Stable derived functors

Written by Claude Opus 5.5 (Anthropic) on 20 October 2026. Self-checked by the writing AI. Public domain (CC0).

The left derived functor $LF$ of a non-additive functor of modules, such as a symmetric power, is not additive and does not commute with shifts: $LS^2(E[1])$ is $L\Lambda^2(E)[2]$, not $LS^2(E)[1]$. Its stabilization corrects both defects. The stable derived functor $LF^{\mathrm{st}}$ is defined on all bounded above complexes, is additive, commutes with the shift and carries distinguished triangles to distinguished triangles. It is the exact functor that best approximates $LF$ on highly connected complexes. Its homology groups are the stable homology groups of Dold and Puppe.

This lesson constructs $LF^{\mathrm{st}}$ for a functor of modules over a ring of a topos that commutes with local inductive limits, and proves that it is an exact functor of triangulated categories. Examples: the stabilization of the free abelian group functor, whose zeroth homology returns the module and whose first homology vanishes, and the shift relations between the stable derived functors of the symmetric, exterior and divided powers, on all bounded above complexes.

Prerequisites. Stabilization of non-additive functors; Sections 2 and 4 of Non-additive functors of complexes in a ringed topos; Theorem 2.1 and Lemma 3.1 of Koszul complexes and the shift formulas; Corollary 7.2 of Homotopy sheaves and the free abelian Whitehead theorem; Corollary 2.2 of Décalage, nerves and the homotopy type of a category. From the AI-integrated Stacks project: triangulated categories, exact functors and the derived category [Stacks, Tag 0145, Tag 014V, Tag 014Q, Tag 0152].

Conventions

$\mathcal C$ is a small site, $T=\operatorname{Sh}(\mathcal C)$, and $A$, $B$ are sheaves of rings. $F\colon\operatorname{Mod}(A)\to\operatorname{Mod}(B)$ commutes with restriction to every $U\in\mathcal C$ and with local inductive limits, as in Section 4 of the lesson on non-additive functors. Its reduced part $F^+$ has the same properties, because local inductive limits are exact. By Corollary 3.7 of the lesson on stabilization, $(F^+)^{\mathrm{st}}\simeq F^{\mathrm{st}}$. So we assume $F$ reduced.

Complexes are cohomological, $X[1]^k=X^{k+1}$ with differential $-d$, as in the derived category. By Dold–Kan, simplicial modules are the complexes concentrated in degrees $\le0$. We write $\operatorname{Simpl}(A)$ for simplicial $A$-modules and $D_\bullet(A)$ for the category obtained from $\operatorname{Simpl}(A)$ by inverting quasi-isomorphisms. A simplicial module is flat if all its terms are.

1. The stable derived functor

Lemma 1.1. The inclusion of complexes concentrated in degrees $\le0$ induces an equivalence of $D_\bullet(A)$ with the full subcategory $D^{\leq0}(A)$ of $D(A)$.

Proof. Let $i$ be the inclusion and $\tau$ the truncation $\tau^{\leq0}X=(\cdots\to X^{-1}\to\ker d^0\to0)$. A map from a complex concentrated in degrees $\le0$ to $X$ lands in $\tau X$, so $\tau$ is right adjoint to $i$. Both functors carry quasi-isomorphisms to quasi-isomorphisms. Hence they induce functors between the localizations, and the unit and counit induce an adjunction between them. The unit $1\to\tau i$ is the identity. The counit $i\tau X\to X$ is a quasi-isomorphism exactly when $X\in D^{\leq0}(A)$. So the induced functor $D_\bullet(A)\to D(A)$ is fully faithful with essential image $D^{\leq0}(A)$. $\square$

Proposition 1.2. Let $u,v\colon X\to Y$ be maps of simplicial $A$-modules.

If $X$ and $Y$ are flat and $u$ is a quasi-isomorphism, then $F^{\mathrm{st}}(u)$ is a quasi-isomorphism.
If $Nu$ and $Nv$ are chain homotopic, then $F^{\mathrm{st}}(u)=F^{\mathrm{st}}(v)$ in $D(B)$.

Proof. (1) By Theorem 3.6 (2) of the lesson on stabilization, $H_iF^{\mathrm{st}}(u)$ is identified with $H_{i+n}F(u[n])$ for $n>i$, naturally. The simplicial modules $X[n]=\Gamma((NX)[n])$ and $Y[n]$ are flat, because $N_kX$ is a direct summand of $X_k$, and $u[n]$ is a quasi-isomorphism. For complexes concentrated in degrees $\le0$, the extension of $F$ in the lesson on non-additive functors is $NF\Gamma$. By Lemma 4.1 there, $F(u[n])$ is a quasi-isomorphism.

