q} = H^p_{II}(H^q_I(K^{\bullet, \bullet}))$.

Proof. Omitted. $\square$

These spectral sequences define two filtrations on $H^n(sK^\bullet)$. We will denote these $F_I$ and $F_{II}$.

Definition 11.19.4. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet, \bullet}$ be a double complex. We say the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$ converges if Definition 11.18.7 applies. In other words, for all $n$ $$\text{gr}_{F_I}(H^n(sK^\bullet)) = \oplus_{p + q = n} {}'E_\infty^{p, q}$$ via the canonical comparison of Lemma 11.18.6. Similarly we say the spectral sequence $({}''E_r, {}''d_r)_{r \geq 0}$ converges if Definition 11.18.7 applies. In other words for all $n$ $$\text{gr}_{F_{II}}(H^n(sK^\bullet)) = \oplus_{p + q = n} {}''E_\infty^{p, q}$$ via the canonical comparison of Lemma 11.18.6.

Same caveats as those following Definition 11.17.6.

Lemma 11.19.5 (First quadrant spectral sequence). Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet, \bullet}$ be a double complex. Assume that for every $n \in \mathbf{Z}$ there are only finitely many nonzero $K^{p, q}$ with $p + q = n$. Then

1. the filtrations $F_I$, $F_{II}$ on each $H^n(K^\bullet)$ are finite,
2. the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$ converges, and
3. the spectral sequence $({}''E_r, {}''d_r)_{r \geq 0}$ converges.

Proof. Follows immediately from Lemma 11.18.9. $\square$

Here is our first application of spectral sequences.

Lemma 11.19.6. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex. Let $A^{\bullet, \bullet}$ be a double complex. Let $\alpha^p : K^p \to A^{p, 0}$ be morphisms. Assume that

1. For every $n \in \mathbf{Z}$ there are only finitely many nonzero $A^{p, q}$ with $p + q = n$.
2. We have $A^{p, q} = 0$ if $q < 0$.
3. The morphisms $\alpha^p$ give rise to a morphism of complexes $\alpha : K^\bullet \to A^{\bullet, 0}$.
4. The complex $A^{p, \bullet}$ is exact in all degrees $q \not = 0$ and the morphism $K^p \to A^{p, 0}$ induces an isomorphism $K^p \to \text{Ker}(d_2^{p, 0})$.
Then $\alpha$ induces a quasi-isomorphism $$K^\bullet \longrightarrow sA^\bullet$$ of complexes. Moreover, there is a variant of this lemma involving the second variable $q$ instead of $p$.

Proof. The map is simply the map given by the morphisms $K^n \to A^{n, 0} \to sA^n$, which are easily seen to define a morphism of complexes. Consider the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$ associated to the double complex $A^{\bullet, \bullet}$. By Lemma 11.19.5 this spectral sequence converges and the induced filtration on $H^n(sA^\bullet)$ is finite for each $n$. By Lemma 11.19.3 and assumption (4) we have ${}'E_1^{p, q} = 0$ unless $q = 0$ and $'E_1^{p, 0} = K^p$ with differential ${}'d_1^{p, 0}$ identified with $d_K^p$. Hence ${}'E_2^{p, 0} = H^p(K^\bullet)$ and zero otherwise. This clearly implies $d_2^{p, q} = d_3^{p, q} = \ldots = 0$ for degree reasons. Hence we conclude that $H^n(sA^\bullet) = H^n(K^\bullet)$. We omit the verification that this identification is given by the morphism of complexes $K^\bullet \to sA^\bullet$ introduced above. $\square$

Remark 11.19.7. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$ be a weak Serre subcategory (see Definition 11.7.1). Suppose that $K^{\bullet, \bullet}$ is a double complex to which Lemma 11.19.5 applies such that for some $r \geq 0$ all the objects ${}'E_r^{p, q}$ belong to $\mathcal{C}$. We claim all the cohomology groups $H^n(sK^\bullet)$ belong to $\mathcal{C}$. Namely, the assumptions imply that the kernels and images of ${}'d_r^{p, q}$ are in $\mathcal{C}$. Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\mathcal{C}$. By induction we see that each ${}'E_\infty^{p, q}$ is in $\mathcal{C}$. Hence each $H^n(sK^\bullet)$ has a finite filtration whose subquotients are in $\mathcal{C}$. Using that $\mathcal{C}$ is closed under extensions we conclude that $H^n(sK^\bullet)$ is in $\mathcal{C}$ as claimed.

