moduli-curves-lemma-geometrically-reduced-curves

There exist an open substack $\Curvesstack^{geomred} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{geomred}$, \item the fibres of the morphism $X \to S$ are geometrically reduced (More on Morphisms of Spaces, Definition \ref{spaces-more-morphisms-definition-geometrically-reduced-fibre}), \end{enumerate} \item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{geomred}$, \item $X$ is geometrically reduced over $k$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-geomred-in-CM

We have $\Curvesstack^{geomred} \subset \Curvesstack^{CM}$ as open substacks of $\Curvesstack$.


moduli-curves-lemma-geometrically-reduced-connected-1-curves

There exist an open substack $\Curvesstack^{grc, 1} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{grc, 1}$, \item the geometric fibres of the morphism $X \to S$ are reduced, connected, and have dimension $1$, \end{enumerate} \item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{grc, 1}$, \item $X$ is geometrically reduced, geometrically connected, and has dimension $1$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-geomredcon-in-h0-1

We have $\Curvesstack^{grc, 1} \subset \Curvesstack^{h0, 1}$ as open substacks of $\Curvesstack$. In particular, given a family of curves $f : X \to S$ whose geometric fibres are reduced, connected and of dimension $1$, then $R^1f_*\mathcal{O}_X$ is a finite locally free $\mathcal{O}_S$-module whose formation commutes with arbitrary base change.


moduli-curves-lemma-one-piece-per-genus

There is a decomposition into open and closed substacks $$ \Curvesstack^{grc, 1} = \coprod\nolimits_{g \geq 0} \Curvesstack^{grc, 1}_g $$ where each $\Curvesstack^{grc, 1}_g$ is characterized as follows: \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{grc, 1}_g$, \item the geometric fibres of the morphism $f : X \to S$ are reduced, connected, of dimension $1$ and $R^1f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_S$-module of rank $g$, \end{enumerate} \item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{grc, 1}_g$, \item $X$ is geometrically reduced, geometrically connected, has dimension $1$, and has genus $g$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-gorenstein-curves

There exist an open substack $\Curvesstack^{Gorenstein} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{Gorenstein}$, \item the morphism $X \to S$ is Gorenstein, \end{enumerate} \item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{Gorenstein}$, \item $X$ is Gorenstein. \end{enumerate} \end{enumerate}


moduli-curves-lemma-gorenstein-1-curves

There exist an open substack $\Curvesstack^{Gorenstein, 1} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{Gorenstein, 1}$, \item the morphism $X \to S$ is Gorenstein and has relative dimension $1$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-dimension}), \end{enumerate} \item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{Gorenstein, 1}$, \item $X$ is Gorenstein and $X$ is equidimensional of dimension $1$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-lci-curves

There exist an open substack $\Curvesstack^{lci} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{lci}$, \item $X \to S$ is a local complete intersection morphism, and \item $X \to S$ is a syntomic morphism. \end{enumerate} \item given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{lci}$, \item $X$ is a local complete intersection over $k$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-isolated-sings-curves

There exist an open substack $\Curvesstack^{+} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{+}$, \item the singular locus of $X \to S$ endowed with any/some closed subspace structure is finite over $S$. \end{enumerate} \item given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{+}$, \item $X \to \Spec(k)$ is smooth except at finitely many points. \end{enumerate} \end{enumerate}


moduli-curves-lemma-in-smooth-locus

In the situation above the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through the open where $\Curvesstack \to \Spec(\mathbf{Z})$ is smooth, \item the deformation category $\Deformationcategory_X$ is unobstructed. \end{enumerate}


moduli-curves-lemma-big-smooth-part-curves

The open substack $$ \Curvesstack^{lci+} = \Curvesstack^{lci} \cap \Curvesstack^{+} \subset \Curvesstack $$ has the following properties \begin{enumerate} \item $\Curvesstack^{lci+} \to \Spec(\mathbf{Z})$ is smooth, \item given a family of curves $X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{lci+}$, \item $X \to S$ is a local complete intersection morphism and the singular locus of $X \to S$ endowed with any/some closed subspace structure is finite over $S$, \end{enumerate} \item given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{lci+}$, \item $X$ is a local complete intersection over $k$ and $X \to \Spec(k)$ is smooth except at finitely many points. \end{enumerate} \end{enumerate}


moduli-curves-lemma-smooth-curves

There exist an open substacks $$ \Curvesstack^{smooth, 1} \subset \Curvesstack^{smooth} \subset \Curvesstack $$ such that \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{smooth}$, resp.\ $\Curvesstack^{smooth, 1}$, \item $f$ is smooth, resp.\ smooth of relative dimension $1$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{smooth}$, resp.\ $\Curvesstack^{smooth, 1}$, \item $X$ is smooth over $k$, resp.\ $X$ is smooth over $k$ and $X$ is equidimensional of dimension $1$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-smooth-curves-h0

There exist an open substack $\Curvesstack^{smooth, h0} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{smooth}$, \item $f_*\mathcal{O}_X = \mathcal{O}_S$, this holds after any base change, and $f$ is smooth of relative dimension $1$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{smooth, h0}$, \item $X$ is smooth, $\dim(X) = 1$, and $k = H^0(X, \mathcal{O}_X)$, \item $X$ is smooth, $\dim(X) = 1$, and $X$ is geometrically connected, \item $X$ is smooth, $\dim(X) = 1$, and $X$ is geometrically integral, and \item $X_{\overline{k}}$ is a smooth curve. \end{enumerate} \end{enumerate}


moduli-curves-lemma-smooth-one-piece-per-genus

There is a decomposition into open and closed substacks $$ \mathcal{M} = \coprod\nolimits_{g \geq 0} \mathcal{M}_g $$ where each $\mathcal{M}_g$ is characterized as follows: \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\mathcal{M}_g$, \item $X \to S$ is smooth, $f_*\mathcal{O}_X = \mathcal{O}_S$, this holds after any base change, and $R^1f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_S$-module of rank $g$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\mathcal{M}_g$, \item $X$ is smooth, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, and $X$ has genus $g$, \item $X$ is smooth, $\dim(X) = 1$, $X$ is geometrically connected, and $X$ has genus $g$, \item $X$ is smooth, $\dim(X) = 1$, $X$ is geometrically integral, and $X$ has genus $g$, and \item $X_{\overline{k}}$ is a smooth curve of genus $g$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-smooth-dense

The inclusion $$ |\Curvesstack^{smooth}| \subset |\Curvesstack^{lci+}| $$ is that of an open dense subset.


moduli-curves-lemma-nodal-curves

There exist an open substack $\Curvesstack^{nodal} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{nodal}$, \item $f$ is at-worst-nodal of relative dimension $1$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{nodal}$, \item the singularities of $X$ are at-worst-nodal and $X$ is equidimensional of dimension $1$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-nodal-curves-smooth

The morphism $\Curvesstack^{nodal} \to \Spec(\mathbf{Z})$ is smooth.


moduli-curves-lemma-CM-dualizing

Let $X \to S$ be a family of curves with Cohen-Macaulay fibres equidimensional of dimension $1$ (Lemma \ref{lemma-CM-1-curves}). Then $\omega_{X/S}^\bullet = \omega_{X/S}[1]$ where $\omega_{X/S}$ is a pseudo-coherent $\mathcal{O}_X$-module flat over $S$ whose formation commutes with arbitrary base change.


moduli-curves-lemma-prestable-curves

There exist an open substack $\Curvesstack^{prestable} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{prestable}$, \item $X \to S$ is a prestable family of curves, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{prestable}$, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, and $k = H^0(X, \mathcal{O}_X)$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-prestable-one-piece-per-genus

There is a decomposition into open and closed substacks $$ \Curvesstack^{prestable} = \coprod\nolimits_{g \geq 0} \Curvesstack^{prestable}_g $$ where each $\Curvesstack^{prestable}_g$ is characterized as follows: \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{prestable}_g$, \item $X \to S$ is a prestable family of curves and $R^1f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_S$-module of rank $g$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{prestable}_g$, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, and the genus of $X$ is $g$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-semistable

Let $f : X \to S$ be a prestable family of curves of genus $g \geq 1$. Let $s \in S$ be a point of the base scheme. Let $m \geq 2$. The following are equivalent \begin{enumerate} \item $X_s$ does not have a rational tail (Algebraic Curves, Example \ref{curves-example-rational-tail}), and \item $f^*f_*\omega_{X/S}^{\otimes m} \to \omega_{X/S}^{\otimes m}$, is surjective over $f^{-1}(U)$ for some $s \in U \subset S$ open. \end{enumerate}


moduli-curves-lemma-semistable-curves

There exist an open substack $\Curvesstack^{semistable} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{semistable}$, \item $X \to S$ is a semistable family of curves, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{semistable}$, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, the genus of $X$ is $\geq 1$, and $X$ has no rational tails, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, and $\omega_{X_s}^{\otimes m}$ is globally generated for $m \geq 2$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-stable

Let $f : X \to S$ be a prestable family of curves of genus $g \geq 2$. Let $s \in S$ be a point of the base scheme. The following are equivalent \begin{enumerate} \item $X_s$ does not have a rational tail and does not have a rational bridge (Algebraic Curves, Examples \ref{curves-example-rational-tail} and \ref{curves-example-rational-bridge}), and \item $\omega_{X/S}$ is ample on $f^{-1}(U)$ for some $s \in U \subset S$ open. \end{enumerate}


moduli-curves-lemma-stable-curves

There exist an open substack $\Curvesstack^{stable} \subset \Curvesstack$ such that \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{stable}$, \item $X \to S$ is a stable family of curves, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{stable}$, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, the genus of $X$ is $\geq 2$, and $X$ has no rational tails or bridges, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, and $\omega_{X_s}$ is ample. \end{enumerate} \end{enumerate}


moduli-curves-lemma-stable-one-piece-per-genus

There is a decomposition into open and closed substacks $$ \overline{\mathcal{M}} = \coprod\nolimits_{g \geq 2} \overline{\mathcal{M}}_g $$ where each $\overline{\mathcal{M}}_g$ is characterized as follows: \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\overline{\mathcal{M}}_g$, \item $X \to S$ is a stable family of curves and $R^1f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_S$-module of rank $g$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\overline{\mathcal{M}}_g$, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, the genus of $X$ is $g$, and $X$ has no rational tails or bridges. \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, the genus of $X$ is $g$, and $\omega_{X_s}$ is ample. \end{enumerate} \end{enumerate}


moduli-curves-lemma-stable-curves-smooth

The morphisms $\overline{\mathcal{M}} \to \Spec(\mathbf{Z})$ and $\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$ are smooth.


moduli-curves-lemma-stable-curves-deligne-mumford

The stacks $\overline{\mathcal{M}}$ and $\overline{\mathcal{M}}_g$ are open substacks of $\Curvesstack^{DM}$. In particular, $\overline{\mathcal{M}}$ and $\overline{\mathcal{M}}_g$ are DM (Morphisms of Stacks, Definition \ref{stacks-morphisms-definition-absolute-separated}) as well as Deligne-Mumford stacks (Algebraic Stacks, Definition \ref{algebraic-definition-deligne-mumford}).


moduli-curves-lemma-smooth-dense-in-stable

Let $g \geq 2$. The inclusion $$ |\mathcal{M}_g| \subset |\overline{\mathcal{M}}_g| $$ is that of an open dense subset.


moduli-curves-lemma-contract

Let $S$ be a scheme and $s \in S$ a point. Let $f : X \to S$ and $g : Y \to S$ be families of curves. Let $c : X \to Y$ be a morphism over $S$. If $c_{s, *}\mathcal{O}_{X_s} = \mathcal{O}_{Y_s}$ and $R^1c_{s, *}\mathcal{O}_{X_s} = 0$, then after replacing $S$ by an open neighbourhood of $s$ we have $\mathcal{O}_Y = c_*\mathcal{O}_X$ and $R^1c_*\mathcal{O}_X = 0$ and this remains true after base change by any morphism $S' \to S$.