(2) Let $\operatorname{Cyl}$ be the mapping cylinder of the identity of $NX$, with inclusions $i_0,i_1\colon NX\to\operatorname{Cyl}$ and projection $p\colon\operatorname{Cyl}\to NX$. Then $pi_0=pi_1=1$, $p$ is a homotopy equivalence, and a chain homotopy from $Nu$ to $Nv$ is a map $h\colon\operatorname{Cyl}\to NY$ with $hi_0=Nu$ and $hi_1=Nv$. Apply $\Gamma$ and $F^{\mathrm{st}}$. By Corollary 3.8 of the lesson on stabilization, $F^{\mathrm{st}}(\Gamma p)$ is an isomorphism in $D(B)$. From $F^{\mathrm{st}}(\Gamma p)F^{\mathrm{st}}(\Gamma i_0)=F^{\mathrm{st}}(\Gamma p)F^{\mathrm{st}}(\Gamma i_1)=1$ we get $F^{\mathrm{st}}(\Gamma i_0)=F^{\mathrm{st}}(\Gamma i_1)$, hence $F^{\mathrm{st}}(u)=F^{\mathrm{st}}(v)$. $\square$

Theorem 1.3 (the stable derived functor). There is a functor $LF^{\mathrm{st}}\colon D_\bullet(A)\to D_\bullet(B)$, the stable derived functor of $F$, with an isomorphism $LF^{\mathrm{st}}(X)\cong F^{\mathrm{st}}(X')$, natural in the quasi-isomorphism $X'\to X$ from a flat simplicial module.

Proof. Let $L_0$ be the functorial flat resolution of Construction 4.3 of the lesson on non-additive functors. It assigns to every complex $Y$ concentrated in degrees $\le0$ a complex $L_0(Y)$ of flat modules concentrated in degrees $\le0$, with a quasi-isomorphism $L_0(Y)\to Y$ natural in $Y$. The functor $Y\mapsto F^{\mathrm{st}}(\Gamma L_0Y)$ carries quasi-isomorphisms to quasi-isomorphisms, by Proposition 1.2 (1). So it factors uniquely through $D_\bullet(A)$. Its values are simplicial modules, so it lands in $D_\bullet(B)$. If $X'\to X$ is a quasi-isomorphism with $X'$ flat, then $\Gamma L_0X'\to X'$ and $\Gamma L_0X'\to\Gamma L_0X$ are quasi-isomorphisms of flat simplicial modules. Proposition 1.2 (1) then gives $F^{\mathrm{st}}(\Gamma L_0X)\cong F^{\mathrm{st}}(X')$, naturally. $\square$

By Lemma 1.1 we regard $LF^{\mathrm{st}}$ as a functor $D^{\leq0}(A)\to D^{\leq0}(B)$. Its homology is the stable homology: $\begin{gathered}H_iLF^{\mathrm{st}}(X)\cong H_{i+n}F(X'[n])\\ (n>i),\end{gathered}$ for a flat simplicial resolution $X'$ of $X$. The right side is $H^{-i-n}LF(X[n])$, computed by Theorem 4.4 of the lesson on non-additive functors.

Proposition 1.4 (additivity). For $X,Y\in D_\bullet(A)$, the map $LF^{\mathrm{st}}(X)\oplus LF^{\mathrm{st}}(Y)\to LF^{\mathrm{st}}(X\oplus Y)$ given by the two inclusions is an isomorphism. Consequently $LF^{\mathrm{st}}$ is an additive functor: $LF^{\mathrm{st}}(f+g)=LF^{\mathrm{st}}(f)+LF^{\mathrm{st}}(g)$.

Proof. The first statement is Proposition 4.1 (1) of the lesson on stabilization, applied to flat resolutions. For the second, $f+g$ is the composite of the diagonal $X\to X\oplus X$, $f\oplus g$, and the sum $Y\oplus Y\to Y$. The diagonal and the sum are determined by their composites with the inclusions and projections of the direct sums. These composites are carried by $LF^{\mathrm{st}}$ to the corresponding composites, and $LF^{\mathrm{st}}(0)=0$. So $LF^{\mathrm{st}}$ carries the diagonal and the sum to the diagonal and the sum, and $f\oplus g$ to $LF^{\mathrm{st}}(f)\oplus LF^{\mathrm{st}}(g)$. $\square$

The shift. Proposition 4.2 of the lesson on stabilization gives an isomorphism $\theta_X\colon F^{\mathrm{st}}(\sigma X)\to F^{\mathrm{st}}(X)[1]$ in $D(B)$, natural in $X$. Here $\sigma X=\Gamma((NX)[1])$ represents $X[1]$. By Theorem 1.3 it gives an isomorphism $\theta_X\colon LF^{\mathrm{st}}(X[1])\to LF^{\mathrm{st}}(X)[1]$, natural in $X\in D_\bullet(A)$, because every morphism of $D_\bullet(A)$ is a composite of maps of flat simplicial modules and inverses of quasi-isomorphisms between them.

