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Graded rings and the spectra of modules

Written by Claude Opus 5.5 (Anthropic) on 21 October 2026. Self-checked by the writing AI. Public domain (CC0).

An abelian sheaf \(G\) on a topos is determined by its spectrum, the family of its iterated nerves \(G\langle n\rangle\), as a spectral diagram of the topos (Multisimplicial objects and spectral diagrams, Section 3). The spectrum records the addition of \(G\) by the faces of the nerves, so it is an object of the topos itself, without additive structure. This lesson shows that bilinear maps, rings and modules are recorded in the same way. A bilinear map \(X\times Y\to Z\) of abelian sheaves is the same as a map of spectra \(X(1)\times Y(1)\to Z(1)\) of the type given by concatenation of coordinates. Hence a ring \(A\) gives a graded monoid \(A(1)^\times\), an \(A\)-module \(M\) gives a graded \(A(1)^\times\)-object \(M(1)^\times\), and \(M\mapsto M(1)^\times\) is fully faithful with an explicit essential image. Combined with Graded monoids and the nerves of their actions, this describes \(A\)-modules by bisimplicial diagrams built from products of copies of \(A\) and \(M\).

On the way the lesson treats graded rings and modules in general: the external tensor product of multisimplicial objects and its compatibility with normalization, reduction and translation; rings and modules of type \(C_\bullet(J)\), their categories of modules and the induced modules; and the equivalence between large simplicial rings and reduced spectral rings.

Prerequisites. Graded monoids and the nerves of their actions (Definitions 1.1–1.5 and 2.1, Propositions 2.3, 2.6 and 3.3, Lemma 3.1); Sections 1–3 of Multisimplicial objects and spectral diagrams (Lemmas 1.1, 1.2 and 3.4, Propositions 1.3, 2.4, 2.5, 3.3 and 3.5); Theorem 1.1 of The cotangent complex (the normalization projection). From the AI-integrated Stacks project: limits and colimits of abelian sheaves [Stacks, Tag 03CO] and generators of sheaves of modules [Tag 03EW].

Conventions

\(T\) is a topos, \(\mathcal A=\operatorname{Ab}(T)\) the abelian category of abelian sheaves on \(T\), that is, of commutative group objects of \(T\), and \(\otimes=\otimes_{\mathbb Z}\). For a small category \(K\), \(\mathcal A^K\) is the category of functors \(K\to\mathcal A\), the abelian sheaves on the topos \(T^K\). The notation for multisimplicial objects, multicomplexes, normalization \(N\), the Dold–Kan functor \(\Gamma\), faces \(\partial^a_\delta\), reduction and translations is that of the lesson on multisimplicial objects; in particular \(\mathbf 0=(0,\ldots,0)\) and \(\mathbf 1=(1,\ldots,1)\). For an abelian sheaf \(G\), the iterated nerve \(G\langle n\rangle\) has \[ \begin{gathered}G\langle n\rangle(p_0,\ldots,p_{n-1})=G^{P},\\ P=\prod_{k<n}\{1,\ldots,p_k\},\end{gathered} \] a product of copies of \(G\) indexed by \(P\). On points \(g=(g_a)_{a\in P}\), an increasing map \(\alpha\colon[p'_k]\to[p_k]\) in the coordinate \(k\) acts by the nerve formula: the component of \(\alpha^*g\) at an index \(a'\) is the sum of the \(g_a\) over the indices \(a\) that agree with \(a'\) outside the coordinate \(k\) and satisfy \(\alpha(a'_k-1)<a_k\leq\alpha(a'_k)\). In particular, for \(1\leq l\leq p_k\), the edge \(e_{l-1,l}\colon[1]\to[p_k]\) acts as the projection to the indices with \(k\)-th coordinate \(l\).

1. External tensor products and graded rings

Let \(\mathcal B\) be an abelian category with a functor \(\otimes\colon\mathcal B\times\mathcal B\to\mathcal B\), additive in each variable. For \(X\in\operatorname{Simpl}_p(\mathcal B)\) and \(Y\in\operatorname{Simpl}_q(\mathcal B)\), the external tensor product \(X\mathbin{\underline\otimes}Y\in\operatorname{Simpl}_{p+q}(\mathcal B)\) has the values \(X(r)\otimes Y(s)\), the coordinates of \(X\) coming first. For multicomplexes \(C\in C_p(\mathcal B)\) and \(D\in C_q(\mathcal B)\), \(C\mathbin{\underline\otimes}D\in C_{p+q}(\mathcal B)\) has the terms \(C_r\otimes D_s\) and the differentials \(\partial_k\otimes1\) for \(k<p\) and \(1\otimes\partial_{k-p}\) for \(k\geq p\); they commute because \(\otimes\) is a functor. More generally, \(F\mathbin{\underline\otimes}F'\) is defined for functors on any two small categories, with values in \(\mathcal B\).

Lemma 1.1. Let \(X\in\operatorname{Simpl}_p(\mathcal B)\), \(Y\in\operatorname{Simpl}_q(\mathcal B)\), \(C\in C_p(\mathcal B)\) and \(D\in C_q(\mathcal B)\).

  1. The diagonal of \(X\mathbin{\underline\otimes}Y\) is the simplicial object \([n]\mapsto\operatorname{diag}X_n\otimes\operatorname{diag}Y_n\).
  2. \(N(X\mathbin{\underline\otimes}Y)=NX\mathbin{\underline\otimes}NY\), as subobjects of \(X\mathbin{\underline\otimes}Y\) in each multidegree, with the same differentials.
  3. There is a unique map \((X\mathbin{\underline\otimes}Y)^{\mathrm{red}}\to X^{\mathrm{red}}\mathbin{\underline\otimes}Y^{\mathrm{red}}\) compatible with the quotient maps from \(X\mathbin{\underline\otimes}Y\), and it is an isomorphism.
  4. \((C\mathbin{\underline\otimes}D)[(r,s)]=C[r]\mathbin{\underline\otimes}D[s]\) for \(r\in\mathbb Z^p\), \(s\in\mathbb Z^q\), whenever both sides are concentrated in \(\mathbb N^{p+q}\). Consequently there are natural isomorphisms \[ \begin{gathered}(X\mathbin{\underline\otimes}Y)[(r,s)]\cong X[r]\mathbin{\underline\otimes}Y[s],\\ (X'\mathbin{\underline\otimes}Y')[-\mathbf 1]\cong X'[-\mathbf 1]\mathbin{\underline\otimes}Y'[-\mathbf 1]\end{gathered} \] for \(r\in\mathbb N^p\), \(s\in\mathbb N^q\), and \(X'\), \(Y'\) reduced, which are identities on normalizations.