The same result holds for the second spectral sequence associated to $K^{\bullet, \bullet}$. Similarly, if $(K^\bullet, F)$ is a filtered complex to which Lemma 11.18.9 applies and for some $r \geq 0$ all the objects $E_r^{p, q}$ belong to $\mathcal{C}$, then each $H^n(K^\bullet)$ is an object of $\mathcal{C}$.

Remark 11.19.8. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet, \bullet}$ be a triple complex. The associated total complex is the complex with terms $$\text{Tot}^n(A^{\bullet, \bullet, \bullet}) = \bigoplus\nolimits_{p + q + r = n} A^{p, q, r}$$ and differential $$d^n_{\text{Tot}(A^{\bullet, \bullet, \bullet})} = \sum\nolimits_{p + q + r = n} d_1^{p, q, r} + (-1)^pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r}$$ With this definition a simple calculation shows that the associated total complex is equal to $$\text{Tot}(A^{\bullet, \bullet, \bullet}) = \text{Tot}(\text{Tot}_{12}(A^{\bullet, \bullet, \bullet})) = \text{Tot}(\text{Tot}_{23}(A^{\bullet, \bullet, \bullet}))$$ In other words, we can either first combine the first two of the variables and then combine sum of those with the last, or we can first combine the last two variables and then combine the first with the sum of the last two.

\section{Spectral sequences: double complexes}
\label{section-double-complex}

\begin{definition}
\label{definition-double-complex}
Let $\mathcal{A}$ be an additive category.
A {\it double complex} in $\mathcal{A}$ is given
by a system $(\{A^{p, q}, d_1^{p, q}, d_2^{p, q}\}_{p, q\in \mathbf{Z}})$,
where each $A^{p, q}$ is an object of $\mathcal{A}$ and
$d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ and
$d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ are morphisms of $\mathcal{A}$
such that the following rules hold:
\begin{enumerate}
\item $d_1^{p + 1, q} \circ d_1^{p, q} = 0$
\item $d_2^{p, q + 1} \circ d_2^{p, q} = 0$
\item $d_1^{p, q + 1} \circ d_2^{p, q} = d_2^{p + 1, q} \circ d_1^{p, q}$
\end{enumerate}
for all $p, q \in \mathbf{Z}$.
\end{definition}

\noindent
This is just the cochain version of the definition.
It says that each $A^{p, \bullet}$ is a cochain complex
and that each $d_1^{p, \bullet}$ is a morphism of complexes
$A^{p, \bullet} \to A^{p + 1, \bullet}$ such that
$d_1^{p + 1, \bullet} \circ d_1^{p, \bullet} = 0$ as morphisms
of complexes. In other words a double complex can be seen as
a complex of complexes. So in the diagram
$$\xymatrix{ \ldots & \ldots & \ldots & \ldots \\ \ldots \ar[r] & A^{p, q + 1} \ar[r]^{d_1^{p, q + 1}} \ar[u] & A^{p + 1, q + 1} \ar[r] \ar[u] & \ldots \\ \ldots \ar[r] & A^{p, q} \ar[r]^{d_1^{p, q}} \ar[u]^{d_2^{p, q}} & A^{p + 1, q} \ar[r] \ar[u]_{d_2^{p + 1, q}} & \ldots \\ \ldots & \ldots \ar[u] & \ldots \ar[u] & \ldots }$$
any square commutes.
Warning: In the literature one encouters a different definition
where a bicomplex'' or  a double complex'' has the property that
the squares in the diagram anti-commute.

\medskip\noindent
It is customary to denote $H^p_I(K^{\bullet, \bullet})$
the complex with terms $\text{Ker}(d_1^{p, q})/\text{Im}(d_1^{p - 1, q})$
(varying $q$) and differential induced by $d_2$.
Then $H^q_{II}(H^p_I(K^{\bullet, \bullet}))$ denotes its cohomology in
degree $q$. It is also customary to denote $H^q_{II}(K^{\bullet, \bullet})$
the complex with terms $\text{Ker}(d_2^{p, q})/\text{Im}(d_2^{p, q - 1})$
(varying $p$) and differential induced by $d_1$.
Then $H^p_I(H^q_{II}(K^{\bullet, \bullet}))$ denotes its cohomology in
degree $q$.

\begin{definition}
\label{definition-associated-simple-complex}
Let $\mathcal{A}$ be an additive category.
Let $A^{\bullet, \bullet}$ be a double complex.
The {\it associated simple complex $sA^\bullet$}, also
sometimes called the {\it associated total complex} is
given by
$$sA^n = \bigoplus\nolimits_{n = p + q} A^{p, q}$$
(if it exists) with differential
$$d_{sA}^n = \sum\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q})$$
Alternatively, we sometimes write $\text{Tot}(A^{\bullet, \bullet})$
to denote this complex.
\end{definition}

\noindent
If countable direct sums exist in $\mathcal{A}$ or if for each $n$ at most
finitely many $A^{p, n - p}$ are nonzero, then $sA^\bullet$ exists. Note that
the definition is {\it not} symmetric in the indices $(p, q)$.