Proposition 1.5 (extension to $D^-$). For $X\in D^-(A)$ with $H^k(X)=0$ for $k>m$, $m\geq0$, put $LF^{\mathrm{st}}(X)=LF^{\mathrm{st}}(X[m])[-m]$. The isomorphisms $\theta$ identify the values for different $m$, compatibly. This defines a functor $LF^{\mathrm{st}}\colon D^-(A)\to D^-(B)$ with a natural isomorphism $\theta_X\colon LF^{\mathrm{st}}(X[1])\cong LF^{\mathrm{st}}(X)[1]$, extending those on $D_\bullet(A)$. It is unique up to unique isomorphism among functors with an isomorphism to the shift that restrict to $(LF^{\mathrm{st}},\theta)$ on $D_\bullet(A)$.

Proof. For $m'=m+1$, the isomorphism $\theta_{X[m]}[-m-1]\colon$ $LF^{\mathrm{st}}(X[m+1])[-m-1]\to$ $LF^{\mathrm{st}}(X[m])[-m]$ identifies the two values. Composites of these identifications give, for any $m\le m'$, an identification independent of the path, because each step is a single $\theta$. On a morphism $f\colon X\to Y$, choose $m$ for both objects and put $LF^{\mathrm{st}}(f)=LF^{\mathrm{st}}(f[m])[-m]$. By the naturality of $\theta$, this is compatible with the identifications. So it defines a functor. The isomorphism $\theta_X$ for $X\in D^-(A)$ is $\theta_{X[m]}[-m]$ for large $m$, under the identification $(X[1])[m]=X[m][1]$. Conversely, let $G$ be a functor on $D^-(A)$ with a natural isomorphism $\theta'\colon G(X[1])\cong G(X)[1]$, and suppose that $(G,\theta')$ restricts to $(LF^{\mathrm{st}},\theta)$ on $D_\bullet(A)$. Then $m$ applications of $\theta'$ give $G(X)\cong G(X[m])[-m]=LF^{\mathrm{st}}(X)$, compatibly with the identifications. $\square$

Theorem 1.6 (exactness). The functor $LF^{\mathrm{st}}\colon D^-(A)\to D^-(B)$, with $\theta$, is an exact functor of triangulated categories: for every distinguished triangle $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}X[1]$, the triangle $\begin{aligned}&LF^{\mathrm{st}}X \to LF^{\mathrm{st}}Y \to LF^{\mathrm{st}}Z\\&\quad \xrightarrow{\theta_X\circ LF^{\mathrm{st}}(h)}LF^{\mathrm{st}}(X)[1]\end{aligned}$ is distinguished.

Proof. Reduction to $D_\bullet$. If $X,Y,Z\in D^{\leq m}(A)$, the triangle $(X[m],Y[m],Z[m],$ $f[m],g[m],(-1)^mh[m])$ is distinguished. This follows from the rotation axiom applied $3m$ times and the isomorphism of triangles $(1,-1,1)$, which changes the signs of the first two maps. Its objects lie in $D_\bullet(A)$. Suppose its image is distinguished, and shift that image back by $[-m]$, which again multiplies the third map by $(-1)^m$. By additivity (Proposition 1.4), $LF^{\mathrm{st}}((-1)^mh[m])=(-1)^mLF^{\mathrm{st}}(h[m])$, so the two signs cancel. By the definitions of Proposition 1.5, the result is the image of the original triangle.

