[18], lecture VII. The underlying reference is [101]; the definitive modern treatment is [93], section 5.5.8. However, we generally follow conventions from [117], tag 0162.
For a concise introduction to simplicial commutative rings, see [96].
In this section, we describe a natural analogue of derived functors for categories which are not necessarily additive. Putting this theory in its proper level of generality involves addressing a lot of technicalities which we elide here.
For \(A \in \Ring\text{,}\) let \(\Ring_A\) be the arrow category (i.e., commutative \(A\)-algebras).
Subsection16.1More on simplicial objects
To introduce this section, we start with a motivating remark.
Remark16.1.1.
Suppose one is trying to write down a functor \(F\) from \(\Mod_A\) to some abelian category which is right exact and commutes with filtered colimits. Then it is enough to specify the values of \(F\) on arbitrary finite free\(A\)-modules: every module is a cokernel of a morphism between two free modules, each of which is itself a filtered colimit of finite free modules. Furthermore, using projective resolutions by free modules, we can compute the left derived functors of \(F\) from this.
The construction of nonabelian derived functors allows us to do something similar starting from the category \(\Ring_A\text{.}\) The free objects in this case (i.e., the essential image of the left adjoint of the forgetful functor to \(\Set\)) are polynomial rings. In order to replace modules to rings, we need to reconceptualize some familiar constructions without reference to the additive structure of the category \(\Mod_A\text{;}\) for example, in \(\Mod_A\) we can form the equalizer of two maps \(f_1, f_2\colon M \to N\) as the kernel of the difference \(f_1 - f_2\text{,}\) but now we need to forgo this shortcut.
The resulting process amounts to the transition from homological algebra to homotopical algebra in the sense of Quillen [101]. Nowadays this is usually done in the framework of \(\infty\)-categories, as in [93]; we will keep ourselves in a very limited part of the picture so as to keep the prerequities for the discussion under control.
To begin, we recall the definition of simplicial objects from Definition 11.2.2, filling some extra details.
Definition16.1.2.
Let \(\Delta\) be the category of finite ordered sets (Definition 11.2.1). Recall (from Definition 11.2.2) that for \(\calC\) an arbitrary category, a simplicial object of a category \(\calC\) is a covariant functor \(U\colon \Delta^{\op} \to \calC\text{,}\) while a cosimplicial object of a category \(\calC\) is a covariant functor \(U\colon \Delta \to \calC\text{.}\) For a simplicial object \(U\text{,}\) we will usually write \(U_n\) as a shorthand for the image object \(U([n])\text{.}\)
The shift operators on derived categories have the following simpicial analogue.
Definition16.1.3.
For \(n \geq 0\text{,}\) let \(\Delta[n]\) denote the simplicial set
For any simplicial set \(U\text{,}\) the morphisms of simplicial sets from \(\Delta[n]\) to \(U\) are naturally in bijection with \(U_n\text{.}\)
Definition16.1.4.
Let \(V\) be a simplicial set such that each \(V_n\) is finite and nonempty. Then for any category \(\calC\) admitting finite coproducts and any simplicial object \(U\) of \(\calC\text{,}\) we define the product \(U \times V\) to be the simplicial object of \(\calC\) with
\begin{equation*}
(U \times V)_n = \coprod_{v \in V_n} U_n
\end{equation*}
corresponding to \(\phi\colon [m] \to [n]\) carries the component indexed by \(v\) to the component indexed by \(v' = V(\phi)(v)\) via \(U(\phi)\text{.}\) (Compare [117], tag 017C.)
Example16.1.5.
In Definition 16.2.5, we will consider the special case of Definition 16.1.4 in which \(V = \Delta[1]\text{.}\) In this case, the two maps \(e_0, e_1\colon \Delta[0] \to \Delta[1]\) corresponding to the two morphisms \([0] \to [1]\) induce morphisms
\begin{equation*}
e_0, e_1\colon U \to U \times \Delta[1].
\end{equation*}
Remark16.1.6.
By way of motivation, you should imagine that \(\Delta[n]\) represents an \(n\)-dimensional simplex and the product \(U \times \Delta[n]\) represents taking the product of some geometric object corresponding to \(U\) with this simplex. This motivates the definition of homotopies between maps of simplicial objects, as in Definition 16.2.5.
Subsection16.2Simplicial resolutions
Definition16.2.1.
A simplicial resolution of an object \(X \in \calC\) is a simplicial object \(U\colon \Delta^{\op} \to \calC\) with colimit \(X\text{.}\) A cosimplicial resolution of \(X\) is a cosimplicial object \(U\colon \Delta \to \calC\) with limit \(X\text{.}\)
Let us see how the previous construction, specialized to the case of modules over a ring, gives resolutions in the homological sense.
We now give a simplicial analogue of a resolution in homological algebra.
Example16.2.2.
Take \(\calC = \Mod_A\) for some \(A \in \Ring\text{.}\) Let \(U\) be a cosimplicial resolution of \(M \in \calC\text{.}\) Then the associated complex \(U([\bullet])\) (Definition 11.2.2) is a resolution of \(M\text{;}\) that is, \(M[0] \to U([\bullet])\) is a quasi-isomorphism.
We describe a trivial example which is not quite so trivial after all.
Definition16.2.3.
For any object \(X \in \calC\text{,}\) the simplicial object \(U\) with \(U([n]) = X\) for all \(n\text{,}\) is a resolution of \(X\text{.}\) We call this the trivial resolution of \(X\text{.}\)
Example16.2.4.
Take \(\calC = \Mod_A\) with \(A \in \Ring\text{.}\) Then the trivial resolution of \(M\) has associated complex
\begin{equation*}
\cdots M \stackrel{0}{\to} M \stackrel{1}{\to} M \stackrel{0}{\to} M
\end{equation*}
which is homotopy equivalent to \(M[0]\text{.}\) Compare this to the proof of Lemma 14.4.8.