moduli-curves-lemma-contract-basic-uniqueness

Let $S$ be a scheme and $s \in S$ a point. Let $f : X \to S$ and $g_i : Y_i \to S$, $i = 1, 2$ be families of curves. Let $c_i : X \to Y_i$ be morphisms over $S$. Assume there is an isomorphism $Y_{1, s} \cong Y_{2, s}$ of fibres compatible with $c_{1, s}$ and $c_{2, s}$. If $c_{1, s, *}\mathcal{O}_{X_s} = \mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\mathcal{O}_{X_s} = 0$, then there exist an open neighbourhood $U$ of $s$ and an isomorphism $Y_{1, U} \cong Y_{2, U}$ of families of curves over $U$ compatible with the given isomorphism of fibres and with $c_1$ and $c_2$.


moduli-curves-lemma-contract-basic

Let $f : X \to S$ be a family of curves. Let $s \in S$ be a point. Let $h_0 : X_s \to Y_0$ be a morphism to a proper scheme $Y_0$ over $\kappa(s)$ such that $h_{0, *}\mathcal{O}_{X_s} = \mathcal{O}_{Y_0}$ and $R^1h_{0, *}\mathcal{O}_{X_s} = 0$. Then there exist an elementary \'etale neighbourhood $(U, u) \to (S, s)$, a family of curves $Y \to U$, and a morphism $h : X_U \to Y$ over $U$ whose fibre in $u$ is isomorphic to $h_0$.


moduli-curves-lemma-contract-prestable-to-stable

Let $f : X \to S$ be a prestable family of curves of genus $g \geq 1$. There is a factorization $X \to Y \to S$ of $f$ where $g : Y \to S$ is a stable family of curves and $c : X \to Y$ has the following properties \begin{enumerate} \item $\mathcal{O}_Y = c_*\mathcal{O}_X$ and $R^1c_*\mathcal{O}_X = 0$ and this remains true after base change by any morphism $S' \to S$, and \item for any $s \in S$ the morphism $c_s : X_s \to Y_s$ is the contraction of rational tails and bridges discussed in Algebraic Curves, Section \ref{curves-section-contracting-to-stable}. \end{enumerate} Moreover $c : X \to Y$ is unique up to unique isomorphism.


moduli-curves-lemma-stable-reduction

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $K = H^0(C, \mathcal{O}_C)$ having genus $g \geq 2$. The following are equivalent \begin{enumerate} \item $C$ has semistable reduction (Semistable Reduction, Definition \ref{models-definition-semistable}), or \item there is a stable family of curves over $R$ with generic fibre $C$. \end{enumerate}


moduli-curves-lemma-unique-stable-model

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth proper curve over $K$ with $K = H^0(C, \mathcal{O}_C)$ and genus $g$. If $X$ and $X'$ are models of $C$ (Semistable Reduction, Section \ref{models-section-models}) and $X$ and $X'$ are stable families of genus $g$ curves over $R$, then there exists an unique isomorphism $X \to X'$ of models.


moduli-curves-lemma-stable-separated

Let $g \geq 2$. The stack $\overline{\mathcal{M}}_g$ is separated.


moduli-curves-lemma-stable-quasi-compact

Let $g \geq 2$. The stack $\overline{\mathcal{M}}_g$ is quasi-compact.


dpa-lemma-silly

Let $A$ be a ring. Let $I$ be an ideal of $A$. \begin{enumerate} \item If $\gamma$ is a divided power structure\footnote{Here and in the following, $\gamma$ stands short for a sequence of maps $\gamma_1, \gamma_2, \gamma_3, \ldots$ from $I$ to $I$.} on $I$, then $n! \gamma_n(x) = x^n$ for $n \geq 1$, $x \in I$. \end{enumerate} Assume $A$ is torsion free as a $\mathbf{Z}$-module. \begin{enumerate} \item[(2)] A divided power structure on $I$, if it exists, is unique. \item[(3)] If $\gamma_n : I \to I$ are maps then $$ \gamma\text{ is a divided power structure} \Leftrightarrow n! \gamma_n(x) = x^n\ \forall x \in I, n \geq 1. $$ \item[(4)] The ideal $I$ has a divided power structure if and only if there exists a set of generators $x_i$ of $I$ as an ideal such that for all $n \geq 1$ we have $x_i^n \in (n!)I$. \end{enumerate}


dpa-lemma-check-on-generators

Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $\gamma_n : I \to I$, $n \geq 1$ be a sequence of maps. Assume \begin{enumerate} \item[(a)] (1), (3), and (4) of Definition \ref{definition-divided-powers} hold for all $x, y \in I$, and \item[(b)] properties (2) and (5) hold for $x$ in some set of generators of $I$ as an ideal. \end{enumerate} Then $\gamma$ is a divided power structure on $I$.


dpa-lemma-two-ideals

Let $A$ be a ring with two ideals $I, J \subset A$. Let $\gamma$ be a divided power structure on $I$ and let $\delta$ be a divided power structure on $J$. Then \begin{enumerate} \item $\gamma$ and $\delta$ agree on $IJ$, \item if $\gamma$ and $\delta$ agree on $I \cap J$ then they are the restriction of a unique divided power structure $\epsilon$ on $I + J$. \end{enumerate}


dpa-lemma-nil

Let $p$ be a prime number. Let $A$ be a ring, let $I \subset A$ be an ideal, and let $\gamma$ be a divided power structure on $I$. Assume $p$ is nilpotent in $A/I$. Then $I$ is locally nilpotent if and only if $p$ is nilpotent in $A$.


dpa-lemma-limits

The category of divided power rings has all limits and they agree with limits in the category of rings.


dpa-lemma-a-version-of-brown

Let $\mathcal{C}$ be the category of divided power rings. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item there exists a cardinal $\kappa$ such that for every $f \in F(A, I, \gamma)$ there exists a morphism $(A', I', \gamma') \to (A, I, \gamma)$ of $\mathcal{C}$ such that $f$ is the image of $f' \in F(A', I', \gamma')$ and $|A'| \leq \kappa$, and \item $F$ commutes with limits. \end{enumerate} Then $F$ is representable, i.e., there exists an object $(B, J, \delta)$ of $\mathcal{C}$ such that $$ F(A, I, \gamma) = \Hom_\mathcal{C}((B, J, \delta), (A, I, \gamma)) $$ functorially in $(A, I, \gamma)$.


dpa-lemma-colimits

The category of divided power rings has all colimits.


dpa-lemma-gamma-extends

Let $(A, I, \gamma)$ be a divided power ring. Let $A \to B$ be a ring map. If $\gamma$ extends to $B$ then it extends uniquely. Assume (at least) one of the following conditions holds \begin{enumerate} \item $IB = 0$, \item $I$ is principal, or \item $A \to B$ is flat. \end{enumerate} Then $\gamma$ extends to $B$.


dpa-lemma-kernel

Let $(A, I, \gamma)$ be a divided power ring. \begin{enumerate} \item If $\varphi : (A, I, \gamma) \to (B, J, \delta)$ is a homomorphism of divided power rings, then $\Ker(\varphi) \cap I$ is preserved by $\gamma_n$ for all $n \geq 1$. \item Let $\mathfrak a \subset A$ be an ideal and set $I' = I \cap \mathfrak a$. The following are equivalent \begin{enumerate} \item $I'$ is preserved by $\gamma_n$ for all $n > 0$, \item $\gamma$ extends to $A/\mathfrak a$, and \item there exist a set of generators $x_i$ of $I'$ as an ideal such that $\gamma_n(x_i) \in I'$ for all $n > 0$. \end{enumerate} \end{enumerate}


dpa-lemma-extend-to-completion

Let $(A, I, \gamma)$ be a divided power ring. Let $p$ be a prime. If $p$ is nilpotent in $A/I$, then \begin{enumerate} \item the $p$-adic completion $A^\wedge = \lim_e A/p^eA$ surjects onto $A/I$, \item the kernel of this map is the $p$-adic completion $I^\wedge$ of $I$, and \item each $\gamma_n$ is continuous for the $p$-adic topology and extends to $\gamma_n^\wedge : I^\wedge \to I^\wedge$ defining a divided power structure on $I^\wedge$. \end{enumerate} If moreover $A$ is a $\mathbf{Z}_{(p)}$-algebra, then \begin{enumerate} \item[(4)] for $e$ large enough the ideal $p^eA \subset I$ is preserved by the divided power structure $\gamma$ and $$ (A^\wedge, I^\wedge, \gamma^\wedge) = \lim_e (A/p^eA, I/p^eA, \bar\gamma) $$ in the category of divided power rings. \end{enumerate}


dpa-lemma-divided-power-polynomial-algebra

Let $(A, I, \gamma)$ be a divided power ring. There exists a unique divided power structure $\delta$ on $$ J = IA\langle x_1, \ldots, x_t \rangle + A\langle x_1, \ldots, x_t \rangle_{+} $$ such that \begin{enumerate} \item $\delta_n(x_i) = x_i^{[n]}$, and \item $(A, I, \gamma) \to (A\langle x_1, \ldots, x_t \rangle, J, \delta)$ is a homomorphism of divided power rings. \end{enumerate} Moreover, $(A\langle x_1, \ldots, x_t \rangle, J, \delta)$ has the following universal property: A homomorphism of divided power rings $\varphi : (A\langle x_1, \ldots, x_t \rangle, J, \delta) \to (C, K, \epsilon)$ is the same thing as a homomorphism of divided power rings $A \to C$ and elements $k_1, \ldots, k_t \in K$.


dpa-lemma-dpdga-good

Let $(A, \text{d}, \gamma)$ and $(B, \text{d}, \gamma)$ be as in Definition \ref{definition-divided-powers-dga}. Let $f : A \to B$ be a map of differential graded algebras compatible with divided power structures. Assume \begin{enumerate} \item $H_k(A) = 0$ for $k > 0$, and \item $f$ is surjective. \end{enumerate} Then $\gamma$ induces a divided power structure on the graded $R$-algebra $H(B)$.


dpa-lemma-base-change-div

Let $(A, \text{d}, \gamma)$ be as in Definition \ref{definition-divided-powers-dga}. Let $R \to R'$ be a ring map. Then $\text{d}$ and $\gamma$ induce similar structures on $A' = A \otimes_R R'$ such that $(A', \text{d}, \gamma)$ is as in Definition \ref{definition-divided-powers-dga}.


dpa-lemma-extend-differential

Let $(A, \text{d}, \gamma)$ be as in Definition \ref{definition-divided-powers-dga}. Let $d \geq 1$ be an integer. Let $A\langle T \rangle$ be the graded divided power polynomial algebra on $T$ with $\deg(T) = d$ constructed in Example \ref{example-adjoining-odd} or \ref{example-adjoining-even}. Let $f \in A_{d - 1}$ be an element with $\text{d}(f) = 0$. There exists a unique differential $\text{d}$ on $A\langle T\rangle$ such that $\text{d}(T) = f$ and such that $\text{d}$ is compatible with the divided power structure on $A\langle T \rangle$.


dpa-lemma-tate-resolution

Let $R \to S$ be a homomorphism of commutative rings. There exists a factorization $$ R \to A \to S $$ with the following properties: \begin{enumerate} \item $(A, \text{d}, \gamma)$ is as in Definition \ref{definition-divided-powers-dga}, \item $A \to S$ is a quasi-isomorphism (if we endow $S$ with the zero differential), \item $A_0 = R[x_j: j\in J] \to S$ is any surjection of a polynomial ring onto $S$, and \item $A$ is a graded divided power polynomial algebra over $R$. \end{enumerate} The last condition means that $A$ is constructed out of $A_0$ by successively adjoining a set of variables $T$ in each degree $> 0$ as in Example \ref{example-adjoining-odd} or \ref{example-adjoining-even}. Moreover, if $R$ is Noetherian and $R\to S$ is of finite type, then $A$ can be taken to have only finitely many generators in each degree.


dpa-lemma-tate-resoluton-pseudo-coherent-ring-map

Let $R \to S$ be a pseudo-coherent ring map (More on Algebra, Definition \ref{more-algebra-definition-pseudo-coherent-perfect}). Then Lemma \ref{lemma-tate-resolution} holds, with the resolution $A$ of $S$ having finitely many generators in each degree.


dpa-lemma-uniqueness-tate-resolution

Let $R$ be a commutative ring. Suppose that $(A, \text{d}, \gamma)$ and $(B, \text{d}, \gamma)$ are as in Definition \ref{definition-divided-powers-dga}. Let $\overline{\varphi} : H_0(A) \to H_0(B)$ be an $R$-algebra map. Assume \begin{enumerate} \item $A$ is a graded divided power polynomial algebra over $R$. \item $H_k(B) = 0$ for $k > 0$. \end{enumerate} Then there exists a map $\varphi : A \to B$ of differential graded $R$-algebras compatible with divided powers that lifts $\overline{\varphi}$.