Triangles in $D_\bullet$. Let $X,Y,Z\in D_\bullet(A)$. In $D(A)$, $f$ is represented by a roof $X\leftarrow X''\to Y$ whose left arrow is a quasi-isomorphism. Applying the truncation $\tau^{\leq0}$ of Lemma 1.1, we may take $X''$ concentrated in degrees $\le0$. Applying the functorial flat resolution $\Gamma L_0$ of the proof of Theorem 1.3, $f$ is represented, up to isomorphisms in $D_\bullet(A)$, by a map $u\colon P\to Q$ of flat simplicial modules. Replace $Q$ by $Q\oplus\gamma P$ and $u$ by $(u,i)$, where $0\to P\xrightarrow{i}\gamma P\to\sigma P\to0$ is the cone sequence. Since $\gamma P$ is homotopy equivalent to $0$, this does not change $f$ in $D_\bullet(A)$. The new map $u$ is split injective in each degree, and its cokernel $R$ is flat. The sequence $0\to P\to Q\to R\to0$ is split in each degree. Its triangle in the derived category [Stacks, Tag 0152] is distinguished, with first map $f$. Two distinguished triangles with the same first map are isomorphic, by the axiom that a commutative square extends to a morphism of distinguished triangles and by [Stacks, Tag 014A]. So it is isomorphic to the given triangle. By Proposition 4.3 of the lesson on stabilization, $F^{\mathrm{st}}$ carries this sequence to a distinguished triangle whose third map is $\theta_P\circ F^{\mathrm{st}}(e)$. There, $e\colon R\to\sigma P$ represents the third map of the triangle of the sequence, which corresponds to $h$. So the image of the given triangle is isomorphic to a distinguished triangle. $\square$

The sign convention matters here. With the conventions of the Stacks project, the third map of the triangle of the cone sequence $0\to X\to\gamma X\to\sigma X\to0$ is $+1$, and $\theta_X$ carries no sign. A convention in which this map is $-1$ requires $\theta_X$ to be defined with a sign. Otherwise distinguished triangles would be carried to triangles in which one map has the wrong sign.

2. Examples

Example 2.1 (the free abelian group). Let $A=B=\mathbb Z$ and $F(M)=\mathbb Z^{(M)}$, the free abelian sheaf on the underlying sheaf of sets of $M$. It commutes with restriction and with local inductive limits. Here $F(0)=\mathbb Z$, and $F^+(M)=I(M)$ is the augmentation ideal.

Invariance. $\mathbb Z^{\mathrm{st}}$ carries every quasi-isomorphism of simplicial abelian sheaves, flat or not, to a quasi-isomorphism. So $L\mathbb Z^{\mathrm{st}}(X)\cong\mathbb Z^{\mathrm{st}}(X)$ for every $X\in D_\bullet(\mathbb Z)$. Indeed, if $u$ is a quasi-isomorphism, so is $u[n]$. Then $\mathbb Z^{(u[n])}$ is a quasi-isomorphism by Corollary 7.2 of the lesson on the free abelian Whitehead theorem, which applies to homomorphisms of simplicial abelian sheaves without any hypothesis on points. So is its summand $I(u[n])$. Apply Theorem 3.6 (2) of the lesson on stabilization.
Augmentation. The sum maps $\mathbb Z^{(G)}\to G$ form a natural transformation from $F$ to the identity functor. The stabilization is functorial in the functor, and the stabilization of the identity is the identity (Corollary 3.7 of the lesson on stabilization). This gives a natural map $\varepsilon\colon\mathbb Z^{\mathrm{st}}(X)\to X$ in $D(\mathbb Z)$.
Degree zero. For an abelian sheaf $X$, $H_0(\varepsilon)\colon H_0\mathbb Z^{\mathrm{st}}(X)\to X$ is an isomorphism. By Exercise 6.2 of the lesson on stabilization, $H_0\mathbb Z^{\mathrm{st}}(X)$ is the cokernel of the map $\operatorname{cr}_2I(X,X)\to I(X)$ induced by the sum $X\oplus X\to X$. On sections over each $U$ before sheafification, $\mathbb Z[M\oplus M']=\mathbb Z[M]\otimes\mathbb Z[M']$ shows $\operatorname{cr}_2I(M,M')=I(M)\otimes I(M')$. The sum map induces the multiplication of the group ring, with image $I(M)^2$. The augmentation induces $I(M)/I(M)^2\to M$, $[m]-1\mapsto m$. Its inverse $m\mapsto[m]-1$ is additive modulo $I(M)^2$, because $([m]-1)([m']-1)=$ $([m+m']-1)-$ $([m]-1)-$ $([m']-1)$. Sheafification is exact.
Degree one. For every abelian sheaf $X$, $H_1\mathbb Z^{\mathrm{st}}(X)=0$. The proof follows Lemma 2.2 below.
Higher degrees. Cartan computed the groups $H_i\mathbb Z^{\mathrm{st}}(X)$, the stable homology groups of the Eilenberg–MacLane spaces. They are torsion for $i>0$, with $H_2=X/2X$ and $H_3=\ker(2\cdot1_X)$. These three statements are proved in Connectivity of powers of simplicial ideals (Theorem 2.7), through the filtration of the group ring by the powers of its augmentation ideal, whose graded pieces are the symmetric powers. Exercise 3.3 computes the groups $H_2$ and $H_3$ of $L(S^2)^{\mathrm{st}}$ that appear there.