Proof. (1) holds by definition.

(2) By Theorem 1.1 of The cotangent complex, the Moore normalization \(N_nV\) of a simplicial object \(V\) is the image of the idempotent \(\pi_n=(1-s_0d_1)\cdots(1-s_{n-1}d_n)\) of \(V_n\). That proof uses only integer combinations of composites of faces and degeneracies, so it holds in every abelian category. The normalization of a multisimplicial object is computed coordinate by coordinate, so \(N(Z)\) in multidegree \(m\) is the image of the product of the commuting idempotents \(\pi^{(k)}_{m_k}\), one for each coordinate \(k\). For \(Z=X\mathbin{\underline\otimes}Y\) and \(k<p\), the faces and degeneracies in the coordinate \(k\) are those of \(X\) tensored with \(1\). Since \(\otimes\) is additive in the first variable, \(\pi^{(k)}\) for \(Z\) is \(\pi^{(k)}\otimes1\); similarly for \(k\geq p\). So the idempotent of \(Z\) in multidegree \((r,s)\) is \(e\otimes e'\), with \(e\) and \(e'\) the idempotents of \(X\) and \(Y\). Additive functors preserve split idempotents and their images, so its image is \(NX_r\otimes NY_s\), embedded in \(X(r)\otimes Y(s)\) as a direct summand. The differential of \(NZ\) in the coordinate \(k<p\) is \(d_0\otimes1\), that of \(NX\mathbin{\underline\otimes}NY\).

(3) The object \(X^{\mathrm{red}}\mathbin{\underline\otimes}Y^{\mathrm{red}}\) is reduced: its face of degree \(0\) in a coordinate \(k<p\) is \(\partial^0_k(X^{\mathrm{red}})\mathbin{\underline\otimes}Y^{\mathrm{red}}=0\), and similarly for \(k\geq p\). By Lemma 1.2 (3) of the lesson on multisimplicial objects, reduction is left adjoint to the inclusion of reduced objects, which gives the unique map. Apply \(N\). By (2) and Lemma 1.2 (3) there, the quotient maps become the projections of \(NX\mathbin{\underline\otimes}NY\) onto \(N(X\mathbin{\underline\otimes}Y)_{\geq\mathbf 1}\) and onto \((NX)_{\geq\mathbf 1}\mathbin{\underline\otimes}(NY)_{\geq\mathbf 1}\). These quotients are equal, because a multidegree \((r,s)\) has all coordinates \(\geq1\) exactly when \(r\geq\mathbf 1\) and \(s\geq\mathbf 1\). So \(N\) of the map is the identity, and the map is an isomorphism because \(N\) is an equivalence (Lemma 1.1 there).

(4) The equality holds term by term, with the same differentials. By definition \(X[r]=\Gamma((NX)[r])\). By (2), \(X[r]\mathbin{\underline\otimes}Y[s]\) and \((X\mathbin{\underline\otimes}Y)[(r,s)]\) have the same normalization \((NX\mathbin{\underline\otimes}NY)[(r,s)]\), and since \(N\) is an equivalence with \(N\Gamma=1\), there is a unique isomorphism between them inducing the identity on normalizations. The same argument applies to \([-\mathbf 1]\) on reduced objects, whose normalizations vanish on the hyperplanes. \(\square\)

Definition 1.2. Let \(K\) be a small strict monoidal category with product \(\otimes\) and unit object \(1\). A \(K\)-graded ring of \(T\) is an object \(A\) of \(\mathcal A^K\) with maps \[ A(k)\otimes A(k')\to A(k\otimes k'), \] natural in \(k,k'\), and a unit \(\mathbb Z\to A(1)\), satisfying the associativity and unit laws. In other words, it is a morphism \(A\mathbin{\underline\otimes}A\to A\) of diagrams of type \(\otimes\) with a unit of type the inclusion of \(1\). A \(K\)-graded \(A\)-module is an object \(M\) of \(\mathcal A^K\) with maps \(A(k)\otimes M(k')\to M(k\otimes k')\), natural, associative and unital. Morphisms are the natural transformations compatible with these maps. For an \(I\)-graded category \(J\) with a monoid structure, a \(J\)-graded ring is an \(\int J\)-graded ring, for the product \(\tilde\mu\), and similarly for modules.

A ring of type \(C_\bullet(J)\) is an object \(R\) of \(\mathcal A^{\int C_\bullet(J)}\), that is, a simplicial object of \(\operatorname{Diagr}(\mathcal A)\) of type \(C_\bullet(J)\), such that every \(R_q\) is a \(C_q(J)\)-graded ring and every \(R(\alpha)\colon R_q\to R_m\circ C(\alpha)\) is compatible with the multiplications and units, through the strict monoidal functor \(C(\alpha)\). An \(R\)-module of type \(C_\bullet(J)\) is an object \(L\) of \(\mathcal A^{\int C_\bullet(J)}\) such that every \(L_q\) is a \(C_q(J)\)-graded \(R_q\)-module and every \(L(\alpha)\) is compatible with the actions through \(R(\alpha)\). They form a category \(R\text{-}\mathrm{Mod}\).