dpa-lemma-get-derivation

Let $R$ be a ring. Let $(A, \text{d}, \gamma)$ be as in Definition \ref{definition-divided-powers-dga}. Let $R' \to R$ be a surjection of rings whose kernel has square zero and is generated by one element $f$. If $A$ is a graded divided power polynomial algebra over $R$ with finitely many variables in each degree, then we obtain a derivation $\theta : A/IA \to A/IA$ where $I$ is the annihilator of $f$ in $R$.


dpa-lemma-compute-theta

Assumption and notation as in Lemma \ref{lemma-get-derivation}. Suppose $S = H_0(A)$ is isomorphic to $R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ for some $n$, $m$, and $f_j \in R[x_1, \ldots, x_n]$. Moreover, suppose given a relation $$ \sum r_j f_j = 0 $$ with $r_j \in R[x_1, \ldots, x_n]$. Choose $r'_j, f'_j \in R'[x_1, \ldots, x_n]$ lifting $r_j, f_j$. Write $\sum r'_j f'_j = gf$ for some $g \in R/I[x_1, \ldots, x_n]$. If $H_1(A) = 0$ and all the coefficients of each $r_j$ are in $I$, then there exists an element $\xi \in H_2(A/IA)$ such that $\theta(\xi) = g$ in $S/IS$.


dpa-lemma-not-finite-pd

Let $R' \to R$ be a surjection of Noetherian rings whose kernel has square zero and is generated by one element $f$. Let $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Let $\sum r_j f_j = 0$ be a relation in $R[x_1, \ldots, x_n]$. Assume that \begin{enumerate} \item each $r_j$ has coefficients in the annihilator $I$ of $f$ in $R$, \item for some lifts $r'_j, f'_j \in R'[x_1, \ldots, x_n]$ we have $\sum r'_j f'_j = gf$ where $g$ is not nilpotent in $S$. \end{enumerate} Then $S$ does not have finite tor dimension over $R$ (i.e., $S$ is not a perfect $R$-algebra).


dpa-lemma-injective

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset J \subset A$ be proper ideals. If $A/J$ has finite tor dimension over $A/I$, then $I/\mathfrak m I \to J/\mathfrak m J$ is injective.


dpa-lemma-regular-sequence

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset J \subset A$ be proper ideals. Assume \begin{enumerate} \item $A/J$ has finite tor dimension over $A/I$, and \item $J$ is generated by a regular sequence. \end{enumerate} Then $I$ is generated by a regular sequence and $J/I$ is generated by a regular sequence.


dpa-lemma-perfect-map-ci

Let $R \to S$ be a local ring map of Noetherian local rings. Let $I \subset R$ and $J \subset S$ be ideals with $IS \subset J$. If $R \to S$ is flat and $S/\mathfrak m_RS$ is regular, then the following are equivalent \begin{enumerate} \item $J$ is generated by a regular sequence and $S/J$ has finite tor dimension as a module over $R/I$, \item $J$ is generated by a regular sequence and $\text{Tor}^{R/I}_p(S/J, R/\mathfrak m_R)$ is nonzero for only finitely many $p$, \item $I$ is generated by a regular sequence and $J/IS$ is generated by a regular sequence in $S/IS$. \end{enumerate}


dpa-lemma-quotient-regular-ring-by-regular-sequence

Let $R$ be a regular ring. Let $\mathfrak p \subset R$ be a prime. Let $f_1, \ldots, f_r \in \mathfrak p$ be a regular sequence. Then the completion of $$ A = (R/(f_1, \ldots, f_r))_\mathfrak p = R_\mathfrak p/(f_1, \ldots, f_r)R_\mathfrak p $$ is a complete intersection in the sense defined above.


dpa-lemma-quotient-regular-ring

Let $R$ be a regular ring. Let $\mathfrak p \subset R$ be a prime. Let $I \subset \mathfrak p$ be an ideal. Set $A = (R/I)_\mathfrak p = R_\mathfrak p/I_\mathfrak p$. The following are equivalent \begin{enumerate} \item the completion of $A$ is a complete intersection in the sense above, \item $I_\mathfrak p \subset R_\mathfrak p$ is generated by a regular sequence, \item the module $(I/I^2)_\mathfrak p$ can be generated by $\dim(R_\mathfrak p) - \dim(A)$ elements, \item add more here. \end{enumerate}


dpa-lemma-ci-good

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $\mathfrak p \subset A$ be a prime ideal. If $A$ is a complete intersection, then $A_\mathfrak p$ is a complete intersection too.


dpa-lemma-check-lci-agrees

Let $S$ be a finite type algebra over a field $k$. \begin{enumerate} \item for a prime $\mathfrak q \subset S$ the local ring $S_\mathfrak q$ is a complete intersection in the sense of Algebra, Definition \ref{algebra-definition-lci-local-ring} if and only if $S_\mathfrak q$ is a complete intersection in the sense of Definition \ref{definition-lci}, and \item $S$ is a local complete intersection in the sense of Algebra, Definition \ref{algebra-definition-lci-field} if and only if $S$ is a local complete intersection in the sense of Definition \ref{definition-lci}. \end{enumerate}


dpa-lemma-avramov

Let $A \to B$ be a flat local homomorphism of Noetherian local rings. Then the following are equivalent \begin{enumerate} \item $B$ is a complete intersection, \item $A$ and $B/\mathfrak m_A B$ are complete intersections. \end{enumerate}


dpa-lemma-ci-map-well-defined

Let $A \to B$ be a local homomorphism of Noetherian complete local rings. The following are equivalent \begin{enumerate} \item for some good factorization $A \to S \to B$ the kernel of $S \to B$ is generated by a regular sequence, and \item for every good factorization $A \to S \to B$ the kernel of $S \to B$ is generated by a regular sequence. \end{enumerate}


dpa-lemma-well-defined-if-you-can-find-good-factorization

Consider a commutative diagram $$ \xymatrix{ S \ar[r] & B \\ & A \ar[lu] \ar[u] } $$ of Noetherian local rings with $S \to B$ surjective, $A \to S$ flat, and $S/\mathfrak m_A S$ a regular local ring. The following are equivalent \begin{enumerate} \item $\Ker(S \to B)$ is generated by a regular sequence, and \item $A^\wedge \to B^\wedge$ is a complete intersection homomorphism as defined above. \end{enumerate}


dpa-lemma-finite-type-lci-map

Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. The following are equivalent \begin{enumerate} \item $A \to B$ is a local complete intersection in the sense of More on Algebra, Definition \ref{more-algebra-definition-local-complete-intersection}, \item for every prime $\mathfrak q \subset B$ and with $\mathfrak p = A \cap \mathfrak q$ the ring map $(A_\mathfrak p)^\wedge \to (B_\mathfrak q)^\wedge$ is a complete intersection homomorphism in the sense defined above. \end{enumerate}


dpa-lemma-local-perfect-diagonal

Let $A \to B$ be a local ring homomorphism of Noetherian local rings such that $B$ is flat and essentially of finite type over $A$. If $$ B \otimes_A B \longrightarrow B $$ is a perfect ring map, i.e., if $B$ has finite tor dimension over $B \otimes_A B$, then $B$ is the localization of a smooth $A$-algebra.


dpa-lemma-perfect-diagonal

Let $A \to B$ be a flat finite type ring map of Noetherian rings. If $$ B \otimes_A B \longrightarrow B $$ is a perfect ring map, i.e., if $B$ has finite tor dimension over $B \otimes_A B$, then $B$ is a smooth $A$-algebra.


dpa-lemma-free-summand-in-ideal-finite-proj-dim

\begin{reference} \cite{Vasconcelos} \end{reference} Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal of finite projective dimension over $R$. If $F \subset I/I^2$ is a direct summand isomorphic to $R/I$, then there exists a nonzerodivisor $x \in I$ such that the image of $x$ in $I/I^2$ generates $F$.


dpa-lemma-vasconcelos

\begin{reference} Local version of \cite[Theorem 1.1]{Vasconcelos} \end{reference} Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal of finite projective dimension over $R$. If $F \subset I/I^2$ is a direct summand free of rank $r$, then there exists a regular sequence $x_1, \ldots, x_r \in I$ such that $x_1 \bmod I^2, \ldots, x_r \bmod I^2$ generate $F$.


dpa-lemma-perfect-NL-lci

Let $A \to B$ be a perfect (More on Algebra, Definition \ref{more-algebra-definition-pseudo-coherent-perfect}) ring homomorphism of Noetherian rings. Then the following are equivalent \begin{enumerate} \item $\NL_{B/A}$ has tor-amplitude in $[-1, 0]$, \item $\NL_{B/A}$ is a perfect object of $D(B)$ with tor-amplitude in $[-1, 0]$, and \item $A \to B$ is a local complete intersection (More on Algebra, Definition \ref{more-algebra-definition-local-complete-intersection}). \end{enumerate}


dpa-lemma-flat-fp-NL-lci

Let $A \to B$ be a flat ring map of finite presentation. Then the following are equivalent \begin{enumerate} \item $\NL_{B/A}$ has tor-amplitude in $[-1, 0]$, \item $\NL_{B/A}$ is a perfect object of $D(B)$ with tor-amplitude in $[-1, 0]$, \item $A \to B$ is syntomic (Algebra, Definition \ref{algebra-definition-lci}), and \item $A \to B$ is a local complete intersection (More on Algebra, Definition \ref{more-algebra-definition-local-complete-intersection}). \end{enumerate}


dpa-proposition-avramov

Let $A \to B$ be a flat local homomorphism of Noetherian local rings. Then the following are equivalent \begin{enumerate} \item $B^\wedge$ is a complete intersection, \item $A^\wedge$ and $(B/\mathfrak m_A B)^\wedge$ are complete intersections. \end{enumerate}


dpa-proposition-avramov-map

Let $A \to B$ be a local homomorphism of Noetherian local rings. Then the following are equivalent \begin{enumerate} \item $B$ is a complete intersection and $\text{Tor}^A_p(B, A/\mathfrak m_A)$ is nonzero for only finitely many $p$, \item $A$ is a complete intersection and $A^\wedge \to B^\wedge$ is a complete intersection homomorphism in the sense defined above. \end{enumerate}


dpa-proposition-regular-ideal

\begin{reference} Variant of \cite[Corollary 1]{Vasconcelos}. See also \cite{Iyengar} and \cite{Ferrand-lci}. \end{reference} Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal which has finite projective dimension and such that $I/I^2$ is finite locally free over $R/I$. Then $I$ is a regular ideal (More on Algebra, Definition \ref{more-algebra-definition-regular-ideal}).


moduli-lemma-coherent-diagonal-affine-fp

The diagonal of $\Cohstack_{X/B}$ over $B$ is affine and of finite presentation.


moduli-lemma-coherent-qs-lfp

The morphism $\Cohstack_{X/B} \to B$ is quasi-separated and locally of finite presentation.


moduli-lemma-coherent-functorial

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be a quasi-finite morphism of algebraic spaces which are separated and of finite presentation over $B$. Then $\pi_*$ induces a morphism $\Cohstack_{X/B} \to \Cohstack_{Y/B}$.


moduli-lemma-coherent-open

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Then the morphism $\Cohstack_{X/B} \to \Cohstack_{Y/B}$ of Lemma \ref{lemma-coherent-functorial} is an open immersion.


moduli-lemma-open-P

In Situation \ref{situation-numerical} the stack $\Cohstack^P_{X/B}$ is algebraic and $$ \Cohstack^P_{X/B} \longrightarrow \Cohstack_{X/B} $$ is a flat closed immersion. If $I$ is finite or $B$ is locally Noetherian, then $\Cohstack^P_{X/B}$ is an open and closed substack of $\Cohstack_{X/B}$.


moduli-lemma-finite-list-perfect-objects

Let $f : X \to B$ be as in the introduction to this section. Let $E_1, \ldots, E_r \in D(\mathcal{O}_X)$ be perfect. Let $I = \mathbf{Z}^{\oplus r}$ and consider the map $$ I \longrightarrow D(\mathcal{O}_X),\quad (n_1, \ldots, n_r) \longmapsto E_1^{\otimes n_1} \otimes \ldots \otimes E_r^{\otimes n_r} $$ Let $P : I \to \mathbf{Z}$ be a map. Then $\Cohstack^P_{X/B} \subset \Cohstack_{X/B}$ as defined in Situation \ref{situation-numerical} is an open and closed substack.