Lemma 2.2. Let $\operatorname{Ner}(\mathbb Z)$ be the nerve of the group $\mathbb Z$. Then $H_q\mathbb Z(\operatorname{Ner}\mathbb Z)=0$ for $q\geq2$, in the topos of sets and, for the constant simplicial sheaf, in every topos.

Proof. The homology sheaves of a constant simplicial sheaf are associated with the constant presheaves, so it suffices to treat sets. Write $G=\mathbb Z$. Let $EG$ be the simplicial set with $(EG)_n=G^{n+1}$, whose faces delete and whose degeneracies repeat a coordinate. It is the nerve of the category with object set $G$ and exactly one arrow between any two objects. That category has a final object, so $EG$ is contractible (Corollary 2.2 of the lesson on décalage), and the chain complex of $\mathbb Z(EG)$ is a resolution of $\mathbb Z$. The group $G$ acts on $EG$ by translation, freely in each degree, so this is a resolution by free $\mathbb Z[G]$-modules. The map $(g_0,\ldots,g_n)\mapsto(g_1-g_0,\ldots,g_n-g_{n-1})$ identifies the quotient $EG/G$ with $\operatorname{Ner}(G)$. Deleting $g_i$ adds two neighbouring differences, or drops the first or the last, and repeating $g_i$ inserts a difference $0$. Hence the chain complex of $\operatorname{Ner}(G)$ is $\mathbb Z\otimes_{\mathbb Z[G]}C(EG)$, and $H_q(\operatorname{Ner}G)=\operatorname{Tor}_q^{\mathbb Z[G]}(\mathbb Z,\mathbb Z)$ [Stacks, Tag 00LZ]. The ring $\mathbb Z[G]=\mathbb Z[t,t^{-1}]$ has the resolution $0\to\mathbb Z[t,t^{-1}]\xrightarrow{t-1}\mathbb Z[t,t^{-1}]\to\mathbb Z\to0$, so $\operatorname{Tor}_q$ vanishes for $q\geq2$. $\square$

Proof of Example 2.1 (4). Write $I=\mathbb Z^+$ for the augmentation ideal functor, so that $\mathbb Z^{(M)}=\mathbb Z\oplus I(M)$.

The case $X=\mathbb Z$. By Theorem 3.6 (2) of the lesson on stabilization with $n=2$, $H_1\mathbb Z^{\mathrm{st}}(\mathbb Z)=H_3I(K(\mathbb Z,2))$, where $K(\mathbb Z,2)=\Gamma(\mathbb Z[2])$. The nerve $\operatorname{Ner}(\mathbb Z)$ is $1$-reduced and normalizes to $\mathbb Z$ in degree $1$ (Lemma 3.4 of the lesson on multisimplicial objects). Apply Theorem 2.1 of the lesson on stabilization to $T=I$ and $Z=\operatorname{Ner}(\mathbb Z)$, with $n=1$ and $m=2=2n$. The suspension $H_2I(\operatorname{Ner}\mathbb Z)\to H_3(\operatorname{Tot}NI(\mathcal B\operatorname{Ner}\mathbb Z))$ is surjective. Its source is $0$ by Lemma 2.2. The bisimplicial object $\mathcal B\operatorname{Ner}(\mathbb Z)$ is the iterated nerve $\mathbb Z\langle2\rangle$, whose diagonal has a homotopy equivalence of normalizations to $K(\mathbb Z,2)$, as in the proof of Lemma 3.5 there. By Lemma 1.3 there, the target is $H_3I(K(\mathbb Z,2))$. So $H_1\mathbb Z^{\mathrm{st}}(\mathbb Z)=0$. The same argument applies in every topos $T/U$, $U\in\mathcal C$.

Flat sheaves. By additivity (Proposition 4.1 (1) of the lesson on stabilization), $H_1\mathbb Z^{\mathrm{st}}(\mathbb Z_U^r)=0$ over every $U$. The functor $\mathbb Z^{\mathrm{st}}$ commutes with local inductive limits: $\mathbb Z^{(-)}$ does, and the other steps are finite sums, $\Gamma$, $N$, diagonals and sums over the strings of $\bar\Delta'$. Local inductive limits are exact and commute with homology. A flat $X$ is a local inductive limit of finite free modules, by Theorem 3.2 of the lesson on non-additive functors. Hence $H_1\mathbb Z^{\mathrm{st}}(X)=0$.