For a \(J\)-graded ring \(A\), the simplicial object \(C_\bullet(A)\) of Definition 2.1 of the lesson on graded monoids is a ring of type \(C_\bullet(J)\), with the multiplications \[ \begin{aligned}&A(i_0,x)\otimes A(j_0,y)\\&\quad \to A\bigl(i_0+j_0,\mu(x,y)\bigr)\end{aligned} \] on the summands of \((\sigma,x)\) and \((\tau,y)\), where \(i_0=\sigma(0)\) and \(j_0=\tau(0)\). For a \(J\)-graded \(A\)-module \(M\), \(C_\bullet(M)\) is a \(C_\bullet(A)\)-module of type \(C_\bullet(J)\) in the same way. The compatibility with the maps \(C(\alpha)\) is the naturality of the multiplication and action, as in 2.4 of that lesson.

Proposition 1.3. Let \(A\) be a \(J\)-graded ring. The functor \(M\mapsto C_\bullet(M)\), from \(J\)-graded \(A\)-modules to \(C_\bullet(A)\)-modules of type \(C_\bullet(J)\), is fully faithful. Its essential image consists of the \(L\) such that, for every \(q\), the map \(L_q\to L_0\circ C(v_0)\) of the vertex \(v_0\colon[0]\to[q]\) is an isomorphism.

Proof. A morphism \(C_\bullet(M)\to C_\bullet(M')\) of modules is in particular a morphism of simplicial objects of \(\operatorname{Diagr}(T)\), so by Proposition 2.3 (2) of the lesson on graded monoids it is \(C_\bullet(u)\) for a unique morphism \(u\colon M\to M'\) of \(J\)-graded objects of \(T\). Each component \(u(i,x)\) is a component of \(C_0(u)\), so \(u\) is additive and \(A\)-linear. Conversely \(C_\bullet(u)\) is a morphism of modules when \(u\) is.

The modules \(C_\bullet(M)\) satisfy the condition by Proposition 2.3 (2) there. Conversely, let \(L\) satisfy it. By that proposition and its proof, there are a \(J\)-graded object \(M\) of \(T\) with \(C_0(M)=L_0\) and an isomorphism \(L\cong C_\bullet(M)\) given by the vertex maps. For \(f\colon i\to i'\), the map \(M(f,\mathrm{id})\) on \(M(i,x)\) is the vertex map of \(v_1\) composed with the inverse of the vertex map of \(v_0\), on the summand of \((f,x)\) of \(C_1(J)\). These vertex maps are homomorphisms, so \(M\) is a \(J\)-graded abelian sheaf and \(L\cong C_\bullet(M)\) in \(\mathcal A^{\int C_\bullet(J)}\). The action of \(C_0(A)\) on \(L_0\) consists of maps \(A(i,x)\otimes M(j,y)\to M(i+j,\mu(x,y))\), natural in \(x\in J_i\) and \(y\in J_j\), associative and unital. Their naturality with respect to the arrows \((f,\mathrm{id})\) and \((g,\mathrm{id})\) of \(\int J\) follows from the compatibility of the vertex maps \(L_1\to L_0\circ C(v_0)\) and \(L_1\to L_0\circ C(v_1)\) with the actions, exactly as in the proof of Proposition 2.6 of the lesson on graded monoids. So \(M\) is a \(J\)-graded \(A\)-module. The vertex maps \(L_q\to C_q(M)\) are compatible with the actions, through the maps \(C(v_0)\colon C_q(A)\to C_0(A)\), which are identities on the summands. So \(L\cong C_\bullet(M)\) as modules. \(\square\)

Proposition 1.4 (induced modules). Assume (H), and let \(R\) be a ring of type \(C_\bullet(J)\).

  1. \(R\text{-}\mathrm{Mod}\) is an abelian category. Kernels, cokernels and all colimits are computed in \(\mathcal A^{\int C_\bullet(J)}\), that is, objectwise, and filtered colimits are exact.
  2. Let \(m\colon\int(C_\bullet(J)\times C_\bullet(J))\to\int C_\bullet(J)\) be the functor induced by the products of the \(C_q(J)\), a discrete opfibration. The forgetful functor \(R\text{-}\mathrm{Mod}\to\mathcal A^{\int C_\bullet(J)}\) has the left adjoint \[ \operatorname{ind}(L)=m_!\bigl(R\mathbin{\underline\otimes}L\bigr), \] where \(R\mathbin{\underline\otimes}L\) has the value \(R_q(c)\otimes L_q(c')\) at \((q,(c,c'))\). Explicitly, \[ \begin{aligned}&\operatorname{ind}(L)_q(c)\\&\quad =\bigoplus_{c=c_1\otimes c_2}R_q(c_1)\otimes L_q(c_2),\end{aligned} \] the sum being over the factorizations of \(c\) in \(C_q(J)\), and \(R\) acts on the left factors.
  3. \(R\text{-}\mathrm{Mod}\) has a generator, so it is a Grothendieck abelian category.
  4. If every \(R_q(c)\) is a flat abelian sheaf, \(\operatorname{ind}\) is exact.

Proof. (1) Let \(u\colon L\to L'\) be a morphism of modules. The objectwise cokernel is a module because \(R_q(c)\otimes-\) is right exact. The objectwise kernel is a module because the action \(R_q(c)\otimes\ker u_q(c')\to L_q(c\otimes c')\) has composite zero to \(L'_q(c\otimes c')\). So kernels and cokernels are computed objectwise, and the image and coimage agree because they agree in \(\mathcal A^{\int C_\bullet(J)}\). Colimits are computed objectwise because \(R_q(c)\otimes-\) commutes with colimits. Filtered colimits of abelian sheaves are exact [Tag 03CO].

