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23  Stable categories of modules with duality

The constructions of this chapter take place over the additive category of modules over a ring, and a form on a module is a morphism of that category. Chapter 24 lists the hypothesis each construction states, and Chapter 22 gives the hyperbolic forms and Witt classes that the zero-dimensional case reproduces.

23.1 Rings with involution and duality

Definition 23.1 (Involutions and the duality functor) An involution on a ring \(A\) is a ring isomorphism \(A\to A^{\mathrm{op}}\), written \(a\mapsto\bar a\), with \(\bar{\bar a}=a\) for every \(a\in A\). It defines the duality functor \[ (-)^{*}\colon A\text{-}\mathbf{Mod}\longrightarrow A\text{-}\mathbf{Mod}, \qquad M^{*}=\operatorname{Hom}_A(M,A), \] where \(M^{*}\) is an \(A\)-module by \((a\cdot f)(x)=f(x)\,\bar a\), and where the dual of \(f\colon M\to N\) is \(f^{*}\colon N^{*}\to M^{*}\), \(g\mapsto g\circ f\). For a finitely generated projective \(A\)-module \(M\) the map \[ \operatorname{can}_M\colon M\longrightarrow M^{**}, \qquad \operatorname{can}_M(x)(f)=\overline{f(x)}, \] is an isomorphism (Ranicki 1998, sec. 20).

Definition 23.2 ($\epsilon$-symmetric and $\epsilon$-quadratic forms) Let \(\epsilon\in A\) be a central unit with \(\bar\epsilon=\epsilon^{-1}\). An \(\epsilon\)-symmetric form \((M,\phi)\) is a finitely generated projective \(A\)-module \(M\) together with an \(A\)-module morphism \(\phi\colon M\to M^{*}\) making the triangle

commute. The form is nonsingular if \(\phi\) is an isomorphism. An \(\epsilon\)-quadratic form \((M,\psi)\) is a finitely generated projective \(A\)-module \(M\) together with a class of morphisms \(\psi\colon M\to M^{*}\) modulo the relation \(\psi\sim\psi+\chi-\epsilon\chi^{*}\) for \(\chi\in\operatorname{Hom}_A(M,M^{*})\); its \(\epsilon\)-symmetrization is \((M,\psi+\epsilon\psi^{*})\) (Ranicki 1998, Def. 20.1). In both formulas the dual \(\chi^{*}\colon M^{**}\to M^{*}\) of a morphism \(M\to M^{*}\) is precomposed with \(\operatorname{can}_M\), as in the triangle above.

Take \(A=R\) commutative with the identity involution and \(\epsilon=1\). Then a \(1\)-symmetric form on \(M\) is the adjoint map \(b^{\sharp}\) of a symmetric \(R\)-bilinear form \(b\colon M\times M\to R\) (Definition 17.3, Definition 17.4), and nonsingularity is the perfectness of Definition 17.4. For \(M\) finitely generated projective, the hyperbolic form of Definition 22.1 is a nonsingular \(1\)-symmetric form by Proposition 22.2, and \(0\oplus M^{*}\) is a lagrangian in the sense of Definition 22.6 by Proposition 22.8.

23.2 Complexes, duality, and \(L\)-groups

Definition 23.3 (Chain complexes in an additive category) Let \(\mathcal A\) be an additive category. A chain complex in \(\mathcal A\) is a family of objects \(C_n\) together with morphisms \(d_n\colon C_n\to C_{n-1}\) satisfying \(d_{n-1}d_n=0\). Chain maps are the degreewise morphisms commuting with the differentials, and \(\mathbf{Ch}(\mathcal A)\) is again additive (Weibel 1994, secs. 1.1–1.2).

Theorem 23.4 (The Dold–Kan correspondence) Let \(\mathcal A\) be an abelian category and let \(X\) be a simplicial object of \(\mathcal A\), with face maps \(d_i\). The normalized chain complex has \[ N(X)_n=\bigcap_{i=1}^{n}\ker\bigl(d_i\colon X_n\to X_{n-1}\bigr) \] and differential induced by \(d_0\). Then \(N\) is an equivalence between the category of simplicial objects of \(\mathcal A\) and \(\mathbf{Ch}_{\geq0}(\mathcal A)\) (Weibel 1994, Thm. 8.4.1).