General $X$. Let $K$ be the kernel of the counit $\mathbb Z^{(X)}\to X$. The sheaf $\mathbb Z^{(X)}$ is flat. By Theorem 1.6, the triangle of $0\to K\to\mathbb Z^{(X)}\to X\to0$ gives an exact sequence $\begin{aligned} &H_1\mathbb Z^{\mathrm{st}}(\mathbb Z^{(X)})\to H_1L\mathbb Z^{\mathrm{st}}(X)\\ &\quad\to H_0L\mathbb Z^{\mathrm{st}}(K)\to H_0L\mathbb Z^{\mathrm{st}}(\mathbb Z^{(X)}). \end{aligned}$ The first group is $0$, and by (3) the last map is the inclusion $K\to\mathbb Z^{(X)}$. So $H_1L\mathbb Z^{\mathrm{st}}(X)=0$, and $L\mathbb Z^{\mathrm{st}}(X)=\mathbb Z^{\mathrm{st}}(X)$ by (1). $\square$

Example 2.3 (symmetric, exterior and divided powers). Let $A=B$ and $n\geq1$. For $X\in D_\bullet(A)$ there are isomorphisms, natural in $X$, $\begin{gathered} L(S^n)^{\mathrm{st}}(X)\cong L(\Lambda^n)^{\mathrm{st}}(X)[n-1],\\ L(\Lambda^n)^{\mathrm{st}}(X)\cong L(\Gamma^n)^{\mathrm{st}}(X)[n-1]. \end{gathered}$

Proof. By Theorem 2.1 of the lesson on Koszul complexes, applied to $0\to M\xrightarrow{1}M\to0\to0$ and to $0\to0\to M\xrightarrow{1}M\to0$, the sequences $\begin{gathered}0 \to\Lambda^nM \to\Lambda^{n-1}M\otimes M \to\cdots\\ \to M\otimes S^{n-1}M \to S^nM \to0,\\ 0 \to\Gamma^nM \to\Gamma^{n-1}M\otimes M \to\cdots\\ \to M\otimes\Lambda^{n-1}M \to\Lambda^nM \to0\end{gathered}$ are exact for every flat module $M$, naturally in $M$. Their middle terms are $G(M)=T(M,M)$ with $T(M,M')=\Lambda^aM\otimes S^{n-a}M'$, respectively $\Gamma^{n-a}M\otimes\Lambda^aM'$, for $1\le a\le n-1$. Such a $T$ vanishes when one variable is $0$, so $G^{\mathrm{st}}$ is acyclic (Example 5.2 of the lesson on stabilization). Let $X$ be a flat simplicial module. The iterated nerves of $X$ are flat in each multidegree, and all constructions after the application of the functor are exact in the functor. So the stabilizations of the terms form an exact sequence of simplicial modules, natural in $X$. Its middle terms are acyclic, and there are $n-1$ of them. By Lemma 3.1 of the lesson on Koszul complexes, the sequence defines an isomorphism from the last term to the first shifted by $n-1$. Theorem 1.3 transports it to $D_\bullet(A)$. $\square$

Proposition 2.4 (Example 2.3 and the shift). Let $\varphi_X\colon L(S^n)^{\mathrm{st}}(X)\to L(\Lambda^n)^{\mathrm{st}}(X)[n-1]$ be the first isomorphism of Example 2.3, for $X\in D_\bullet(A)$.

$\theta^{\Lambda^n}_X[n-1]\circ\varphi_{X[1]}=$ $(-1)^{n-1}\,\varphi_X[1]\circ\theta^{S^n}_X$.
For $X\in D^-(A)$ with $H^k(X)=0$ for $k>m$, $m\geq0$, put $\Phi_X=(-1)^{m(n-1)}\varphi_{X[m]}[-m]$, through the identifications of Proposition 1.5. Then $\Phi_X$ does not depend on $m$, and it is an isomorphism $L(S^n)^{\mathrm{st}}(X)\to L(\Lambda^n)^{\mathrm{st}}(X)[n-1]$, natural in $X\in D^-(A)$, which satisfies (1) on $D^-(A)$.

The same holds for the second isomorphism of Example 2.3, from $L(\Lambda^n)^{\mathrm{st}}$ to $L(\Gamma^n)^{\mathrm{st}}[n-1]$.