moduli-lemma-quot-diagonal-closed

The diagonal of $\Quotfunctor_{\mathcal{F}/X/B} \to B$ is a closed immersion. If $\mathcal{F}$ is of finite type, then the diagonal is a closed immersion of finite presentation.


moduli-lemma-quot-s-lfp

The morphism $\Quotfunctor_{\mathcal{F}/X/B} \to B$ is separated. If $\mathcal{F}$ is of finite presentation, then it is also locally of finite presentation.


moduli-lemma-quot-existence-part

Assume $X \to B$ is proper as well as of finite presentation and $\mathcal{F}$ quasi-coherent of finite type. Then $\Quotfunctor_{\mathcal{F}/X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-valuative-criterion}).


moduli-lemma-quot-functorial

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an affine quasi-finite morphism of algebraic spaces which are separated and of finite presentation over $B$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\pi_*$ induces a morphism $\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}$.


moduli-lemma-quot-open

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an affine open immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then the morphism $\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}$ of Lemma \ref{lemma-quot-functorial} is an open immersion.


moduli-lemma-quot-better-open

Let $B$ be an algebraic space. Let $j : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module and set $\mathcal{F} = j^*\mathcal{G}$. Then there is an open immersion $$ \Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor_{\mathcal{G}/Y/B} $$ of algebraic spaces over $B$.


moduli-lemma-quot-closed

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then the morphism $\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}$ of Lemma \ref{lemma-quot-functorial} is an isomorphism.


moduli-lemma-quot-quotient

Let $X \to B$ be as in the introduction to this section. Let $\mathcal{F} \to \mathcal{G}$ be a surjection of quasi-coherent $\mathcal{O}_X$-modules. Then there is a canonical closed immersion $\Quotfunctor_{\mathcal{G}/X/B} \to \Quotfunctor_{\mathcal{F}/X/B}$.


moduli-lemma-quot-tensor-invertible

Let $f : X \to B$ and $\mathcal{F}$ be as in the introduction to this section. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then tensoring with $\mathcal{L}$ defines an isomophism $$ \Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor_{\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}/X/B} $$ Given a numerical polynomial $P(t)$, then setting $P'(t) = P(t + 1)$ this map induces an isomorphism $\Quotfunctor^P_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor^{P'}_{\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}/X/B}$ of open and closed substacks.


moduli-lemma-quot-power-invertible

Let $f : X \to B$ and $\mathcal{F}$ be as in the introduction to this section. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $$ \Quotfunctor^{P, \mathcal{L}}_{\mathcal{F}/X/B} = \Quotfunctor^{P', \mathcal{L}^{\otimes n}}_{\mathcal{F}/X/B} $$ where $P'(t) = P(nt)$.


moduli-lemma-quot-Pn

Let $n \geq 0$, $r \geq 1$, $P \in \mathbf{Q}[t]$. The algebraic space $$ X = \Quotfunctor^P_{\mathcal{O}^{\oplus r}_{\mathbf{P}^n_\mathbf{Z}}/ \mathbf{P}^n_\mathbf{Z}/\mathbf{Z}} $$ parametrizing quotients of $\mathcal{O}_{\mathbf{}P^n_\mathbf{Z}}^{\oplus r}$ with Hilbert polynomial $P$ is proper over $\Spec(\mathbf{Z})$.


moduli-lemma-quot-Pn-over-base

Let $B$ be an algebraic space. Let $X = B \times \mathbf{P}^n_\mathbf{Z}$. Let $\mathcal{L}$ be the pullback of $\mathcal{O}_{\mathbf{P}^n}(1)$ to $X$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation. The algebraic space $\Quotfunctor^P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.


moduli-lemma-quot-proper-over-base

Let $f : X \to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\Quotfunctor^P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.


moduli-lemma-quot-qc-over-base

Let $f : X \to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\Quotfunctor^P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is separated of finite presentation over $B$.


moduli-lemma-hilb-diagonal-closed

The diagonal of $\Hilbfunctor_{X/B} \to B$ is a closed immersion of finite presentation.


moduli-lemma-hilb-open

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Then $\pi$ induces an open immersion $\Hilbfunctor_{X/B} \to \Hilbfunctor_{Y/B}$.


moduli-lemma-hilb-proper-over-base

Let $f : X \to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\Hilbfunctor^P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.


moduli-lemma-hilb-qc-over-base

Let $f : X \to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\Hilbfunctor^P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is separated of finite presentation over $B$.


moduli-lemma-pic-diagonal-affine-fp

The diagonal of $\Picardstack_{X/B}$ over $B$ is affine and of finite presentation.


moduli-lemma-pic-qs-lfp

The morphism $\Picardstack_{X/B} \to B$ is quasi-separated and locally of finite presentation.


moduli-lemma-pic-inertia

Assume $f_{T, *}\mathcal{O}_{X_T} \cong \mathcal{O}_T$ for all schemes $T$ over $B$. Then the inertia stack of $\Picardstack_{X/B}$ is equal to $\mathbf{G}_m \times \Picardstack_{X/B}$.


moduli-lemma-pic-curves-smooth

Assume $f : X \to B$ has relative dimension $\leq 1$ in addition to the other assumptions in this section. Then $\Picardstack_{X/B} \to B$ is smooth.


moduli-lemma-pic-gerbe-over-pic-functor

The morphism $\Picardstack_{X/B} \to \Picardfunctor_{X/B}$ turns the Picard stack into a gerbe over the Picard functor.


moduli-lemma-pic-functor-diagonal-qc-immersion

The diagonal of $\Picardfunctor_{X/B}$ over $B$ is a quasi-compact immersion.


moduli-lemma-Mor-diagonal-closed

The diagonal of $\mathit{Mor}_B(Y, X) \to B$ is a closed immersion of finite presentation.


moduli-lemma-Mor-s-lfp

The morphism $\mathit{Mor}_B(Y, X) \to B$ is separated and locally of finite presentation.


moduli-lemma-Isom-in-Mor

With $B, X, Y$ as in the introduction of this section, in addition assume $X \to B$ is proper. Then the subfunctor $\mathit{Isom}_B(Y, X) \subset \mathit{Mor}_B(Y, X)$ of isomorphisms is an open subspace.


moduli-lemma-Mor-qc-over-base

With $B, X, Y$ as in the introduction of this section, let $\mathcal{L}$ be ample on $X/B$ and let $\mathcal{N}$ be ample on $Y/B$. See Divisors on Spaces, Definition \ref{spaces-divisors-definition-relatively-ample}. Let $P$ be a numerical polynomial. Then $$ \mathit{Mor}^{P, \mathcal{M}}_B(Y, X) \longrightarrow B $$ is separated and of finite presentation where $\mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes_{\mathcal{O}_{Y \times_B X}} \text{pr}_2^*\mathcal{L}$.


moduli-lemma-polarized-diagonal-separated-fp

The diagonal of $\Polarizedstack$ is separated and of finite presentation.


moduli-lemma-polarized-qs-lfp

The morphism $\Polarizedstack \to \Spec(\mathbf{Z})$ is quasi-separated and locally of finite presentation.


moduli-lemma-bounded-polarized

Let $n \geq 1$ be an integer and let $P$ be a numerical polynomial. Let $$ T \subset |\Polarizedstack| $$ be a subset with the following property: for every $\xi \in T$ there exists a field $k$ and an object $(X, \mathcal{L})$ of $\Polarizedstack$ over $k$ representing $\xi$ such that \begin{enumerate} \item the Hilbert polynomial of $\mathcal{L}$ on $X$ is $P$, and \item there exists a closed immersion $i : X \to \mathbf{P}^n_k$ such that $i^*\mathcal{O}_{\mathbf{P}^n}(1) \cong \mathcal{L}$. \end{enumerate} Then $T$ is a Noetherian topological space, in particular quasi-compact.


moduli-lemma-complexes-diagonal-affine-fp

The diagonal of $\Complexesstack_{X/B}$ over $B$ is affine and of finite presentation.


derived-lemma-composition-zero

Let $\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle. Then $g \circ f = 0$, $h \circ g = 0$ and $f[1] \circ h = 0$.


derived-lemma-representable-homological

Let $\mathcal{D}$ be a pre-triangulated category. For any object $W$ of $\mathcal{D}$ the functor $\Hom_\mathcal{D}(W, -)$ is homological, and the functor $\Hom_\mathcal{D}(-, W)$ is cohomological.


derived-lemma-third-isomorphism-triangle

Let $\mathcal{D}$ be a pre-triangulated category. Let $$ (a, b, c) : (X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h') $$ be a morphism of distinguished triangles. If two among $a, b, c$ are isomorphisms so is the third.


derived-lemma-third-map-square-zero

Let $\mathcal{D}$ be a pre-triangulated category. Let $$ (0, b, 0), (0, b', 0) : (X, Y, Z, f, g, h) \to (X, Y, Z, f, g, h) $$ be endomorphisms of a distinguished triangle. Then $bb' = 0$.


derived-lemma-cone-triangle-unique-isomorphism

Let $\mathcal{D}$ be a pre-triangulated category. Let $f : X \to Y$ be a morphism of $\mathcal{D}$. There exists a distinguished triangle $(X, Y, Z, f, g, h)$ which is unique up to (nonunique) isomorphism of triangles. More precisely, given a second such distinguished triangle $(X, Y, Z', f, g', h')$ there exists an isomorphism $$ (1, 1, c) : (X, Y, Z, f, g, h) \longrightarrow (X, Y, Z', f, g', h') $$


derived-lemma-uniqueness-third-arrow

Let $\mathcal{D}$ be a pre-triangulated category. Let $$ (a, b, c) : (X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h') $$ be a morphism of distinguished triangles. If one of the following conditions holds \begin{enumerate} \item $\Hom(Y, X') = 0$, \item $\Hom(Z, Y') = 0$, \item $\Hom(X, X') = \Hom(Z, X') = 0$, \item $\Hom(Z, X') = \Hom(Z, Z') = 0$, or \item $\Hom(X[1], Z') = \Hom(Z, X') = 0$ \end{enumerate} then $b$ is the unique morphism from $Y \to Y'$ such that $(a, b, c)$ is a morphism of triangles.


derived-lemma-third-object-zero

Let $\mathcal{D}$ be a pre-triangulated category. Let $f : X \to Y$ be a morphism of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item $f$ is an isomorphism, \item $(X, Y, 0, f, 0, 0)$ is a distinguished triangle, and \item for any distinguished triangle $(X, Y, Z, f, g, h)$ we have $Z = 0$. \end{enumerate}


derived-lemma-direct-sum-triangles

Let $\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ be triangles. The following are equivalent \begin{enumerate} \item $(X \oplus X', Y \oplus Y', Z \oplus Z', f \oplus f', g \oplus g', h \oplus h')$ is a distinguished triangle, \item both $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ are distinguished triangles. \end{enumerate}


derived-lemma-split

Let $\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle. \begin{enumerate} \item If $h = 0$, then there exists a right inverse $s : Z \to Y$ to $g$. \item For any right inverse $s : Z \to Y$ of $g$ the map $f \oplus s : X \oplus Z \to Y$ is an isomorphism. \item For any objects $X', Z'$ of $\mathcal{D}$ the triangle $(X', X' \oplus Z', Z', (1, 0), (0, 1), 0)$ is distinguished. \end{enumerate}


derived-lemma-when-split

Let $\mathcal{D}$ be a pre-triangulated category. Let $f : X \to Y$ be a morphism of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item $f$ has a kernel, \item $f$ has a cokernel, \item $f$ is the isomorphic to a composition $K \oplus Z \to Z \to Z \oplus Q$ of a projection and coprojection for some objects $K, Z, Q$ of $\mathcal{D}$. \end{enumerate}


derived-lemma-projectors-have-images-triangulated

Let $\mathcal{D}$ be a pre-triangulated category. If $\mathcal{D}$ has countable products, then $\mathcal{D}$ is Karoubian. If $\mathcal{D}$ has countable coproducts, then $\mathcal{D}$ is Karoubian.