\medskip\noindent
There are two natural filtrations on the simple complex $sA^\bullet$
associated to the double complex $A^{\bullet, \bullet}$. Namely, we
define
$$F_I^p(sA^n) = \bigoplus\nolimits_{i + j = n, \ i \geq p} A^{i, j} \quad \text{and} \quad F_{II}^p(sA^n) = \bigoplus\nolimits_{i + j = n, \ j \geq p} A^{i, j}.$$
It is immediately verified that $(sA^\bullet, F_I)$ and
$(sA^\bullet, F_{II})$ are filtered complexes.
By Section \ref{section-filtered-complex}
we obtain two spectral sequences. It is customary to
denote $({}'E_r, {}'d_r)_{r \geq 0}$ the spectral sequence associated
to the filtration $F_I$ and to denote $({}''E_r, {}''d_r)_{r \geq 0}$
the spectral sequence associated to the filtration $F_{II}$.
Here is a description of these spectral sequences.

\begin{lemma}
\label{lemma-ss-double-complex}
Let $\mathcal{A}$ be an abelian category.
Let $K^{\bullet, \bullet}$ be a double complex.
The spectral sequences associated to $K^{\bullet, \bullet}$
have the following terms:
\begin{enumerate}
\item ${}'E_0^{p, q} = K^{p, q}$ with
${}'d_0^{p, q} = (-1)^p d_2^{p, q} : K^{p, q} \to K^{p, q + 1}$,
\item ${}''E_0^{p, q} = K^{q, p}$ with
${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \to K^{q + 1, p}$,
\item ${}'E_1^{p, q} = H^q(K^{p, \bullet})$ with
${}'d_1^{p, q} = H^q(d_1^{p, \bullet})$,
\item ${}''E_1^{p, q} = H^q(K^{\bullet, p})$ with
${}''d_1^{p, q} = (-1)^q H^q(d_2^{\bullet, p})$,
\item ${}'E_2^{p, q} = H^p_I(H^q_{II}(K^{\bullet, \bullet}))$,
\item ${}''E_2^{p, q} = H^p_{II}(H^q_I(K^{\bullet, \bullet}))$.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\noindent
These spectral sequences define two filtrations on $H^n(sK^\bullet)$.
We will denote these $F_I$ and $F_{II}$.

\begin{definition}
\label{definition-ss-double-complex-converge}
Let $\mathcal{A}$ be an abelian category.
Let $K^{\bullet, \bullet}$ be a double complex.
We say the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$
{\it converges} if Definition \ref{definition-filtered-complex-ss-converges}
applies. In other words, for all $n$
$$\text{gr}_{F_I}(H^n(sK^\bullet)) = \oplus_{p + q = n} {}'E_\infty^{p, q}$$
via the canonical comparison of
Lemma \ref{lemma-compute-cohomology-filtered-complex}.
Similarly we say the spectral sequence $({}''E_r, {}''d_r)_{r \geq 0}$
{\it converges} if Definition \ref{definition-filtered-complex-ss-converges}
applies. In other words for all $n$
$$\text{gr}_{F_{II}}(H^n(sK^\bullet)) = \oplus_{p + q = n} {}''E_\infty^{p, q}$$
via the canonical comparison of
Lemma \ref{lemma-compute-cohomology-filtered-complex}.
\end{definition}

\noindent
Same caveats as those following
Definition \ref{definition-filtered-differential-ss-converges}.

Let $\mathcal{A}$ be an abelian category.
Let $K^{\bullet, \bullet}$ be a double complex.
Assume that for every $n \in \mathbf{Z}$ there are only finitely many nonzero
$K^{p, q}$ with $p + q = n$.
Then
\begin{enumerate}
\item the filtrations $F_I$, $F_{II}$ on each $H^n(K^\bullet)$ are finite,
\item the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$ converges, and
\item the spectral sequence $({}''E_r, {}''d_r)_{r \geq 0}$ converges.
\end{enumerate}
\end{lemma}

\begin{proof}
Follows immediately from Lemma \ref{lemma-biregular-ss-converges}.
\end{proof}

\noindent
Here is our first application of spectral sequences.