Proof. (1) Both sides are natural in $X$, so we may take $X$ flat; then $\sigma X$ is flat and represents $X[1]$. Write the exact sequence of the proof of Example 2.3 as $0\to P\to I_{n-2}\to\cdots\to I_0\to Q\to0$, with $P=\Lambda^n$ and $Q=S^n$, and let $K_j$ be the kernel of $I_{j-1}\to I_{j-2}$, with $K_0=Q$ and $K_{n-1}=P$. These are reduced functors, and $0\to K_{j+1}\to I_j\to K_j\to0$ is exact on flat modules. By Lemma 3.1 (1) of the lesson on Koszul complexes, $\varphi_X=\delta_{n-2}(X)[n-2]\circ\cdots$ $\circ\,\delta_1(X)[1]\circ\delta_0(X)$, where $\delta_j(X)\colon K_j^{\mathrm{st}}(X)\to K_{j+1}^{\mathrm{st}}(X)[1]$ is the third map of the triangle of the stabilized sequence. By Proposition 4.5 of the lesson on stabilization, applied with $\mathcal P$ the flat modules, $\theta^{K_{j+1}}_X[1]\circ\delta_j(\sigma X)=-\,\delta_j(X)[1]\circ\theta^{K_j}_X.$ Moving $\theta$ through the $n-1$ factors of $\varphi_{\sigma X}$, one at a time, gives (1).

(2) For $X\in D^{\leq m}(A)$, the identification of the values for $m$ and $m+1$ in Proposition 1.5 is $\theta_{X[m]}[-m-1]$. By (1), applied to $X[m]$, $\begin{aligned}&\theta^{\Lambda^n}_{X[m]}[n-1][-m-1]\\&\quad\circ(-1)^{(m+1)(n-1)}\varphi_{X[m+1]}[-m-1]\\&=(-1)^{m(n-1)}\varphi_{X[m]}[-m]\\&\quad\circ\theta^{S^n}_{X[m]}[-m-1].\end{aligned}$ So the two definitions of $\Phi_X$ agree. For a morphism $f\colon X\to Y$ of $D^-(A)$, choose $m$ for both objects; naturality follows from that of $\varphi$ on $D_\bullet(A)$. The relation (1) for $\Phi$ follows from (1) for $\varphi$ at $X[m]$, by the definition of $\theta$ on $D^-(A)$ in Proposition 1.5. The second isomorphism is treated in the same way, with the second exact sequence of Example 2.3. $\square$

The sign in (1) is the one with which the shift $[n-1]$ carries distinguished triangles to distinguished triangles: with the shift isomorphism $(-1)^{n-1}\theta^{\Lambda^n}[n-1]$ on $L(\Lambda^n)^{\mathrm{st}}[n-1]$, the isomorphism $\Phi$ commutes with the shift isomorphisms.

For $A=\mathbb Z$ and $X=\mathbb Z$, the groups computed in Example 5.3 of the lesson on stabilization illustrate the shifts. The stable homology of $\Gamma^2$ is $\mathbb Z/2$ in degrees $0,2,4$, that of $\Lambda^2$ in degrees $1,3$, and that of $S^2$ in degrees $2,4$, with $0$ in the other degrees $\le4$. Likewise $H_0(\Gamma^3)^{\mathrm{st}}(\mathbb Z)=\mathbb Z/3=H_2(\Lambda^3)^{\mathrm{st}}(\mathbb Z)$.

3. Exercises

3.1. Let $F$ be additive. Show that $LF^{\mathrm{st}}$ is the left derived functor $LF$ on $D_\bullet(A)$, and that $\theta$ is the canonical isomorphism $LF(X[1])=LF(X)[1]$.

3.2. Let $F$ be reduced. Show that $H_0LF^{\mathrm{st}}$ is right exact on modules: for an exact sequence $M'\to M\to M''\to0$, the sequence $H_0LF^{\mathrm{st}}(M')\to$ $H_0LF^{\mathrm{st}}(M)\to$ $H_0LF^{\mathrm{st}}(M'')\to$ $0$ is exact.

3.3. Let $T$ be the topos of sets, $A=B=\mathbb Z$, and $X$ an abelian group. Show that $H_0L(\Gamma^2)^{\mathrm{st}}(X)=X/2X$ and $H_1L(\Gamma^2)^{\mathrm{st}}(X)=\ker(2\cdot1_X)$, and deduce the groups $H_2$ and $H_3$ of $L(S^2)^{\mathrm{st}}(X)$.

4. Complete solutions

3.1. For additive $F$, Corollary 3.7 of the lesson on stabilization gives a quasi-isomorphism $F^{\mathrm{st}}(X')\to F(X')$, natural in $X'$. For a flat resolution $X'$ of $X$, $F(X')$ computes $LF(X)$ by Theorem 4.4 of the lesson on non-additive functors. By Proposition 4.2 of the lesson on stabilization, $\theta$ is the identity for the identity functor. The same argument applies to an additive $F$, because $F$ carries the cone sequence of $X$ to the cone sequence of $F(X)$.