derived-lemma-easier-axiom-four

Let $\mathcal{D}$ be a pre-triangulated category. In order to prove TR4 it suffices to show that given any pair of composable morphisms $f : X \to Y$ and $g : Y \to Z$ there exist \begin{enumerate} \item isomorphisms $i : X' \to X$, $j : Y' \to Y$ and $k : Z' \to Z$, and then setting $f' = j^{-1}fi : X' \to Y'$ and $g' = k^{-1}gj : Y' \to Z'$ there exist \item distinguished triangles $(X', Y', Q_1, f', p_1, d_1)$, $(X', Z', Q_2, g' \circ f', p_2, d_2)$ and $(Y', Z', Q_3, g', p_3, d_3)$, such that the assertion of TR4 holds. \end{enumerate}


derived-lemma-triangulated-subcategory

Let $\mathcal{D}$ be a pre-triangulated category. Assume that $\mathcal{D}'$ is an additive full subcategory of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item there exists a set of triangles $\mathcal{T}'$ such that $(\mathcal{D}', \mathcal{T}')$ is a pre-triangulated subcategory of $\mathcal{D}$, \item $\mathcal{D}'$ is preserved under $[1], [-1]$ and given any morphism $f : X \to Y$ in $\mathcal{D}'$ there exists a distinguished triangle $(X, Y, Z, f, g, h)$ in $\mathcal{D}$ such that $Z$ is isomorphic to an object of $\mathcal{D}'$. \end{enumerate} In this case $\mathcal{T}'$ as in (1) is the set of distinguished triangles $(X, Y, Z, f, g, h)$ of $\mathcal{D}$ such that $X, Y, Z \in \Ob(\mathcal{D}')$. Finally, if $\mathcal{D}$ is a triangulated category, then (1) and (2) are also equivalent to \begin{enumerate} \item[(3)] $\mathcal{D}'$ is a triangulated subcategory. \end{enumerate}


derived-lemma-exact-equivalence

Let $F : \mathcal{D} \to \mathcal{D}'$ be a fully faithful exact functor of pre-triangulated categories. Then a triangle $(X, Y, Z, f, g, h)$ of $\mathcal{D}$ is distinguished if and only if $(F(X), F(Y), F(Z), F(f), F(g), F(h))$ is distinguished in $\mathcal{D}'$.


derived-lemma-composition-exact

Let $\mathcal{D}, \mathcal{D}', \mathcal{D}''$ be pre-triangulated categories. Let $F : \mathcal{D} \to \mathcal{D}'$ and $F' : \mathcal{D}' \to \mathcal{D}''$ be exact functors. Then $F' \circ F$ is an exact functor.


derived-lemma-exact-compose-homological-functor

Let $\mathcal{D}$ be a pre-triangulated category. Let $\mathcal{A}$ be an abelian category. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor. \begin{enumerate} \item Let $\mathcal{D}'$ be a pre-triangulated category. Let $F : \mathcal{D}' \to \mathcal{D}$ be an exact functor. Then the composition $H \circ F$ is a homological functor as well. \item Let $\mathcal{A}'$ be an abelian category. Let $G : \mathcal{A} \to \mathcal{A}'$ be an exact functor. Then $G \circ H$ is a homological functor as well. \end{enumerate}


derived-lemma-exact-compose-delta-functor

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A}$ be an abelian category. Let $G : \mathcal{A} \to \mathcal{D}$ be a $\delta$-functor. \begin{enumerate} \item Let $\mathcal{D}'$ be a triangulated category. Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor. Then the composition $F \circ G$ is a $\delta$-functor as well. \item Let $\mathcal{A}'$ be an abelian category. Let $H : \mathcal{A}' \to \mathcal{A}$ be an exact functor. Then $G \circ H$ is a $\delta$-functor as well. \end{enumerate}


derived-lemma-compose-delta-functor-homological

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $G : \mathcal{A} \to \mathcal{D}$ be a $\delta$-functor. Let $H : \mathcal{D} \to \mathcal{B}$ be a homological functor. Assume that $H^{-1}(G(A)) = 0$ for all $A$ in $\mathcal{A}$. Then the collection $$ \{H^n \circ G, H^n(\delta_{A \to B \to C})\}_{n \geq 0} $$ is a $\delta$-functor from $\mathcal{A} \to \mathcal{B}$, see Homology, Definition \ref{homology-definition-cohomological-delta-functor}.


derived-lemma-localization-conditions

Let $\mathcal{D}$ be a pre-triangulated category. Let $S$ be a set of morphisms of $\mathcal{D}$ and assume that axioms MS1, MS5, MS6 hold (see Categories, Definition \ref{categories-definition-multiplicative-system} and Definition \ref{definition-localization}). Then MS2 holds.


derived-lemma-triangle-functor-localize

Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories.  Let $$ S = \{f \in \text{Arrows}(\mathcal{D}) \mid F(f)\text{ is an isomorphism}\} $$ Then $S$ is a saturated (see Categories, Definition \ref{categories-definition-saturated-multiplicative-system}) multiplicative system compatible with the triangulated structure on $\mathcal{D}$.


derived-lemma-homological-functor-localize

Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor between a pre-triangulated category and an abelian category. Let $$ S = \{f \in \text{Arrows}(\mathcal{D}) \mid H^i(f)\text{ is an isomorphism for all }i \in \mathbf{Z}\} $$ Then $S$ is a saturated (see Categories, Definition \ref{categories-definition-saturated-multiplicative-system}) multiplicative system compatible with the triangulated structure on $\mathcal{D}$.


derived-lemma-universal-property-localization

Let $\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative system compatible with the triangulated category. Let $Q : \mathcal{D} \to S^{-1}\mathcal{D}$ be the localization functor, see Proposition \ref{proposition-construct-localization}. \begin{enumerate} \item If $H : \mathcal{D} \to \mathcal{A}$ is a homological functor into an abelian category $\mathcal{A}$ such that $H(s)$ is an isomorphism for all $s \in S$, then the unique factorization $H' : S^{-1}\mathcal{D} \to \mathcal{A}$ such that $H = H' \circ Q$ (see Categories, Lemma \ref{categories-lemma-properties-left-localization}) is a homological functor too. \item If $F : \mathcal{D} \to \mathcal{D}'$ is an exact functor into a pre-triangulated category $\mathcal{D}'$ such that $F(s)$ is an isomorphism for all $s \in S$, then the unique factorization $F' : S^{-1}\mathcal{D} \to \mathcal{D}'$ such that $F = F' \circ Q$ (see Categories, Lemma \ref{categories-lemma-properties-left-localization}) is an exact functor too. \end{enumerate}


derived-lemma-kernel-localization

Let $\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative system compatible with the triangulated structure. Let $Z$ be an object of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item $Q(Z) = 0$ in $S^{-1}\mathcal{D}$, \item there exists $Z' \in \Ob(\mathcal{D})$ such that $0 : Z \to Z'$ is an element of $S$, \item there exists $Z' \in \Ob(\mathcal{D})$ such that $0 : Z' \to Z$ is an element of $S$, and \item there exists an object $Z'$ and a distinguished triangle $(X, Y, Z \oplus Z', f, g, h)$ such that $f \in S$. \end{enumerate} If $S$ is saturated, then these are also equivalent to \begin{enumerate} \item[(5)] the morphism $0 \to Z$ is an element of $S$, \item[(6)] the morphism $Z \to 0$ is an element of $S$, \item[(7)] there exists a distinguished triangle $(X, Y, Z, f, g, h)$ such that $f \in S$. \end{enumerate}


derived-lemma-limit-triangles

Let $\mathcal{D}$ be a triangulated category. Let $S$ be a saturated multiplicative system in $\mathcal{D}$ that is compatible with the triangulated structure. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle in $\mathcal{D}$. Consider the category of morphisms of triangles $$ \mathcal{I} = \{(s, s', s'') : (X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h') \mid s, s', s'' \in S\} $$ Then $\mathcal{I}$ is a filtered category and the functors $\mathcal{I} \to X/S$, $\mathcal{I} \to Y/S$, and $\mathcal{I} \to Z/S$ are cofinal.


derived-lemma-triangle-functor-kernel

Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories. Let $\mathcal{D}''$ be the full subcategory of $\mathcal{D}$ with objects $$ \Ob(\mathcal{D}'') = \{X \in \Ob(\mathcal{D}) \mid F(X) = 0\} $$ Then $\mathcal{D}''$ is a strictly full saturated pre-triangulated subcategory of $\mathcal{D}$. If $\mathcal{D}$ is a triangulated category, then $\mathcal{D}''$ is a triangulated subcategory.


derived-lemma-homological-functor-kernel

Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor of a pre-triangulated category into an abelian category. Let $\mathcal{D}'$ be the full subcategory of $\mathcal{D}$ with objects $$ \Ob(\mathcal{D}') = \{X \in \Ob(\mathcal{D}) \mid H(X[n]) = 0\text{ for all }n \in \mathbf{Z}\} $$ Then $\mathcal{D}'$ is a strictly full saturated pre-triangulated subcategory of $\mathcal{D}$. If $\mathcal{D}$ is a triangulated category, then $\mathcal{D}'$ is a triangulated subcategory.


derived-lemma-homological-functor-bounded

Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor of a pre-triangulated category into an abelian category. Let $\mathcal{D}_H^{+}, \mathcal{D}_H^{-}, \mathcal{D}_H^b$ be the full subcategory of $\mathcal{D}$ with objects $$ \begin{matrix} \Ob(\mathcal{D}_H^{+}) = \{X \in \Ob(\mathcal{D}) \mid H(X[n]) = 0\text{ for all }n \ll 0\} \\ \Ob(\mathcal{D}_H^{-}) = \{X \in \Ob(\mathcal{D}) \mid H(X[n]) = 0\text{ for all }n \gg 0\} \\ \Ob(\mathcal{D}_H^b) = \{X \in \Ob(\mathcal{D}) \mid H(X[n]) = 0\text{ for all }|n| \gg 0\} \end{matrix} $$ Each of these is a strictly full saturated pre-triangulated subcategory of $\mathcal{D}$. If $\mathcal{D}$ is a triangulated category, then each is a triangulated subcategory.


derived-lemma-construct-multiplicative-system

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. Set \begin{equation} \label{equation-multiplicative-system} S = \left\{ \begin{matrix} f \in \text{Arrows}(\mathcal{D}) \text{ such that there exists a distinguished triangle }\\ (X, Y, Z, f, g, h) \text{ of }\mathcal{D}\text{ with } Z\text{ isomorphic to an object of }\mathcal{D}' \end{matrix} \right\} \end{equation} Then $S$ is a multiplicative system compatible with the triangulated structure on $\mathcal{D}$. In this situation the following are equivalent \begin{enumerate} \item $S$ is a saturated multiplicative system, \item $\mathcal{D}'$ is a saturated triangulated subcategory. \end{enumerate}


derived-lemma-universal-property-quotient

\begin{slogan} The universal property of the Verdier quotient. \end{slogan} Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{B}$ be a full triangulated subcategory of $\mathcal{D}$. Let $Q : \mathcal{D} \to \mathcal{D}/\mathcal{B}$ be the quotient functor. \begin{enumerate} \item If $H : \mathcal{D} \to \mathcal{A}$ is a homological functor into an abelian category $\mathcal{A}$ such that $\mathcal{B} \subset \Ker(H)$ then there exists a unique factorization $H' : \mathcal{D}/\mathcal{B} \to \mathcal{A}$ such that $H = H' \circ Q$ and $H'$ is a homological functor too. \item If $F : \mathcal{D} \to \mathcal{D}'$ is an exact functor into a pre-triangulated category $\mathcal{D}'$ such that $\mathcal{B} \subset \Ker(F)$ then there exists a unique factorization $F' : \mathcal{D}/\mathcal{B} \to \mathcal{D}'$ such that $F = F' \circ Q$ and $F'$ is an exact functor too. \end{enumerate}


derived-lemma-kernel-quotient

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{B}$ be a full triangulated subcategory. The kernel of the quotient functor $Q : \mathcal{D} \to \mathcal{D}/\mathcal{B}$ is the strictly full subcategory of $\mathcal{D}$ whose objects are $$ \Ob(\Ker(Q)) = \left\{ \begin{matrix} Z \in \Ob(\mathcal{D}) \text{ such that there exists a }Z' \in \Ob(\mathcal{D}) \\ \text{ such that }Z \oplus Z'\text{ is isomorphic to an object of }\mathcal{B} \end{matrix} \right\} $$ In other words it is the smallest strictly full saturated triangulated subcategory of $\mathcal{D}$ containing $\mathcal{B}$.