(2) The functor \(m\) is the type of the last-vertex morphism \(D_1\to D_0\) of Proposition 3.3 of the lesson on graded monoids, for any \(G\) and \(X\), so it is a discrete opfibration by part (1) of that proposition. For functors with values in \(\mathcal A\), the left Kan extension along a discrete opfibration is the direct sum over the fibres, by the argument of Lemma 3.1 (2) there; this gives the explicit formula. An element \(r\otimes\xi\) of the summand of \((c_1,c_2)\), multiplied on the left by a section of \(R_q(c'')\), goes to the summand of \((c''\otimes c_1,c_2)\) through the multiplication of \(R_q\). This action is associative and unital, and it is compatible with the maps of \(\int C_\bullet(J)\) because every \(C(\alpha)\) is strict monoidal and \(R(\alpha)\) is multiplicative. The unit \(\eta\colon L\to\operatorname{ind}(L)\) maps \(L_q(c)\) to the summand of \((1,c)\) through \(\mathbb Z\to R_q(1)\). For a module \(L'\) and a morphism \(g\colon L\to L'\) of \(\mathcal A^{\int C_\bullet(J)}\), the morphism \(\hat g\colon\operatorname{ind}(L)\to L'\) that is \(R_q(c_1)\otimes L_q(c_2)\to\) \(R_q(c_1)\otimes L'_q(c_2)\to\) \(L'_q(c_1\otimes c_2)\) on the summand of \((c_1,c_2)\) is \(R\)-linear, and it is the only \(R\)-linear morphism with \(\hat g\eta=g\), because the summands are generated by \(\eta(L)\) under the action. So \(g\mapsto\hat g\) is inverse to composition with \(\eta\).

(3) The topos \(T^{\int C_\bullet(J)}\) is the topos of sheaves on a site, and every abelian sheaf on it is a quotient of a direct sum of sheaves from a set \(\{U_\lambda\}\) of abelian sheaves [Tag 03EW]. For a module \(L\), choose an epimorphism \(\bigoplus U_\lambda\to L\) of abelian sheaves. Its adjoint \(\operatorname{ind}(\bigoplus U_\lambda)\to L\) is an epimorphism, because its composite with \(\eta\) is. So \(\bigoplus_\lambda\operatorname{ind}(U_\lambda)\) is a generator, and with (1) the category is Grothendieck abelian.

(4) The functor \(L\mapsto R_q(c_1)\otimes L_q(c_2)\) is exact when \(R_q(c_1)\) is flat, and direct sums are exact. \(\square\)

2. Large simplicial rings and spectral rings

Let \(J=S_a\) on \(I=\bar\Delta'\), with the concatenation of coordinates (Examples 1.2 and 1.4 of the lesson on graded monoids). Unwinding Definition 1.2, an \(S_a\)-graded ring of \(T\) is a large diagram \(X\) of type \(a\) of abelian sheaves (Definition 2.1 of the lesson on multisimplicial objects) with maps of \((m+n)\)-simplicial objects \[ X^{(m)}\mathbin{\underline\otimes}X^{(n)}\to X^{(m+n)} \] and a unit \(\mathbb Z\to X^{(0)}\), associative and unital, such that for \(\delta\colon[m-1]\to[m'-1]\) and \(\varepsilon\colon[n-1]\to[n'-1]\) the square \[ \begin{array}{ccc}X^{(m)}\mathbin{\underline\otimes}X^{(n)}&\longrightarrow&X^{(m+n)}\\ \downarrow{\scriptstyle X_\delta\otimes X_\varepsilon}&&\downarrow{\scriptstyle X_{\delta+\varepsilon}}\\ \partial^a_\delta X^{(m')}\mathbin{\underline\otimes}\partial^a_\varepsilon X^{(n')}&\longrightarrow&\partial^a_{\delta+\varepsilon}X^{(m'+n')}\end{array} \] commutes. Here \(\partial^a_\delta Y\mathbin{\underline\otimes}\partial^a_\varepsilon Y'=\partial^a_{\delta+\varepsilon}(Y\mathbin{\underline\otimes}Y')\), because \((\delta+\varepsilon)_a=\delta_a\times\varepsilon_a\). An \(S_1\)-graded ring is reduced if its underlying spectral diagram is reduced. Modules are described in the same way.

Proposition 2.1.

  1. The functors \(X\mapsto X(1)\) and \(Y\mapsto Y(-1)\) of Proposition 2.4 of the lesson on multisimplicial objects are quasi-inverse equivalences between \(S_0\)-graded rings and reduced \(S_1\)-graded rings. The multiplication of \(X(1)\) is \[ \begin{aligned}&X^{(m)}[\mathbf 1]\mathbin{\underline\otimes}X^{(n)}[\mathbf 1]\\ &\quad \cong(X^{(m)}\mathbin{\underline\otimes}X^{(n)})[\mathbf 1]\\ &\quad \to X^{(m+n)}[\mathbf 1],\end{aligned} \] with the isomorphism of Lemma 1.1 (4). For an \(S_0\)-graded ring \(A\), the same functors are quasi-inverse equivalences between \(S_0\)-graded \(A\)-modules and reduced \(S_1\)-graded \(A(1)\)-modules.
  2. For an \(S_1\)-graded ring \(B\), the reduction \(B^{\mathrm{red}}\) of Proposition 2.5 of the lesson on multisimplicial objects is a ring, with the multiplication \(B^{\mathrm{red}}\mathbin{\underline\otimes}B^{\mathrm{red}}\cong(B\mathbin{\underline\otimes}B)^{\mathrm{red}}\to B^{\mathrm{red}}\), and \(B\mapsto B^{\mathrm{red}}\) is left adjoint to the inclusion of reduced \(S_1\)-graded rings. A reduced \(S_1\)-graded \(B\)-module is the same as a reduced \(S_1\)-graded \(B^{\mathrm{red}}\)-module, and \(M\mapsto M^{\mathrm{red}}\) is left adjoint to the inclusion of reduced \(B\)-modules into \(B\)-modules.

Proof. (1) Apply \(N\) to all the data. By Lemma 1.1 (2) and (4) and Proposition 1.3 (2) of the lesson on multisimplicial objects, the normalization of \(X(1)\) is \(NX\) translated by \(\mathbf 1\) in each component. Its structure maps go into the restrictions to the multidegrees with \(q_l=1\) for the coordinates \(l\) outside the image of \(\delta\), which are the translates of the restrictions to \(q_l=0\) (Lemma 1.2 (1) and (4) there). The multiplication of \(X(1)\) has as normalization the translate by \(\mathbf 1\) of the normalization of the multiplication of \(X\), and all the isomorphisms used are identities on normalizations. Each axiom of a ring, written for \(X(1)\), therefore has as normalization the translate of the same axiom for \(X\). Since \(N\) is faithful, \(X(1)\) is a reduced \(S_1\)-graded ring. In the same way \(Y(-1)\) is an \(S_0\)-graded ring for a reduced \(S_1\)-graded ring \(Y\). The functors are quasi-inverse on the underlying diagrams (Proposition 2.4 there), and the multiplications correspond, since maps \(X^{(m)}[\mathbf 1]\mathbin{\underline\otimes}X^{(n)}[\mathbf 1]\to X^{(m+n)}[\mathbf 1]\) and maps \(X^{(m)}\mathbin{\underline\otimes}X^{(n)}\to X^{(m+n)}\) both correspond bijectively to maps \(NX^{(m)}\mathbin{\underline\otimes}NX^{(n)}\to NX^{(m+n)}\) of multicomplexes. The same argument applies to modules.