\begin{lemma}
\label{lemma-double-complex-gives-resolution}
Let $\mathcal{A}$ be an abelian category.
Let $K^\bullet$ be a complex.
Let $A^{\bullet, \bullet}$ be a double complex.
Let $\alpha^p : K^p \to A^{p, 0}$ be morphisms.
Assume that
\begin{enumerate}
\item For every $n \in \mathbf{Z}$ there are only finitely many nonzero
$A^{p, q}$ with $p + q = n$.
\item We have $A^{p, q} = 0$ if $q < 0$.
\item The morphisms $\alpha^p$ give rise to a morphism
of complexes $\alpha : K^\bullet \to A^{\bullet, 0}$.
\item The complex $A^{p, \bullet}$ is exact in all degrees
$q \not = 0$ and the morphism $K^p \to A^{p, 0}$ induces
an isomorphism $K^p \to \text{Ker}(d_2^{p, 0})$.
\end{enumerate}
Then $\alpha$ induces a quasi-isomorphism
$$K^\bullet \longrightarrow sA^\bullet$$
of complexes.
Moreover, there is a variant of this lemma involving the second
variable $q$ instead of $p$.
\end{lemma}

\begin{proof}
The map is simply the map given by the morphisms
$K^n \to A^{n, 0} \to sA^n$, which are easily seen to define
a morphism of complexes.
Consider the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$
associated to the double complex $A^{\bullet, \bullet}$.
By Lemma \ref{lemma-first-quadrant-ss} this spectral sequence converges
and the induced filtration on $H^n(sA^\bullet)$ is finite for each $n$.
By Lemma \ref{lemma-ss-double-complex} and assumption (4) we have
${}'E_1^{p, q} = 0$ unless $q = 0$ and $'E_1^{p, 0} = K^p$
with differential ${}'d_1^{p, 0}$ identified with $d_K^p$.
Hence ${}'E_2^{p, 0} = H^p(K^\bullet)$ and zero otherwise.
This clearly implies $d_2^{p, q} = d_3^{p, q} = \ldots = 0$
for degree reasons. Hence we conclude that $H^n(sA^\bullet) = H^n(K^\bullet)$.
We omit the verification that this identification is given by the
morphism of complexes $K^\bullet \to sA^\bullet$ introduced above.
\end{proof}

\begin{remark}
\label{remark-weak-serre-subcategory}
Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$
be a weak Serre subcategory (see
Definition \ref{definition-serre-subcategory}).
Suppose that $K^{\bullet, \bullet}$ is a double complex to which
applies such that for some $r \geq 0$ all the objects
${}'E_r^{p, q}$ belong to $\mathcal{C}$. We claim all the cohomology groups
$H^n(sK^\bullet)$ belong to $\mathcal{C}$. Namely, the assumptions imply
that the kernels and images of ${}'d_r^{p, q}$ are in $\mathcal{C}$.
Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\mathcal{C}$.
By induction we see that each ${}'E_\infty^{p, q}$ is in $\mathcal{C}$.
Hence each $H^n(sK^\bullet)$ has a finite filtration whose subquotients
are in $\mathcal{C}$. Using that $\mathcal{C}$ is closed under extensions
we conclude that $H^n(sK^\bullet)$ is in $\mathcal{C}$ as claimed.

\medskip\noindent
The same result holds for the second spectral sequence associated
to $K^{\bullet, \bullet}$. Similarly, if $(K^\bullet, F)$ is a filtered
complex to which
Lemma \ref{lemma-biregular-ss-converges}
applies and for some $r \geq 0$ all the objects $E_r^{p, q}$
belong to $\mathcal{C}$, then each $H^n(K^\bullet)$ is
an object of $\mathcal{C}$.
\end{remark}

\begin{remark}
\label{remark-triple-complex}
Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet, \bullet}$
be a triple complex. The associated total complex is the complex with
terms
$$\text{Tot}^n(A^{\bullet, \bullet, \bullet}) = \bigoplus\nolimits_{p + q + r = n} A^{p, q, r}$$
and differential
$$d^n_{\text{Tot}(A^{\bullet, \bullet, \bullet})} = \sum\nolimits_{p + q + r = n} d_1^{p, q, r} + (-1)^pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r}$$
With this definition a simple calculation shows that the associated total
complex is equal to
$$\text{Tot}(A^{\bullet, \bullet, \bullet}) = \text{Tot}(\text{Tot}_{12}(A^{\bullet, \bullet, \bullet})) = \text{Tot}(\text{Tot}_{23}(A^{\bullet, \bullet, \bullet}))$$
In other words, we can either first combine the first two of the variables
and then combine sum of those with the last, or we can first combine the
last two variables and then combine the first with the sum of the last two.
\end{remark}


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