Definition 23.5 (Comonad resolutions and the bar construction) Let \(F\colon\mathcal C\to\mathcal A\) be left adjoint to \(U\colon\mathcal A\to\mathcal C\). The composite \(\bot=FU\), with counit \(\varepsilon\colon\bot\Rightarrow\operatorname{id}_{\mathcal A}\) and comultiplication built from the unit, is a comonad on \(\mathcal A\) (Weibel 1994, Def. 8.6.1, Main Application 8.6.2). For an object \(X\) of \(\mathcal A\) the associated augmented simplicial object has \(\bot^{\,n+1}X\) in degree \(n\), with faces and degeneracies from \(\varepsilon\) and the unit, and its chain complex is a resolution of \(X\) (Weibel 1994, sec. 8.6). For \(\mathcal A=A\text{-}\mathbf{Mod}\) and \(F\) the free module functor this is the bar resolution, and it computes the derived functors of an additive functor out of \(\mathcal A\).

Definition 23.6 (The $n$-dual of a complex) Let \(A\) be a ring with involution and \(C\) an \(A\)-module chain complex, and write \(C^{r}=(C_r)^{*}\). The \(n\)-dual \(C^{n-*}\) is the \(A\)-module chain complex with \[ (C^{n-*})_r=C^{\,n-r}, \qquad d_{C^{n-*}}=(-1)^{r}(d_C)^{*}, \] (Ranicki 1998, sec. 20).

Definition 23.7 (Algebraic Poincaré complexes and $L$-groups) An \(n\)-dimensional \(\epsilon\)-symmetric Poincaré complex over a ring with involution \(A\) is a pair \((C,\phi)\) consisting of an \(n\)-dimensional \(A\)-module chain complex \(C\) and an \(\epsilon\)-symmetric structure \(\phi\) on \(C\) whose component \(\phi_0\colon C^{n-*}\to C\) is a chain equivalence; the \(\epsilon\)-quadratic case replaces \(\phi\) by an \(\epsilon\)-quadratic structure \(\psi\) and requires \((1+T_{\epsilon})\psi_0\) to be a chain equivalence, where \(T_{\epsilon}\) is the \(\epsilon\)-transposition involution on \(C\otimes_AC\) given by \(x\otimes_A y\mapsto(-1)^{pq}\epsilon\,y\otimes_A x\) for \(x\in C_p\) and \(y\in C_q\) (Ranicki 1998, sec. 20). For a \(*\)-invariant subgroup \(U\subseteq\widetilde K_0(A)\), the \(n\)-dimensional \(U\)-intermediate \(\epsilon\)-symmetric \(L\)-group \(L^{n}_{U}(A,\epsilon)\) and its \(\epsilon\)-quadratic counterpart \(L^{U}_{n}(A,\epsilon)\) are the cobordism groups of such complexes with \([C]\in U\) (Ranicki 1998, Def. 20.10).

Theorem 23.8 (The zero-dimensional $L$-group) \(L^{0}_{U}(A,\epsilon)\) is the Witt group of nonsingular \(\epsilon\)-symmetric forms over \(A\), and \(L^{U}_{0}(A,\epsilon)\) the Witt group of nonsingular \(\epsilon\)-quadratic forms; a form admitting a lagrangian is zero in it (Ranicki 1998, Ex. 20.11).

For \(A=R\) a Dedekind domain with the identity involution and \(\epsilon=1\), the group \(L^{0}_{\widetilde K_0(R)}(R,1)\) is the additive group of the Witt ring \(W(R)\) of Theorem 22.15, and by Proposition 22.8 every hyperbolic form of Definition 22.1 has the zero class in it.

23.3 Model presentation

Definition 23.9 (Model categories) A model category is a category together with three classes of maps in it, the fibrations, the cofibrations, and the weak equivalences, satisfying the axioms of (Quillen 1967, Ch. I §1, Def. 1). Its homotopy category is the localization at the weak equivalences (Quillen 1967, Ch. I §1, Def. 6).