(2) By Lemma 1.1 (3), applied componentwise, the multiplication of \(B\) induces the stated multiplication. Each axiom for \(B^{\mathrm{red}}\) holds after composition with the epimorphism \(B\mathbin{\underline\otimes}B\to B^{\mathrm{red}}\mathbin{\underline\otimes}B^{\mathrm{red}}\) (or its triple analogue), where it follows from the axiom for \(B\). So \(B^{\mathrm{red}}\) is a ring and \(B\to B^{\mathrm{red}}\) is a homomorphism. A homomorphism \(B\to C\) with \(C\) reduced factors uniquely through \(B^{\mathrm{red}}\) as a map of spectral diagrams (Proposition 2.5 there), and the factorization is multiplicative, as one sees after composing with the same epimorphism. Let \(M\) be a reduced \(B\)-module. The action \(B^{(m)}\mathbin{\underline\otimes}M^{(n)}\to M^{(m+n)}\) has a reduced target, so it factors through \((B^{(m)}\mathbin{\underline\otimes}M^{(n)})^{\mathrm{red}}\cong(B^{(m)})^{\mathrm{red}}\mathbin{\underline\otimes}M^{(n)}\). This makes \(M\) a \(B^{\mathrm{red}}\)-module, and conversely a \(B^{\mathrm{red}}\)-module is a \(B\)-module through \(B\to B^{\mathrm{red}}\). For any \(B\)-module \(M\), the action induces \((B\mathbin{\underline\otimes}M)^{\mathrm{red}}\to M^{\mathrm{red}}\), and with \(B^{\mathrm{red}}\mathbin{\underline\otimes}M^{\mathrm{red}}\cong(B\mathbin{\underline\otimes}M)^{\mathrm{red}}\) this makes \(M^{\mathrm{red}}\) a \(B^{\mathrm{red}}\)-module, hence a \(B\)-module. A linear map from \(M\) to a reduced module factors uniquely through \(M^{\mathrm{red}}\), and the factorization is linear, by the same epimorphism argument. \(\square\)

2.2. Type \(C_\bullet(S_a)\). An object \(L\) of \(\mathcal A^{\int C_\bullet(S_a)}\) consists, for each \(q\) and each \(q\)-simplex \(\sigma=([k_0-1]\to\cdots\to[k_q-1])\) of the nerve of \(\bar\Delta'\), of a \(k_0\)-simplicial object \(L_q(\sigma)\) of \(\mathcal A\), with maps \(L_q(\sigma)\to\partial^a_{\sigma(0,\alpha(0))}L_m(\sigma\alpha)\) for \(\alpha\colon[m]\to[q]\), satisfying the composition rule. Call \(L\) reduced, for \(a=1\), if every \(L_q(\sigma)\) is reduced. For \(a=0\) put \(L(1)_q(\sigma)=L_q(\sigma)[\mathbf 1]\), with the maps corresponding to those of \(L\) under the first bijection of Proposition 1.3 (3) of the lesson on multisimplicial objects. For \(a=1\) and \(L\) reduced, define \(L(-1)\) in the same way with \([-\mathbf 1]\) and the second bijection. For \(a=1\), define \(L^{\mathrm{red}}\) term by term, with the structure maps of Proposition 2.5 there.

Proposition 2.2.

  1. \(L\mapsto L(-1)\) and \(L'\mapsto L'(1)\) are quasi-inverse equivalences between the reduced objects of \(\mathcal A^{\int C_\bullet(S_1)}\) and \(\mathcal A^{\int C_\bullet(S_0)}\), and they restrict to quasi-inverse equivalences between reduced rings of type \(C_\bullet(S_1)\) and rings of type \(C_\bullet(S_0)\), and between the corresponding modules.
  2. \(L\mapsto L^{\mathrm{red}}\) is left adjoint to the inclusion of the reduced objects of \(\mathcal A^{\int C_\bullet(S_1)}\), and the statements of Proposition 2.1 (2) hold for rings and modules of type \(C_\bullet(S_1)\).
  3. \(C_\bullet\) commutes with these functors: \(C_\bullet(X(1))=C_\bullet(X)(1)\), \(C_\bullet(Y(-1))=C_\bullet(Y)(-1)\) and \(C_\bullet(Y^{\mathrm{red}})=C_\bullet(Y)^{\mathrm{red}}\), as rings or modules when \(X\) and \(Y\) are rings or modules.

Proof. The proofs of Propositions 2.4 and 2.5 of the lesson on multisimplicial objects and of Proposition 2.1 use only the bijections of Proposition 1.3 (3) there, their compatibility with composition (Proposition 1.3 (2)), Lemma 1.2 there and Lemma 1.1, one structure map at a time. They apply without change to the maps \(L_q(\sigma)\to\partial^a_{\sigma(0,\alpha(0))}L_m(\sigma\alpha)\), which satisfy the same composition rule. For (3): on the summand of \(\sigma\), \(C_q(X(1))\) is \(X(1)^{(k_0)}=X^{(k_0)}[\mathbf 1]\), and its map for \(\alpha\) is \(X(1)_{\sigma(0,\alpha(0))}\), which corresponds to \(X_{\sigma(0,\alpha(0))}\) under the bijection of Proposition 1.3 (3) there; this is the definition of \(C_\bullet(X)(1)\). The other two cases are the same, and the multiplications agree by construction. \(\square\)

3. Spectra of modules

For an abelian sheaf \(G\), let \(G(1)^\times\) be its spectrum \(G(1)\), with \(G(1)^{(n)}=G\langle n\rangle\), regarded as an \(S_1\)-graded object of \(T\), that is, as a functor \(\mathbb S_1\to T\). Its structure maps are identities (Section 3 of the lesson on multisimplicial objects).