derived-lemma-operations

Let $\mathcal{D}$ be a triangulated category. The operations described above have the following properties \begin{enumerate} \item $S(\mathcal{B}(S))$ is the ``saturation'' of $S$, i.e., it is the smallest saturated multiplicative system in $\mathcal{D}$ containing $S$, and \item $\mathcal{B}(S(\mathcal{B}))$ is the ``saturation'' of $\mathcal{B}$, i.e., it is the smallest strictly full saturated triangulated subcategory of $\mathcal{D}$ containing $\mathcal{B}$. \end{enumerate} In particular, the constructions define mutually inverse maps between the (partially ordered) set of saturated multiplicative systems in $\mathcal{D}$ compatible with the triangulated structure on $\mathcal{D}$ and the (partially ordered) set of strictly full saturated triangulated subcategories of $\mathcal{D}$.


derived-lemma-acyclic-general

Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor from a triangulated category $\mathcal{D}$ to an abelian category $\mathcal{A}$, see Definition \ref{definition-homological}. The subcategory $\Ker(H)$ of $\mathcal{D}$ is a strictly full saturated triangulated subcategory of $\mathcal{D}$ whose corresponding saturated multiplicative system (see Lemma \ref{lemma-operations}) is the set $$ S = \{f \in \text{Arrows}(\mathcal{D}) \mid H^i(f)\text{ is an isomorphism for all }i \in \mathbf{Z}\}. $$ The functor $H$ factors through the quotient functor $Q : \mathcal{D} \to \mathcal{D}/\Ker(H)$.


derived-lemma-adjoint-is-exact

Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor between triangulated categories. If $F$ admits a right adjoint $G: \mathcal{D'} \to \mathcal{D}$, then $G$ is also an exact functor.


derived-lemma-fully-faithful-adjoint-kernel-zero

Let $\mathcal{D}$, $\mathcal{D}'$ be triangulated categories. Let $F : \mathcal{D} \to \mathcal{D}'$ and $G : \mathcal{D}' \to \mathcal{D}$ be functors. Assume that \begin{enumerate} \item $F$ and $G$ are exact functors, \item $F$ is fully faithful, \item $G$ is a right adjoint to $F$, and \item the kernel of $G$ is zero. \end{enumerate} Then $F$ is an equivalence of categories.


derived-lemma-functorial-cone

Suppose that $$ \xymatrix{ K_1^\bullet \ar[r]_{f_1} \ar[d]_a & L_1^\bullet \ar[d]^b \\ K_2^\bullet \ar[r]^{f_2} & L_2^\bullet } $$ is a diagram of morphisms of complexes which is commutative up to homotopy. Then there exists a morphism $c : C(f_1)^\bullet \to C(f_2)^\bullet$ which gives rise to a morphism of triangles $(a, b, c) : (K_1^\bullet, L_1^\bullet, C(f_1)^\bullet, f_1, i_1, p_1) \to (K_2^\bullet, L_2^\bullet, C(f_2)^\bullet, f_2, i_2, p_2)$ of $K(\mathcal{A})$.


derived-lemma-map-from-cone

Suppose that $f: K^\bullet \to L^\bullet$ and $g : L^\bullet \to M^\bullet$ are morphisms of complexes such that $g \circ f$ is homotopic to zero. Then \begin{enumerate} \item $g$ factors through a morphism $C(f)^\bullet \to M^\bullet$, and \item $f$ factors through a morphism $K^\bullet \to C(g)^\bullet[-1]$. \end{enumerate}


derived-lemma-make-commute-map

Let $\mathcal{A}$ be an additive category. Let $$ \xymatrix{ A^\bullet \ar[r]_f \ar[d]_a & B^\bullet \ar[d]^b \\ C^\bullet \ar[r]^g & D^\bullet } $$ be a diagram of morphisms of complexes commuting up to homotopy. If $f$ is a termwise split injection, then $b$ is homotopic to a morphism which makes the diagram commute. If $g$ is a split surjection, then $a$ is homotopic to a morphism which makes the diagram commute.


derived-lemma-make-injective

Let $\mathcal{A}$ be an additive category. Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of complexes of $\mathcal{A}$. There exists a factorization $$ \xymatrix{ K^\bullet \ar[r]^{\tilde \alpha} \ar@/_1pc/[rr]_\alpha & \tilde L^\bullet \ar[r]^\pi & L^\bullet } $$ such that \begin{enumerate} \item $\tilde \alpha$ is a termwise split injection (see Definition \ref{definition-termwise-split-map}), \item there is a map of complexes $s : L^\bullet \to \tilde L^\bullet$ such that $\pi \circ s = \text{id}_{L^\bullet}$ and such that $s \circ \pi$ is homotopic to $\text{id}_{\tilde L^\bullet}$. \end{enumerate} Moreover, if both $K^\bullet$ and $L^\bullet$ are in $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, or $K^b(\mathcal{A})$, then so is $\tilde L^\bullet$.


derived-lemma-make-surjective

Let $\mathcal{A}$ be an additive category. Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of complexes of $\mathcal{A}$. There exists a factorization $$ \xymatrix{ K^\bullet \ar[r]^i \ar@/_1pc/[rr]_\alpha & \tilde K^\bullet \ar[r]^{\tilde \alpha} & L^\bullet } $$ such that \begin{enumerate} \item $\tilde \alpha$ is a termwise split surjection (see Definition \ref{definition-termwise-split-map}), \item there is a map of complexes $s : \tilde K^\bullet \to K^\bullet$ such that $s \circ i = \text{id}_{K^\bullet}$ and such that $i \circ s$ is homotopic to $\text{id}_{\tilde K^\bullet}$. \end{enumerate} Moreover, if both $K^\bullet$ and $L^\bullet$ are in $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, or $K^b(\mathcal{A})$, then so is $\tilde K^\bullet$.


derived-lemma-nilpotent

Let $\mathcal{A}$ be an additive category. Let $0 \to A_i^\bullet \to B_i^\bullet \to C_i^\bullet \to 0$, $i = 1, 2, 3$ be termwise split exact sequences of complexes. Let $b : B_1^\bullet \to B_2^\bullet$ and $b' : B_2^\bullet \to B_3^\bullet$ be morphisms of complexes such that $$ \vcenter{ \xymatrix{ A_1^\bullet \ar[d]_0 \ar[r] & B_1^\bullet \ar[r] \ar[d]_b & C_1^\bullet \ar[d]_0 \\ A_2^\bullet \ar[r] & B_2^\bullet \ar[r] & C_2^\bullet } } \quad\text{and}\quad \vcenter{ \xymatrix{ A_2^\bullet \ar[d]^0 \ar[r] & B_2^\bullet \ar[r] \ar[d]^{b'} & C_2^\bullet \ar[d]^0 \\ A_3^\bullet \ar[r] & B_3^\bullet \ar[r] & C_3^\bullet } } $$ commute in $K(\mathcal{A})$. Then $b' \circ b = 0$ in $K(\mathcal{A})$.


derived-lemma-third-isomorphism

Let $\mathcal{A}$ be an additive category. Let $f_1 : K_1^\bullet \to L_1^\bullet$ and $f_2 : K_2^\bullet \to L_2^\bullet$ be morphisms of complexes. Let $$ (a, b, c) : (K_1^\bullet, L_1^\bullet, C(f_1)^\bullet, f_1, i_1, p_1) \longrightarrow (K_2^\bullet, L_2^\bullet, C(f_2)^\bullet, f_2, i_2, p_2) $$ be any morphism of triangles of $K(\mathcal{A})$. If $a$ and $b$ are homotopy equivalences then so is $c$.


derived-lemma-the-same-up-to-isomorphisms

Let $\mathcal{A}$ be an additive category. \begin{enumerate} \item Given a termwise split sequence of complexes $(\alpha : A^\bullet \to B^\bullet, \beta : B^\bullet \to C^\bullet, s^n, \pi^n)$ there exists a homotopy equivalence $C(\alpha)^\bullet \to C^\bullet$ such that the diagram $$ \xymatrix{ A^\bullet \ar[r] \ar[d] & B^\bullet \ar[d] \ar[r] & C(\alpha)^\bullet \ar[r]_{-p} \ar[d] & A^\bullet[1] \ar[d] \\ A^\bullet \ar[r] & B^\bullet \ar[r] & C^\bullet \ar[r]^\delta & A^\bullet[1] } $$ defines an isomorphism of triangles in $K(\mathcal{A})$. \item Given a morphism of complexes $f : K^\bullet \to L^\bullet$ there exists an isomorphism of triangles $$ \xymatrix{ K^\bullet \ar[r] \ar[d] & \tilde L^\bullet \ar[d] \ar[r] & M^\bullet \ar[r]_{\delta} \ar[d] & K^\bullet[1] \ar[d] \\ K^\bullet \ar[r] & L^\bullet \ar[r] & C(f)^\bullet \ar[r]^{-p} & K^\bullet[1] } $$ where the upper triangle is the triangle associated to a termwise split exact sequence $K^\bullet \to \tilde L^\bullet \to M^\bullet$. \end{enumerate}


derived-lemma-sequence-maps-split

Let $\mathcal{A}$ be an additive category. Let $A_1^\bullet \to A_2^\bullet \to \ldots \to A_n^\bullet$ be a sequence of composable morphisms of complexes. There exists a commutative diagram $$ \xymatrix{ A_1^\bullet \ar[r] & A_2^\bullet \ar[r] & \ldots \ar[r] & A_n^\bullet \\ B_1^\bullet \ar[r] \ar[u] & B_2^\bullet \ar[r] \ar[u] & \ldots \ar[r] & B_n^\bullet \ar[u] } $$ such that each morphism $B_i^\bullet \to B_{i + 1}^\bullet$ is a split injection and each $B_i^\bullet \to A_i^\bullet$ is a homotopy equivalence. Moreover, if all $A_i^\bullet$ are in $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, or $K^b(\mathcal{A})$, then so are the $B_i^\bullet$.


derived-lemma-rotate-triangle

Let $\mathcal{A}$ be an additive category. Let $(\alpha : A^\bullet \to B^\bullet, \beta : B^\bullet \to C^\bullet, s^n, \pi^n)$ be a termwise split sequence of complexes. Let $(A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta)$ be the associated triangle. Then the triangle $(C^\bullet[-1], A^\bullet, B^\bullet, \delta[-1], \alpha, \beta)$ is isomorphic to the triangle $(C^\bullet[-1], A^\bullet, C(\delta[-1])^\bullet, \delta[-1], i, p)$.


derived-lemma-rotate-cone

Let $\mathcal{A}$ be an additive category. Let $f : K^\bullet \to L^\bullet$ be a morphism of complexes. The triangle $(L^\bullet, C(f)^\bullet, K^\bullet[1], i, p, f[1])$ is the triangle associated to the termwise split sequence $$ 0 \to L^\bullet \to C(f)^\bullet \to K^\bullet[1] \to 0 $$ coming from the definition of the cone of $f$.


derived-lemma-two-split-injections

Let $\mathcal{A}$ be an additive category. Suppose that $\alpha : A^\bullet \to B^\bullet$ and $\beta : B^\bullet \to C^\bullet$ are split injections of complexes. Then there exist distinguished triangles $(A^\bullet, B^\bullet, Q_1^\bullet, \alpha, p_1, d_1)$, $(A^\bullet, C^\bullet, Q_2^\bullet, \beta \circ \alpha, p_2, d_2)$ and $(B^\bullet, C^\bullet, Q_3^\bullet, \beta, p_3, d_3)$ for which TR4 holds.


derived-lemma-bounded-triangulated-subcategories

Let $\mathcal{A}$ be an additive category. The categories $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, and $K^b(\mathcal{A})$ are full triangulated subcategories of $K(\mathcal{A})$.


derived-lemma-improve-distinguished-triangle-homotopy

Let $\mathcal{A}$ be an additive category. Let $(A^\bullet, B^\bullet, C^\bullet, a, b, c)$ be a distinguished triangle in $K(\mathcal{A})$. Then there exists an isomorphic distinguished triangle $(A^\bullet, (B')^\bullet, C^\bullet, a', b', c)$ such that $0 \to A^n \to (B')^n \to C^n \to 0$ is a split short exact sequence for all $n$.