Theorem 23.10 (The projective model structure on complexes) Let \(\mathcal A\) be an abelian category with enough projectives and let \(C_{+}(\mathcal A)\) be the category of chain complexes in \(\mathcal A\) that are bounded below. Then \(C_{+}(\mathcal A)\) is a model category in which

  • the weak equivalences are the maps inducing isomorphisms on homology,

  • the fibrations are the epimorphisms, and

  • the cofibrations are the injective maps \(i\) for which \(\operatorname{coker}i\) has a projective object of \(\mathcal A\) in each degree

(Quillen 1967, Ch. I §1, Ex. B). The cofibrant replacement of an object of \(\mathcal A\) concentrated in degree \(0\) is a projective resolution of that object.

In a pointed model category the loop and suspension functors and the fibration and cofibration sequences are constructed on the homotopy category (Quillen 1967, Ch. I §§2–3). The corresponding \(\infty\)-categorical constructions are Definition 6.2, and their adjunction is Section 6.2.

23.4 Stabilization

Definition 23.11 (Stable $\infty$-categories) An \(\infty\)-category \(\mathcal C\) is stable if it has a zero object, every morphism of \(\mathcal C\) admits a kernel and a cokernel, and a triangle in \(\mathcal C\) is exact if and only if it is coexact (Lurie 2009, Def. 2.9). The homotopy category of a stable \(\infty\)-category is triangulated (Lurie 2009, Thm. 3.11), a stable \(\infty\)-category admits all finite limits and colimits (Lurie 2009, Prop. 4.4), and on it the suspension and loop functors of Definition 6.2 are mutually inverse equivalences (Lurie 2009, sec. 2).

For a pointed \(\infty\)-category \(\mathcal C\) admitting the relevant finite limits and colimits, Section 6.2 gives \[ \adj{\mathcal C}{\mathcal C}{\Sigma}{\Omega}, \qquad \Sigma\dashv\Omega. \] A pointed \(\mathcal C\) is stable if and only if it admits finite limits and colimits and a square in \(\mathcal C\) is a pushout if and only if it is a pullback (Lurie 2009, Prop. 4.4).

Definition 23.12 (Prespectrum and spectrum objects) Let \(\mathcal C\) be an \(\infty\)-category. A prespectrum object of \(\mathcal C\) is a functor \(X\colon\mathrm N(\mathbf Z\times\mathbf Z)\to\mathcal C\) whose value \(X(i,j)\) is a zero object of \(\mathcal C\) whenever \(i\neq j\); write \(E_n=X(n,n)\) (Lurie 2009, Def. 8.1, Rmk. 8.3). The square of \(X\) on the corners \((n,n)\), \((n,n+1)\), \((n+1,n)\), \((n+1,n+1)\) is

By Definition 6.2 the pushout of the span \(0\leftarrow E_n\to0\) is \(\Sigma E_n\), and the pullback of the cospan \(0\to E_{n+1}\leftarrow0\) is \(\Omega E_{n+1}\), so the square determines the structure maps \[ \sigma_n\colon\Sigma E_n\longrightarrow E_{n+1}, \qquad \varepsilon_n\colon E_n\longrightarrow\Omega E_{n+1}, \] which correspond to one another under \(\Sigma\dashv\Omega\). The prespectrum is a spectrum object when every \(\varepsilon_n\) is an equivalence; the spectrum objects span a full subcategory \(\mathrm{Sp}(\mathcal C)\subseteq\mathrm{PSp}(\mathcal C)\) (Lurie 2009, Def. 8.4).

Theorem 23.13 (The universal property of the stabilization) For an \(\infty\)-category \(\mathcal C\) the stabilization \(\mathrm{Stab}(\mathcal C)\) is universal among stable \(\infty\)-categories equipped with a left exact functor to \(\mathcal C\) (Lurie 2009, Prop. 10.12). The \(\infty\)-category of spectra is \(\mathrm{Sp}=\mathrm{Sp}(\mathcal S_{*})=\mathrm{Stab}(\mathcal S)\) (Lurie 2009, Def. 9.1), and it is freely generated under colimits, as a stable \(\infty\)-category, by the sphere spectrum (Lurie 2009, Cor. 15.6).