For abelian sheaves \(X\), \(Y\), \(Z\) and a map \(f\colon X\times Y\to Z\) of \(T\), let \(f(1,1)^\times\) be the family of maps \[ \begin{gathered}X\langle m\rangle(p)\times Y\langle n\rangle(p')\\ \to Z\langle m+n\rangle(p,p'),\\ (x,y)\longmapsto\bigl(f(x_a,y_b)\bigr)_{(a,b)\in P\times P'},\end{gathered} \] where \(P\) and \(P'\) index the factors of \(X\langle m\rangle(p)\) and \(Y\langle n\rangle(p')\). The concatenation of coordinates \(\pi=\tilde\mu\colon\mathbb S_1\times\mathbb S_1\to\mathbb S_1\) is the product of \(\int S_1=\mathbb S_1\) (Example 1.4 of the lesson on graded monoids). A morphism of type \(\pi\) from \(X(1)^\times\times Y(1)^\times\) to \(Z(1)^\times\) is a morphism of \(\operatorname{Diagr}(T)\) of this type.

Proposition 3.1. The map \(f\mapsto f(1,1)^\times\) is a bijection from the bilinear maps \(X\times Y\to Z\), that is, the homomorphisms \(X\otimes Y\to Z\), to the morphisms \(X(1)^\times\times Y(1)^\times\to Z(1)^\times\) of type \(\pi\).

Proof. Maps of \(T\) can be compared on points with values in arbitrary objects, by the Yoneda lemma, and we argue on points.

If \(f\) is bilinear, \(f(1,1)^\times\) is a morphism of type \(\pi\). Its components are natural in \((p,p')\in(\Delta^{\mathrm{op}})^{m+n}\). Indeed, an increasing map in a coordinate \(k<m\) acts on \(X\langle m\rangle(p)\) by replacing each component with the sum of the components over a set of indices that agree outside the coordinate \(k\) (Conventions), and on \(Z\langle m+n\rangle(p,p')\) by the same rule on the first part of the index \((a,b)\). Since \(f\) is additive in the first variable, \(f(\sum_ax_a,y_b)=\sum_af(x_a,y_b)\), and the two composites agree. The coordinates of \(Y\) are treated in the same way. The structure maps of \(X(1)^\times\), \(Y(1)^\times\) and \(Z(1)^\times\) are identities, and for \(\delta\colon[m-1]\to[m'-1]\) and \(\varepsilon\colon[n-1]\to[n'-1]\), the face \(\partial^1_{\delta+\varepsilon}\) fixes the coordinates outside the image of \(\delta+\varepsilon\) at \([1]\). It selects the components of index \(1\) in those coordinates, and so carries the component of \(f(1,1)^\times\) at \((m',n')\) to the component at \((m,n)\). This is the compatibility with the maps \(((\delta,\mathrm{id}),(\varepsilon,\mathrm{id}))\) of \(\mathbb S_1\times\mathbb S_1\). Naturality with respect to all arrows follows.

Every morphism \(F\) of type \(\pi\) is \(f(1,1)^\times\) for a unique bilinear \(f\). Let \(f\colon X\times Y\to Z\) be the component of \(F\) at \(((0,()),(0,()))\), using \(X\langle0\rangle=X\). For \(m,n\geq0\), the empty maps \(\varnothing\to[m-1]\) and \(\varnothing\to[n-1]\) give arrows of \(\mathbb S_1\) to \((m,\mathbf 1)\) and \((n,\mathbf 1)\), and naturality with respect to them shows that the component of \(F\) at \(((m,\mathbf 1),(n,\mathbf 1))\) is \(f\). For \(a\in P\) and \(b\in P'\), the product of the edges \(e_{a_k-1,a_k}\) acts as the projection to the factor \(a\) (Conventions). Naturality with respect to these arrows gives \[ \operatorname{pr}_{(a,b)}\circ F_{(m,p),(n,p')}=f\circ(\operatorname{pr}_a\times\operatorname{pr}_b). \] So \(F=f(1,1)^\times\), and \(f\) is determined by \(F\). It remains to see that \(f\) is bilinear. The component of \(F\) at \(((1,(2)),(0,()))\) is the map \(X^2\times Y\to Z^2\), \((x_1,x_2,y)\mapsto(f(x_1,y),f(x_2,y))\). The face \(d_1\colon X\langle1\rangle(2)\to X\langle1\rangle(1)\) is the addition \((x_1,x_2)\mapsto x_1+x_2\), and on \(Z\langle1\rangle\) it is the addition as well. Naturality with respect to it gives \(f(x_1+x_2,y)=f(x_1,y)+f(x_2,y)\). The same argument with the component at \(((0,()),(1,(2)))\) gives additivity in \(y\). \(\square\)

Definition 3.2. Let \(A\) be a ring of \(T\), with multiplication \(f\colon A\times A\to A\). Its spectrum \(A(1)^\times\) is the \(S_1\)-graded monoid of \(T\) whose underlying object is the spectrum of the abelian sheaf \(A\), with the multiplication \(f(1,1)^\times\) and the unit \(e\to A=A\langle0\rangle\) given by \(1\). For an \(A\)-module \(M\) with action \(g\colon A\times M\to M\), its spectrum \(M(1)^\times\) is the \(S_1\)-graded \(A(1)^\times\)-object with the underlying object the spectrum of \(M\) and the action \(g(1,1)^\times\).

The multiplication is a morphism of type \(\pi\) by Proposition 3.1. It is associative because on points \(((x_ay_b)z_c)_{(a,b,c)}=(x_a(y_bz_c))_{(a,b,c)}\), with the index sets \((P\times P')\times P''\) and \(P\times(P'\times P'')\) identified. It is unital because the index set of \(A\langle0\rangle\) has one element. The same computations show that \(M(1)^\times\) is an \(A(1)^\times\)-object. A linear map \(u\colon M\to M'\) gives the morphism \(u(1)\), and \(M\mapsto M(1)^\times\) is a functor.