derived-lemma-cohomology-homological

Let $\mathcal{A}$ be an abelian category. The functor $$ H^0 : K(\mathcal{A}) \longrightarrow \mathcal{A} $$ is homological.


derived-lemma-acyclic

Let $\mathcal{A}$ be an abelian category. The full subcategory $\text{Ac}(\mathcal{A})$ of $K(\mathcal{A})$ consisting of acyclic complexes is a strictly full saturated triangulated subcategory of $K(\mathcal{A})$. The corresponding saturated multiplicative system (see Lemma \ref{lemma-operations}) of $K(\mathcal{A})$ is the set $\text{Qis}(\mathcal{A})$ of quasi-isomorphisms. In particular, the kernel of the localization functor $Q : K(\mathcal{A}) \to \text{Qis}(\mathcal{A})^{-1}K(\mathcal{A})$ is $\text{Ac}(\mathcal{A})$ and the functor $H^0$ factors through $Q$.


derived-lemma-complex-cohomology-bounded

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex. \begin{enumerate} \item If $H^n(K^\bullet) = 0$ for all $n \ll 0$, then there exists a quasi-isomorphism $K^\bullet \to L^\bullet$ with $L^\bullet$ bounded below. \item If $H^n(K^\bullet) = 0$ for all $n \gg 0$, then there exists a quasi-isomorphism $M^\bullet \to K^\bullet$ with $M^\bullet$ bounded above. \item If $H^n(K^\bullet) = 0$ for all $|n| \gg 0$, then there exists a commutative diagram of morphisms of complexes $$ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u] } $$ where all the arrows are quasi-isomorphisms, $L^\bullet$ bounded below, $M^\bullet$ bounded above, and $N^\bullet$ a bounded complex. \end{enumerate}


derived-lemma-bounded-derived

Let $\mathcal{A}$ be an abelian category. The subcategories $\text{Ac}^{+}(\mathcal{A})$, $\text{Ac}^{-}(\mathcal{A})$, resp.\ $\text{Ac}^b(\mathcal{A})$ are strictly full saturated triangulated subcategories of $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, resp.\ $K^b(\mathcal{A})$. The corresponding saturated multiplicative systems (see Lemma \ref{lemma-operations}) are the sets $\text{Qis}^{+}(\mathcal{A})$, $\text{Qis}^{-}(\mathcal{A})$, resp.\ $\text{Qis}^b(\mathcal{A})$. \begin{enumerate} \item The kernel of the functor $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is $\text{Ac}^{+}(\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^{+}(\mathcal{A})/\text{Ac}^{+}(\mathcal{A}) = \text{Qis}^{+}(\mathcal{A})^{-1}K^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{A}) $$ \item The kernel of the functor $K^{-}(\mathcal{A}) \to D^{-}(\mathcal{A})$ is $\text{Ac}^{-}(\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^{-}(\mathcal{A})/\text{Ac}^{-}(\mathcal{A}) = \text{Qis}^{-}(\mathcal{A})^{-1}K^{-}(\mathcal{A}) \longrightarrow D^{-}(\mathcal{A}) $$ \item The kernel of the functor $K^b(\mathcal{A}) \to D^b(\mathcal{A})$ is $\text{Ac}^b(\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^b(\mathcal{A})/\text{Ac}^b(\mathcal{A}) = \text{Qis}^b(\mathcal{A})^{-1}K^b(\mathcal{A}) \longrightarrow D^b(\mathcal{A}) $$ \end{enumerate}


derived-lemma-derived-canonical-delta-functor

Let $\mathcal{A}$ be an abelian category. The functor $\text{Comp}(\mathcal{A}) \to D(\mathcal{A})$ defined has the natural structure of a $\delta$-functor, with $$ \delta_{A^\bullet \to B^\bullet \to C^\bullet} = - p \circ q^{-1} $$ with $p$ and $q$ as explained above. The same construction turns the functors $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$, $\text{Comp}^{-}(\mathcal{A}) \to D^{-}(\mathcal{A})$, and $\text{Comp}^b(\mathcal{A}) \to D^b(\mathcal{A})$ into $\delta$-functors.


derived-lemma-derived-compare-triangles-ses

Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ 0 \ar[r] & A^\bullet \ar[r] \ar[d] & B^\bullet \ar[r] \ar[d] & C^\bullet \ar[r] \ar[d] & 0 \\ 0 \ar[r] & D^\bullet \ar[r] & E^\bullet \ar[r] & F^\bullet \ar[r] & 0 } $$ be a commutative diagram of morphisms of complexes such that the rows are short exact sequences of complexes, and the vertical arrows are quasi-isomorphisms. The $\delta$-functor of Lemma \ref{lemma-derived-canonical-delta-functor} above maps the short exact sequences $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ and $0 \to D^\bullet \to E^\bullet \to F^\bullet \to 0$ to isomorphic distinguished triangles.


derived-lemma-trick-vanishing-composition

Let $\mathcal{A}$ be an abelian category. Let $$ K_0^\bullet \to K_1^\bullet \to \ldots \to K_n^\bullet $$ be maps of complexes such that \begin{enumerate} \item $H^i(K_0^\bullet) = 0$ for $i > 0$, \item $H^{-j}(K_j^\bullet) \to H^{-j}(K_{j + 1}^\bullet)$ is zero. \end{enumerate} Then the composition $K_0^\bullet \to K_n^\bullet$ factors through $\tau_{\leq -n}K_n^\bullet \to K_n^\bullet$ in $D(\mathcal{A})$. Dually, given maps of complexes $$ K_n^\bullet \to K_{n - 1}^\bullet \to \ldots \to K_0^\bullet $$ such that \begin{enumerate} \item $H^i(K_0^\bullet) = 0$ for $i < 0$, \item $H^j(K_{j + 1}^\bullet) \to H^j(K_j^\bullet)$ is zero, \end{enumerate} then the composition $K_n^\bullet \to K_0^\bullet$ factors through $K_n^\bullet \to \tau_{\geq n}K_n^\bullet$ in $D(\mathcal{A})$.


derived-lemma-filtered-cohomology-homological

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item The functor $K(\text{Fil}^f(\mathcal{A})) \longrightarrow \text{Gr}(\mathcal{A})$, $K^\bullet \longmapsto H^0(\text{gr}(K^\bullet))$ is homological. \item The functor $K(\text{Fil}^f(\mathcal{A})) \rightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0(\text{gr}^p(K^\bullet))$ is homological. \item The functor $K(\text{Fil}^f(\mathcal{A})) \longrightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0((\text{forget }F)K^\bullet)$ is homological. \end{enumerate}


derived-lemma-filtered-acyclic

Let $\mathcal{A}$ be an abelian category. The full subcategory $\text{FAc}(\mathcal{A})$ of $K(\text{Fil}^f(\mathcal{A}))$ consisting of filtered acyclic complexes is a strictly full saturated triangulated subcategory of $K(\text{Fil}^f(\mathcal{A}))$. The corresponding saturated multiplicative system (see Lemma \ref{lemma-operations}) of $K(\text{Fil}^f(\mathcal{A}))$ is the set $\text{FQis}(\mathcal{A})$ of filtered quasi-isomorphisms. In particular, the kernel of the localization functor $$ Q : K(\text{Fil}^f(\mathcal{A})) \longrightarrow \text{FQis}(\mathcal{A})^{-1}K(\text{Fil}^f(\mathcal{A})) $$ is $\text{FAc}(\mathcal{A})$ and the functor $H^0 \circ \text{gr}$ factors through $Q$.


derived-lemma-filtered-complex-cohomology-bounded

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet \in K(\text{Fil}^f(\mathcal{A}))$. \begin{enumerate} \item If $H^n(\text{gr}(K^\bullet)) = 0$ for all $n < a$, then there exists a filtered quasi-isomorphism $K^\bullet \to L^\bullet$ with $L^n = 0$ for all $n < a$. \item If $H^n(\text{gr}(K^\bullet)) = 0$ for all $n > b$, then there exists a filtered quasi-isomorphism $M^\bullet \to K^\bullet$ with $M^n = 0$ for all $n > b$. \item If $H^n(\text{gr}(K^\bullet)) = 0$ for all $|n| \gg 0$, then there exists a commutative diagram of morphisms of complexes $$ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u] } $$ where all the arrows are filtered quasi-isomorphisms, $L^\bullet$ bounded below, $M^\bullet$ bounded above, and $N^\bullet$ a bounded complex. \end{enumerate}


derived-lemma-derived-functor

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $f : X \to Y$ be a morphism of $\mathcal{D}$. \begin{enumerate} \item If $RF$ is defined at $X$ and $Y$ then there exists a unique morphism $RF(f) : RF(X) \to RF(Y)$ between the values such that for any commutative diagram $$ \xymatrix{ X \ar[d]_f \ar[r]_s & X' \ar[d]^{f'} \\ Y \ar[r]^{s'} & Y' } $$ with $s, s' \in S$ the diagram $$ \xymatrix{ F(X) \ar[d] \ar[r] & F(X') \ar[d] \ar[r] & RF(X) \ar[d] \\ F(Y) \ar[r] & F(Y') \ar[r] & RF(Y) } $$ commutes. \item If $LF$ is defined at $X$ and $Y$ then there exists a unique morphism $LF(f) : LF(X) \to LF(Y)$ between the values such that for any commutative diagram $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_s & X \ar[d]^f \\ Y' \ar[r]^{s'} & Y } $$ with $s, s'$ in $S$ the diagram $$ \xymatrix{ LF(X) \ar[d] \ar[r] & F(X') \ar[d] \ar[r] & F(X) \ar[d] \\ LF(Y) \ar[r] & F(Y') \ar[r] & F(Y) } $$ commutes. \end{enumerate}


derived-lemma-derived-inverts

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $s : X \to Y$ be an element of $S$. \begin{enumerate} \item $RF$ is defined at $X$ if and only if it is defined at $Y$. In this case the map $RF(s) : RF(X) \to RF(Y)$ between values is an isomorphism. \item $LF$ is defined at $X$ if and only if it is defined at $Y$. In this case the map $LF(s) : LF(X) \to LF(Y)$ between values is an isomorphism. \end{enumerate}


derived-lemma-derived-shift

\begin{slogan} Derived functors are compatible with shifts \end{slogan} Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $X$ be an object of $\mathcal{D}$ and $n \in \mathbf{Z}$. \begin{enumerate} \item $RF$ is defined at $X$ if and only if it is defined at $X[n]$. In this case there is a canonical isomorphism $RF(X)[n]= RF(X[n])$ between values. \item $LF$ is defined at $X$ if and only if it is defined at $X[n]$. In this case there is a canonical isomorphism $LF(X)[n] \to LF(X[n])$ between values. \end{enumerate}


derived-lemma-2-out-of-3-defined

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\mathcal{D}$. If $RF$ is defined at two out of three of $X, Y, Z$, then it is defined at the third. Moreover, in this case $$ (RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h)) $$ is a distinguished triangle in $\mathcal{D}'$. Similarly for $LF$.


derived-lemma-direct-sum-defined

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $X, Y$ be objects of $\mathcal{D}$. \begin{enumerate} \item If $RF$ is defined at $X$ and $Y$, then $RF$ is defined at $X \oplus Y$. \item If $\mathcal{D}'$ is Karoubian and $RF$ is defined at $X \oplus Y$, then $RF$ is defined at both $X$ and $Y$. \end{enumerate} In either case we have $RF(X \oplus Y) = RF(X) \oplus RF(Y)$. Similarly for $LF$.


derived-lemma-computes-shift

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $X$ be an object of $\mathcal{D}$ and $n \in \mathbf{Z}$. \begin{enumerate} \item $X$ computes $RF$ if and only if $X[n]$ computes $RF$. \item $X$ computes $LF$ if and only if $X[n]$ computes $LF$. \end{enumerate}


derived-lemma-2-out-of-3-computes

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\mathcal{D}$. If $X, Y$ compute $RF$ then so does $Z$. Similar for $LF$.