3.2. Let $K$ be the image of $M'\to M$ and $K'$ the kernel of $M'\to K$. The short exact sequences $0\to K\to M\to M''\to0$ and $0\to K'\to M'\to K\to0$ give distinguished triangles, which $LF^{\mathrm{st}}$ carries to distinguished triangles by Theorem 1.6. Since $LF^{\mathrm{st}}$ takes values in $D_\bullet$, the groups $H_{-1}$ vanish. The long exact homology sequences give an exact sequence $H_0LF^{\mathrm{st}}(K)\to$ $H_0LF^{\mathrm{st}}(M)\to$ $H_0LF^{\mathrm{st}}(M'')\to$ $0$ and a surjection $H_0LF^{\mathrm{st}}(M')\to H_0LF^{\mathrm{st}}(K)$. Combining them gives the claim.

3.3. For a free abelian group $P$, Exercise 6.2 of the lesson on stabilization gives $H_0(\Gamma^2)^{\mathrm{st}}(P)=$ $\Gamma^2(P)/\operatorname{im}(P\otimes P\to\Gamma^2P)$, where the cross-effect $\operatorname{cr}_2\Gamma^2(P,P)=P\otimes P$ maps by $x\otimes y\mapsto xy$. The quotient is $P/2P$ through $\gamma_2(x)\mapsto x$: modulo the products, $\gamma_2$ becomes additive, because $\gamma_2(x+y)=\gamma_2(x)+xy+\gamma_2(y)$, and $2\gamma_2(x)=x\cdot x$ becomes $0$. Since $\gamma_2(ax)=a^2\gamma_2(x)$ and $a^2\equiv a$ modulo $2$, the map $P\to\Gamma^2(P)/\operatorname{im}$, $x\mapsto\gamma_2(x)$, is additive and induces an inverse $P/2P\to\Gamma^2(P)/\operatorname{im}$. The groups $H_0$ and $H_1$ for a general $X$ follow from a free resolution $0\to P_1\to P_0\to X\to0$ and Theorem 1.6, provided $H_1(\Gamma^2)^{\mathrm{st}}(P)=0$ for free $P$. For that, use the exact sequence $0\to S^2P\to\Gamma^2P\to P/2P\to0$, natural in free $P$, where $S^2P\to\Gamma^2P$ is $xy\mapsto xy$ and $\Gamma^2P\to P/2P$ is $\gamma_2(x)\mapsto x$, $xy\mapsto0$. Stabilization turns it into a triangle with third term $P/2P$, whose stabilization is $P/2P$ in degree $0$ (Corollary 3.7 of the lesson on stabilization). By Example 2.3, $(S^2)^{\mathrm{st}}(P)\cong(\Gamma^2)^{\mathrm{st}}(P)[2]$, which has no homology in degrees $0$ and $1$. The long exact sequence gives $H_0(\Gamma^2)^{\mathrm{st}}(P)=P/2P$ and $H_1(\Gamma^2)^{\mathrm{st}}(P)=0$. For $X$, the triangle of $0\to P_1\to P_0\to X\to0$ gives the exact sequence $\begin{aligned}&0 \to H_1L(\Gamma^2)^{\mathrm{st}}(X) \to P_1/2P_1\\&\quad \to P_0/2P_0 \to H_0L(\Gamma^2)^{\mathrm{st}}(X)\\&\quad \to0,\end{aligned}$ so $H_0=X/2X$ and $H_1=\operatorname{Tor}_1(X,\mathbb Z/2)=\ker(2\cdot1_X)$. By Example 2.3 with $n=2$, $L(S^2)^{\mathrm{st}}(X)\cong L(\Gamma^2)^{\mathrm{st}}(X)[2]$. So $H_2L(S^2)^{\mathrm{st}}(X)=X/2X$ and $H_3L(S^2)^{\mathrm{st}}(X)=\ker(2\cdot1_X)$.

Sources

Stable derived functors were introduced by A. Dold and D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Annales de l'Institut Fourier 11 (1961), 201–312 (Numdam). Section 8 there defines them and shows they are additive, and Section 12 studies the symmetric powers. The construction through stabilization, which gives an exact functor on the derived category, is due to Illusie, in his work on the cotangent complex. The stable homology of Eilenberg–MacLane spaces was determined in H. Cartan, Séminaire Henri Cartan 7 (1954–1955), Algèbres d'Eilenberg–MacLane et homotopie (Numdam). L. Breen, Extensions du groupe additif, Publications Mathématiques de l'IHÉS 48 (1978), 39–125 (Numdam), uses the stable functors of the symmetric and divided powers.