Remark 3.3. \(A(1)^\times\) is the multiplicative monoid of the reduced \(S_1\)-graded ring \((cA)(1)\) of Proposition 2.1, where \(cA\) is the trivial \(S_0\)-graded ring with value \(A\). Indeed \((cA)(1)^{(n)}=A[\mathbf 1]\), which is identified with \(A\langle n\rangle\) through their common normalization, \(A\) placed in multidegree \(\mathbf 1\) (Lemma 3.4 of the lesson on multisimplicial objects). The map of \((m+n)\)-simplicial abelian sheaves \(A\langle m\rangle\mathbin{\underline\otimes}A\langle n\rangle\to A\langle m+n\rangle\) defined by \(f(1,1)^\times\) has as normalization the multiplication \(A\otimes A\to A\) in multidegree \(\mathbf 1\), because the index sets of multidegree \(\mathbf 1\) have one element. This is also the normalization of the multiplication of \((cA)(1)\). Since \(N\) is faithful, the two multiplications agree.

Proposition 3.4. The functor \(M\mapsto M(1)^\times\), from \(A\)-modules of \(T\) to \(S_1\)-graded \(A(1)^\times\)-objects of \(T\), is fully faithful. Its essential image consists of the objects whose underlying spectral diagram satisfies the conditions (i) and (ii) of Proposition 3.5 of the lesson on multisimplicial objects: every component of positive dimension \(n\) is isomorphic to \(H\langle n\rangle\) for some abelian sheaf \(H\), and every structure map is an isomorphism.

Proof. A morphism \(M(1)^\times\to M'(1)^\times\) is in particular a morphism of spectral diagrams of \(T\), so by Proposition 3.5 of the lesson on multisimplicial objects it is \(\varphi(1)\) for a unique homomorphism \(\varphi\colon M\to M'\). Its compatibility with the actions, at \(((0,()),(0,()))\), says \(\varphi(ax)=a\varphi(x)\). Conversely \(\varphi(1)\) is compatible with the actions for a linear \(\varphi\), by the formula for \(g(1,1)^\times\).

The objects \(M(1)^\times\) satisfy the conditions. Conversely, let \(Y\) be an \(S_1\)-graded \(A(1)^\times\)-object whose spectral diagram satisfies them. By Proposition 3.5 there, there is an isomorphism of spectral diagrams \(Y\cong M(1)\) for an abelian sheaf \(M\). Transported along it, the action of \(Y\) becomes a morphism \(A(1)^\times\times M(1)^\times\to M(1)^\times\) of type \(\pi\). By Proposition 3.1 it is \(g(1,1)^\times\) for a unique bilinear \(g\colon A\times M\to M\). The associativity and unit laws of the action, at the index \(((0,()),(0,()))\) and its triple analogue, say that \(g\) is an action of the ring \(A\). So \(Y\cong M(1)^\times\). \(\square\)

Proposition 3.5. The functor \[ M\longmapsto\operatorname{Ner}\bigl(C_\bullet(A(1)^\times),C_\bullet(M(1)^\times)\bigr), \] from \(A\)-modules of \(T\) to the category \(\mathcal Y_{A(1)^\times}\) of Proposition 2.6 of the lesson on graded monoids (for \(J=S_1\)), is fully faithful. Its essential image consists of the \(Y\) satisfying conditions 1 and 2 of that proposition and the following two conditions.

  1. For every coface \(\delta^i\colon[n-1]\to[n]\) of \(\bar\Delta'\), \(0\le i\le n\), the map of \(n\)-simplicial objects \[ Y_{0,0}|_{(\Delta^{\mathrm{op}})^n}\to\partial^1_i\bigl(Y_{0,0}|_{(\Delta^{\mathrm{op}})^{n+1}}\bigr) \] is an isomorphism. Here \(|\) denotes restriction to a summand of \(C_0(S_1)=\coprod_n(\Delta^{\mathrm{op}})^n\), and the map is the face \(d_0\colon Y_{0,1}\to Y_{0,0}\) in the second index on the summand of \(\delta^i\) of \(C_1(S_1)\), composed with the inverse of the isomorphism of condition 2 for \(q=1\) on that summand.
  2. For every \(n\geq1\), \(Y_{0,0}|_{(\Delta^{\mathrm{op}})^n}\) is isomorphic to \(H\langle n\rangle\) for some abelian sheaf \(H\).

Proof. The functor is the composite of \(M\mapsto M(1)^\times\) and \(X\mapsto\operatorname{Ner}(C_\bullet(A(1)^\times),C_\bullet(X))\), both fully faithful (Proposition 3.4 and Proposition 2.6 of the lesson on graded monoids). By the latter, \(Y\) is isomorphic to \(\operatorname{Ner}(C_\bullet(A(1)^\times),C_\bullet(X))\) for an \(S_1\)-graded \(A(1)^\times\)-object \(X\) if and only if conditions 1 and 2 hold. Then \(X\) is determined up to isomorphism, with \(X^{(n)}=Y_{0,0}|_{(\Delta^{\mathrm{op}})^n}\), and by Proposition 2.3 (1) there and its proof, the structure map \(X_{\delta^i}\) is the map of condition 3. The maps \(X_\delta\) are composites of the \(X_{\delta^i}\) and their faces (Section 2 of the lesson on multisimplicial objects), so all of them are isomorphisms exactly when condition 3 holds. By Proposition 3.4, \(X\cong M(1)^\times\) for an \(A\)-module \(M\) exactly when conditions 3 and 4 hold. \(\square\)

For \(n\geq2\), condition 4 can be stated without \(H\): by Proposition 3.3 of the lesson on multisimplicial objects, it says that every simplicial object obtained by fixing all coordinates but one satisfies the conditions (i)–(iii) of Proposition 3.1 there. For \(n=1\) these conditions describe the nerves of all groups, not only of the commutative ones; but under condition 3 the case \(n=1\) follows from the case \(n=2\), since \(Y_{0,0}|_{\Delta^{\mathrm{op}}}\cong\) \(\partial^1_0(Y_{0,0}|_{(\Delta^{\mathrm{op}})^2})\cong\) \(\partial^1_0H\langle2\rangle=\) \(H\langle1\rangle\).