derived-lemma-existence-computes

Assumptions and notation as in Situation \ref{situation-derived-functor}. \begin{enumerate} \item If for every object $X \in \Ob(\mathcal{D})$ there exists an arrow $s : X \to X'$ in $S$ such that $X'$ computes $RF$, then $RF$ is everywhere defined. \item If for every object $X \in \Ob(\mathcal{D})$ there exists an arrow $s : X' \to X$ in $S$ such that $X'$ computes $LF$, then $LF$ is everywhere defined. \end{enumerate}


derived-lemma-find-existence-computes

Assumptions and notation as in Situation \ref{situation-derived-functor}. If there exists a subset $\mathcal{I} \subset \Ob(\mathcal{D})$ such that \begin{enumerate} \item for all $X \in \Ob(\mathcal{D})$ there exists $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$, and \item for every arrow $s : X \to X'$ in $S$ with $X, X' \in \mathcal{I}$ the map $F(s) : F(X) \to F(X')$ is an isomorphism, \end{enumerate} then $RF$ is everywhere defined and every $X \in \mathcal{I}$ computes $RF$. Dually, if there exists a subset $\mathcal{P} \subset \Ob(\mathcal{D})$ such that \begin{enumerate} \item for all $X \in \Ob(\mathcal{D})$ there exists $s : X' \to X$ in $S$ with $X' \in \mathcal{P}$, and \item for every arrow $s : X \to X'$ in $S$ with $X, X' \in \mathcal{P}$ the map $F(s) : F(X) \to F(X')$ is an isomorphism, \end{enumerate} then $LF$ is everywhere defined and every $X \in \mathcal{P}$ computes $LF$.


derived-lemma-compose-derived-functors-general

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be triangulated categories. Let $S$, resp.\ $S'$ be a saturated multiplicative system in $\mathcal{A}$, resp.\ $\mathcal{B}$ compatible with the triangulated structure. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be exact functors. Denote $F' : \mathcal{A} \to (S')^{-1}\mathcal{B}$ the composition of $F$ with the localization functor. \begin{enumerate} \item If $RF'$, $RG$, $R(G \circ F)$ are everywhere defined, then there is a canonical transformation of functors $t : R(G \circ F) \longrightarrow RG \circ RF'$. \item If $LF'$, $LG$, $L(G \circ F)$ are everywhere defined, then there is a canonical transformation of functors $t : LG \circ LF' \to L(G \circ F)$. \end{enumerate}


derived-lemma-subcategory-left-resolution

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset containing $0$ such that every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$. Let $a \in \mathbf{Z}$. \begin{enumerate} \item Given $K^\bullet$ with $K^n = 0$ for $n > a$ there exists a quasi-isomorphism $P^\bullet \to K^\bullet$ with $P^n \in \mathcal{P}$ and $P^n \to K^n$ surjective for all $n$ and $P^n = 0$ for $n > a$. \item Given $K^\bullet$ with $H^n(K^\bullet) = 0$ for $n > a$ there exists a quasi-isomorphism $P^\bullet \to K^\bullet$ with $P^n \in \mathcal{P}$ for all $n$ and $P^n = 0$ for $n > a$. \end{enumerate}


derived-lemma-subcategory-right-resolution

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{I} \subset \Ob(\mathcal{A})$ be a subset containing $0$ such that every object of $\mathcal{A}$ is a subobject of an element of $\mathcal{I}$. Let $a \in \mathbf{Z}$. \begin{enumerate} \item Given $K^\bullet$ with $K^n = 0$ for $n < a$ there exists a quasi-isomorphism $K^\bullet \to I^\bullet$ with $K^n \to I^n$ injective and $I^n \in \mathcal{I}$ for all $n$ and $I^n = 0$ for $n < a$, \item Given $K^\bullet$ with $H^n(K^\bullet) = 0$ for $n < a$ there exists a quasi-isomorphism $K^\bullet \to I^\bullet$ with $I^n \in \mathcal{I}$ and $I^n = 0$ for $n < a$. \end{enumerate}


derived-lemma-subcategory-right-acyclics

In Situation \ref{situation-classical}. Let $\mathcal{I} \subset \Ob(\mathcal{A})$ be a subset with the following properties: \begin{enumerate} \item every object of $\mathcal{A}$ is a subobject of an element of $\mathcal{I}$, \item for any short exact sequence $0 \to P \to Q \to R \to 0$ of $\mathcal{A}$ with $P, Q \in \mathcal{I}$, then $R \in \mathcal{I}$, and $0 \to F(P) \to F(Q) \to F(R) \to 0$ is exact. \end{enumerate} Then every object of $\mathcal{I}$ is acyclic for $RF$.


derived-lemma-subcategory-left-acyclics

In Situation \ref{situation-classical}. Let $\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset with the following properties: \begin{enumerate} \item every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$, \item for any short exact sequence $0 \to P \to Q \to R \to 0$ of $\mathcal{A}$ with $Q, R \in \mathcal{P}$, then $P \in \mathcal{P}$, and $0 \to F(P) \to F(Q) \to F(R) \to 0$ is exact. \end{enumerate} Then every object of $\mathcal{P}$ is acyclic for $LF$.


derived-lemma-negative-vanishing

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories. Let $K^\bullet$ be a complex of $\mathcal{A}$ and $a \in \mathbf{Z}$. \begin{enumerate} \item If $H^i(K^\bullet) = 0$ for all $i < a$ and $RF$ is defined at $K^\bullet$, then $H^i(RF(K^\bullet)) = 0$ for all $i < a$. \item If $RF$ is defined at $K^\bullet$ and $\tau_{\leq a}K^\bullet$, then $H^i(RF(\tau_{\leq a}K^\bullet)) = H^i(RF(K^\bullet))$ for all $i \leq a$. \end{enumerate}


derived-lemma-left-exact-higher-derived

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. \begin{enumerate} \item We have $R^iF = 0$ for $i < 0$, \item $R^0F$ is left exact, \item the map $F \to R^0F$ is an isomorphism if and only if $F$ is left exact. \end{enumerate}


derived-lemma-F-acyclic

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $A$ be an object of $\mathcal{A}$. \begin{enumerate} \item $A$ is right acyclic for $F$ if and only if $F(A) \to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$, \item if $F$ is left exact, then $A$ is right acyclic for $F$ if and only if $R^iF(A) = 0$ for all $i > 0$. \end{enumerate}


derived-lemma-right-derived-delta-functor

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. \begin{enumerate} \item The functors $R^iF$, $i \geq 0$ come equipped with a canonical structure of a $\delta$-functor from $\mathcal{A} \to \mathcal{B}$, see Homology, Definition \ref{homology-definition-cohomological-delta-functor}. \item If every object of $\mathcal{A}$ is a subobject of a right acyclic object for $F$, then $\{R^iF, \delta\}_{i \geq 0}$ is a universal $\delta$-functor, see Homology, Definition \ref{homology-definition-universal-delta-functor}. \end{enumerate}


derived-lemma-leray-acyclicity

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $A^\bullet$ be a bounded below complex of $F$-acyclic objects. The canonical map $$ F(A^\bullet) \longrightarrow RF(A^\bullet) $$ is an isomorphism in $D^{+}(\mathcal{B})$, i.e., $A^\bullet$ computes $RF$.


derived-lemma-right-derived-exact-functor

Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor of abelian categories. Then \begin{enumerate} \item every object of $\mathcal{A}$ is right acyclic for $F$, \item $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined, \item $RF : D(\mathcal{A}) \to D(\mathcal{B})$ is everywhere defined, \item every complex computes $RF$, in other words, the canonical map $F(K^\bullet) \to RF(K^\bullet)$ is an isomorphism for all complexes, and \item $R^iF = 0$ for $i \not = 0$. \end{enumerate}


derived-lemma-cohomology-in-serre-subcategory

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{B} \subset \mathcal{A}$ be a weak Serre subcategory. The category $D_\mathcal{B}(\mathcal{A})$ is a strictly full saturated triangulated subcategory of $D(\mathcal{A})$. Similarly for the bounded versions.


derived-lemma-derived-of-quotient

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{B} \subset \mathcal{A}$ be a Serre subcategory. Then $D(\mathcal{A}) \to D(\mathcal{A}/\mathcal{B})$ is essentially surjective.


derived-lemma-quotient-by-serre-easy

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{B} \subset \mathcal{A}$ be a Serre subcategory. Suppose that the functor $v : \mathcal{A} \to \mathcal{A}/\mathcal{B}$ has a left adjoint $u : \mathcal{A}/\mathcal{B} \to \mathcal{A}$ such that $vu \cong \text{id}$. Then $$ D(\mathcal{A})/D_\mathcal{B}(\mathcal{A}) = D(\mathcal{A}/\mathcal{B}) $$ and similarly for the bounded versions.


derived-lemma-fully-faithful-embedding

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{B} \subset \mathcal{A}$ be a Serre subcategory. Assume that for every surjection $X \to Y$ with $X \in \Ob(\mathcal{A})$ and $Y \in \Ob(\mathcal{B})$ there exists $X' \subset X$, $X' \in \Ob(\mathcal{B})$ which surjects onto $Y$. Then the functor $D^-(\mathcal{B}) \to D^-_\mathcal{B}(\mathcal{A})$ of (\ref{equation-compare}) is an equivalence.


derived-lemma-cohomology-bounded-below

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex of $\mathcal{A}$. \begin{enumerate} \item If $K^\bullet$ has an injective resolution then $H^n(K^\bullet) = 0$ for $n \ll 0$. \item If $H^n(K^\bullet) = 0$ for all $n \ll 0$ then there exists a quasi-isomorphism $K^\bullet \to L^\bullet$ with $L^\bullet$ bounded below. \end{enumerate}


derived-lemma-injective-resolutions-exist

Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. \begin{enumerate} \item Any object of $\mathcal{A}$ has an injective resolution. \item If $H^n(K^\bullet) = 0$ for all $n \ll 0$ then $K^\bullet$ has an injective resolution. \item If $K^\bullet$ is a complex with $K^n = 0$ for $n < a$, then there exists an injective resolution $\alpha : K^\bullet \to I^\bullet$ with $I^n = 0$ for $n < a$ such that each $\alpha^n : K^n \to I^n$ is injective. \end{enumerate}


derived-lemma-acyclic-is-zero

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be an acyclic complex. Let $I^\bullet$ be bounded below and consisting of injective objects. Any morphism $K^\bullet \to I^\bullet$ is homotopic to zero.


derived-lemma-morphisms-lift

Let $\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_\gamma & L^\bullet \ar@{-->}[dl]^\beta \\ I^\bullet } $$ where $I^\bullet$ is bounded below and consists of injective objects, and $\alpha$ is a quasi-isomorphism. \begin{enumerate} \item There exists a map of complexes $\beta$ making the diagram commute up to homotopy. \item If $\alpha$ is injective in every degree then we can find a $\beta$ which makes the diagram commute. \end{enumerate}


derived-lemma-morphisms-equal-up-to-homotopy

Let $\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_\gamma & L^\bullet \ar@{-->}[dl]^{\beta_i} \\ I^\bullet } $$ where $I^\bullet$ is bounded below and consists of injective objects, and $\alpha$ is a quasi-isomorphism. Any two morphisms $\beta_1, \beta_2$ making the diagram commute up to homotopy are homotopic.


derived-lemma-morphisms-into-injective-complex

Let $\mathcal{A}$ be an abelian category. Let $I^\bullet$ be bounded below complex consisting of injective objects. Let $L^\bullet \in K(\mathcal{A})$. Then $$ \Mor_{K(\mathcal{A})}(L^\bullet, I^\bullet) = \Mor_{D(\mathcal{A})}(L^\bullet, I^\bullet). $$


derived-lemma-injective-resolution-ses

Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. For any short exact sequence $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ of $\text{Comp}^{+}(\mathcal{A})$ there exists a commutative diagram in $\text{Comp}^{+}(\mathcal{A})$ $$ \xymatrix{ 0 \ar[r] & A^\bullet \ar[r] \ar[d] & B^\bullet \ar[r] \ar[d] & C^\bullet \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I_1^\bullet \ar[r] & I_2^\bullet \ar[r] & I_3^\bullet \ar[r] & 0 } $$ where the vertical arrows are injective resolutions and the rows are short exact sequences of complexes. In fact, given any injective resolution $A^\bullet \to I^\bullet$ we may assume $I_1^\bullet = I^\bullet$.