Theorem 23.14 (The derived $\infty$-category) Let \(\mathcal A\) be an abelian category with enough projectives. Then \(\mathcal D^{-}(\mathcal A)\) is a stable \(\infty\)-category, the subcategories of complexes with vanishing homology in negative and in positive degrees determine a \(t\)-structure on it, and its heart is equivalent to the nerve of \(\mathcal A\) (Lurie 2009, Prop. 13.10).

Taking \(\mathcal A=A\text{-}\mathbf{Mod}\) places the complexes of Definition 23.7 in the stable \(\infty\)-category \(\mathcal D^{-}(A\text{-}\mathbf{Mod})\), whose heart is \(A\text{-}\mathbf{Mod}\), and the duality functor of Definition 23.1 extends to complexes by Definition 23.6.

23.5 Ring spectra and trace invariants

Definition 23.15 (Ring spectra and their modules) Let \(B\) be a commutative \(S\)-algebra. A \(B\)-algebra is a monoid in the symmetric monoidal category of \(B\)-modules under \(\wedge_B\), and a module spectrum over it is a module for that monoid (Elmendorf et al. 2007). The enveloping \(B\)-algebra of a \(B\)-algebra \(A\) is \(A^{e}=A\wedge_B A^{\mathrm{op}}\).

Definition 23.16 (Topological Hochschild homology) For a \(B\)-algebra \(A\) and an \((A,A)\)-bimodule \(M\), the topological Hochschild homology of \(A\) with coefficients in \(M\) is the derived smash product \[ THH^{B}(A;M)=M\wedge_{A^{e}}A \] (Elmendorf et al. 2007, Ch. IX). When \(B\) is a discrete commutative ring and \(A\) is a \(B\)-algebra that is flat as a \(B\)-module, \(\pi_n THH^{B}(A;M)\cong HH^{B}_n(A;M)\) (Elmendorf et al. 2007, Ch. IX). The simplicial construction of \(THH^{B}(A)\) is cyclic, and \(THH^{B}(A)\) is an \(S^{1}\)-equivariant \(B\)-module (Elmendorf et al. 2007, Ch. IX).

Algebraic \(K\)-theory of a ring and of an exact category, and the groups \(K_n(A)=\pi_nK(A)\), are those of (Weibel 2013).

23.6 Open statements

A stabilization of \(\mathbf{Lat}_R\) by an endofunctor \(F_M\) of Definition 22.9 is asked to do two things at once: to become invertible, so that it presents a stable category, and to move the Witt class, so that it shifts an invariant. Theorem 24.6 shows that on \(\mathbf{Lat}_R\) these two demands are incompatible: an \(M\) with \([M]\neq0\) has \(b_M\neq0\) and so \(F_M\) is not an equivalence, while an \(M\) for which \(F_M\) is an equivalence has \([M]=0\) and induces the identity on \(W(R)\). The three statements below are not proved in this book.

Question 23.17 Let \(R\) be a Dedekind domain with \(4\cdot1_R\neq0\) and let \(S=F_{H(R)}\) be the hyperbolic stabilization of \(\mathbf{Lat}_R\) (Definition 22.9). Is there a pointed \(\infty\)-category \(\mathcal C\), a functor \(\iota\colon\mathbf{Lat}_R^{\simeq}\to\mathcal C^{\simeq}\), and an equivalence \(\iota\circ S\simeq\Sigma\circ\iota\) with \(\Sigma\) the suspension of Definition 6.2?

Question 23.18 Is there a symmetric monoidal structure on a category built from \(\mathbf{Lat}_R\), other than the orthogonal sum of Definition 17.6, with an invertible object \(M\) whose class in \(W(R)\) is nonzero? Theorem 24.6 answers this for \((\mathbf{Lat}_R,\perp,0)\) itself.

Question 23.19 Does the discriminant construction of Chapter 19 factor through the \(\epsilon\)-quadratic Poincaré complexes of Definition 23.7 over \(\mathbb Z\), and through which functor?