4. Exercises

4.1. Let \(T\) be the topos of sets and \(X=Y=\mathbb Z\langle1\rangle\), the nerve of the group \(\mathbb Z\), a simplicial abelian group. Use Lemma 1.1 (2) and Lemma 3.4 of the lesson on multisimplicial objects to show that the map \(\mathbb Z\langle1\rangle\mathbin{\underline\otimes}\mathbb Z\langle1\rangle\to\mathbb Z\langle2\rangle\) defined by the multiplication of \(\mathbb Z\) (Remark 3.3) is an isomorphism of bisimplicial abelian groups, and describe it in bidegree \((p_0,p_1)\).

4.2. Let \(X,Y,Z\) be abelian groups and \(f\colon X\times Y\to Z\) a map of sets. Show that the maps \((x,y)\mapsto(f(x_a,y_b))\) are natural in \((\Delta^{\mathrm{op}})^{1+0}\), on \(X\langle1\rangle\times Y\to Z\langle1\rangle\), only if \(f\) is additive in the first variable. Give an example with \(X=Y=Z=\mathbb Z\) where naturality fails, and name a face at which it fails.

4.3. Let \(I\) be the strict monoidal category with one object and one arrow and \(J_o\) the one-point category, so that (H) holds. Show that a ring of type \(C_\bullet(J)\) is a simplicial ring \(R\) of \(T\), that an \(R\)-module of type \(C_\bullet(J)\) is a simplicial \(R\)-module, and that \(\operatorname{ind}(L)=R\otimes L\) degreewise. Show that \(\operatorname{ind}\) is not exact for the constant simplicial ring \(R=\mathbb Z/2\) of the topos of sets.

5. Complete solutions

4.1. By Lemma 3.4 of the lesson on multisimplicial objects, \(N\mathbb Z\langle1\rangle\) is \(\mathbb Z\) in degree \(1\), and \(N\mathbb Z\langle2\rangle\) is \(\mathbb Z\) in bidegree \((1,1)\). By Lemma 1.1 (2), \(N(\mathbb Z\langle1\rangle\mathbin{\underline\otimes}\mathbb Z\langle1\rangle)=\mathbb Z\otimes\mathbb Z=\mathbb Z\) in bidegree \((1,1)\). The map induced by the multiplication has, in bidegree \((1,1)\), the multiplication \(\mathbb Z\otimes\mathbb Z\to\mathbb Z\), an isomorphism, because the index sets there have one element. A map of bisimplicial abelian groups whose normalization is an isomorphism is an isomorphism, since \(N\) is an equivalence. In bidegree \((p_0,p_1)\) the map is \(\mathbb Z^{p_0}\otimes\mathbb Z^{p_1}\to\mathbb Z^{p_0p_1}\), \(x\otimes y\mapsto(x_ay_b)_{(a,b)}\), the identification of the tensor product of two free modules with bases indexed by \(\{1,\ldots,p_0\}\) and \(\{1,\ldots,p_1\}\) with the free module on the product of the index sets.

4.2. In degree \(2\) the maps are \(X^2\times Y\to Z^2\), \((x_1,x_2,y)\mapsto(f(x_1,y),f(x_2,y))\). The face \(d_1\) acts by addition on \(X\langle1\rangle(2)=X^2\) and on \(Z\langle1\rangle(2)=Z^2\). Naturality with respect to \(d_1\) says \(f(x_1+x_2,y)=f(x_1,y)+f(x_2,y)\). For \(f(x,y)=x^2y\) on \(\mathbb Z\), with \(x_1=x_2=y=1\), the two composites give \(f(2,1)=4\) and \(f(1,1)+f(1,1)=2\). So naturality fails at \(d_1\colon X\langle1\rangle(2)\to X\langle1\rangle(1)\). (The faces \(d_0\) and \(d_2\), which delete a component, and the degeneracies, which insert \(0\), commute with these maps when \(f(0,y)=0\); here \(f(0,y)=0\).)

4.3. Every \(C_q(J)\) is the one-point category and every \(C(\alpha)\) is the identity, so \(\int C_\bullet(J)=\Delta^{\mathrm{op}}\). An object of \(\mathcal A^{\Delta^{\mathrm{op}}}\) is a simplicial abelian sheaf, and the product of \(C_q(J)\) is the identity of the point, so a ring of type \(C_\bullet(J)\) is a simplicial abelian sheaf \(R\) with multiplications \(R_q\otimes R_q\to R_q\) and units compatible with the simplicial maps: a simplicial ring. Likewise a module is a simplicial abelian sheaf \(L\) with \(R_q\)-module structures compatible with the simplicial maps. The functor \(m\) is the identity of \(\Delta^{\mathrm{op}}\), every fibre has one element, and \(\operatorname{ind}(L)_q=R_q\otimes L_q\). For \(R=\mathbb Z/2\) constant, the injection \(2\colon\mathbb Z\to\mathbb Z\) of constant simplicial abelian groups goes to \(\mathbb Z/2\otimes\mathbb Z\to\mathbb Z/2\otimes\mathbb Z\), the zero map of \(\mathbb Z/2\) in each degree, which is not injective. So \(\operatorname{ind}\) is not exact; \(\mathbb Z/2\) is not flat.

Sources

Rings and modules graded by monoids in categories, the rings of type \(C_\bullet(J)\) and their induced modules, the equivalence between large simplicial rings and reduced spectral rings, and the description of modules by the spectra \(A(1)^\times\) and \(M(1)^\times\) are due to Illusie, in his work on the cotangent complex. The compatibility of normalization with tensor products of multisimplicial objects goes back to Dold and Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Annales de l'Institut Fourier 11 (1961), 201–312 (Numdam).