When working with resolutions, we would like to be able to compare these, in the same way that we can show that any two injective/projective resolutions of an object of \(\Mod_A\) are homotopy equivalent. Here is the key definition.
Definition16.2.5.
Suppose that the category \(\calC\) has finite coproducts. Let \(U,V\) be simplicial objects of \(\calC\) and let \(a,b\colon U \to V\) be two morphisms. A homotopy from \(a\) to \(b\) is a morphism
\begin{equation*}
h\colon U \times \Delta[1] \to V
\end{equation*}
(interpreting the source as per Definition 16.1.4) such that \(a = h \circ e_0\) and \(b = h \circ e_1\text{.}\) The property that such a homotopy exists, for a given pair \(a,b\) is reflexive but not necessarily symmetric or transitive.
We say that \(a\) and \(b\) are homotopic if they belong to the same equivalence class under the equivalence relation generated by homotopies. We say that a single morphism \(a\colon U \to V\) is a homotopy equivalence if there exists a second morphism \(b\colon V \to U\) such that \(b \circ a\) is homotopic to \(\id_U\) and \(a \circ b\) is homotopic to \(\id_V\text{.}\)
Example16.2.6.
In Definition 16.2.5, the maps \(e_0, e_1\colon U \times \Delta[1] \to U\) are themselves homotopy equivalences. See Exercise 16.6.2.
Example16.2.7.
Take \(\calC = \Mod_A\) with \(A \in \Ring\text{.}\) Then a homotopy between morphisms \(a,b\colon U \to V\) of simplicial objects gives rise to a homotopy of the corresponding complexes in \(\Comp(A)\text{.}\) In particular, if two simplicial objects \(U,V\) are homotopy equivalent, then the corresponding objects in \(K(A)\) are isomorphic (and similarly for cosimplicial objects). For a converse to this assertion, see [117], tag 01A1.
Remark16.2.8.
Just as in homological algebra, one would like to work in the derived category to enforce that any object is “interchangeable” with a sufficiently nice resolution, in the simplicial realm one wants to to replace objects with simplicial objects that are more flexible (in the sense of being fibrant or cofibrant). The general story is out of scope for these notes (in part due to the need to develop robust combinatorial formalism, as in the language of \(\infty\)-categories, to keep track of homotopy coherence); here we limit ourselves to a few critical examples, such as Example 16.2.9.
Example16.2.9.
Let \(A \to B\) be a morphism in \(\Ring\text{.}\) Choose a simplicial resolution \(U\) of \(B\) by free \(A\)-algebras (e.g., the standard resolution; see Example 16.3.4). Then for any morphism \(A \to C\) of rings, we may define the simplicial tensor product \(B \otimes^L_A C\) to be the simplicial ring \(U \otimes_A C\text{;}\) any two choices of \(U\) will give rise to homotopy equivalent objects. Similarly, we may define the simplicial tensor product of two simplicial \(A\)-algebras.
Subsection16.3Standard resolution
The following construction gives a functorial construction of simplicial resolutions; see [117], tag 08N8.
Definition16.3.1.
Let \(V\colon \calC_1 \to \calC_2\) be a functor with a left adjoint \(U\colon \calC_2 \to \calC_1\text{.}\) By definition, this means we have natural transformations
\begin{equation*}
\eta\colon \id_{C_2} \to V \circ U, \qquad \epsilon\colon U \circ V \to \id_{C_1}
\end{equation*}
(the unit and counit).
For \(n \geq 0\text{,}\) let \(X_n\) be the \((n+1)\)-fold composition of \(U \circ V\text{,}\) with \(X_{-1} = \id_{C_1}\text{;}\) note that we have a natural identification \(X_{n+m+1} = X_n \circ X_m\text{.}\) Define the natural transformations
(writing \(\star\) for composition of natural transformations to distinguish it from \(\circ\) for composition of functors).
By Lemma 16.3.2, for any \(Y \in \calC_1\text{,}\) the objects \(X_n(Y)\) form a simplicial resolution of \(Y\text{.}\) We call this the standard resolution of \(Y\) with respect to the functor \(V\text{.}\)
Lemma16.3.2.
In Definition 16.3.1, \(X\) is a simplicial resolution of the constant functor \(\id_{\calC_1}\) via \(\epsilon\text{.}\) Consequently, for any \(Y \in \calC\text{,}\) the objects \(X_n(Y)\) form a simplicial resolution of \(Y\) in \(\calC_1\text{.}\)
For example, this construction can be used to construct functorial projective resolutions of modules over a ring.
Example16.3.3.
In Definition 16.3.1, take \(V\) to be the forgetful functor \(\Mod_A \to \Set\) for some \(A \in \Ring\text{;}\) we may then take \(U\) to be the functor taking \(S \in \Set\) to the free \(A\)-module \(A^S\text{.}\) For \(M \in \Mod_A\text{,}\) we obtain a simplicial resolution \(P_n\) with \(P_{-1} = M\) and \(P_{n+1} = A^{P_n}\) for \(n \geq -1\text{.}\) This in particular gives rise to a projective resolution of \(M\) using the dual construction of the one in Definition 11.2.2.
Here is the natural analogue for algebras over a ring.
Example16.3.4.
In Definition 16.3.1, take \(V\) to be the forgetful functor \(\Ring_A \to \Set\) for some \(A \in \Ring\text{;}\) we may then take \(U\) to be the functor taking \(S \in \Set\) to the free polynomial ring \(A[S]\text{.}\) For \(B \in \Ring_A\text{,}\) we obtain a simplicial resolution \(P_n\) with \(P_{-1} = B\) and \(P_{n+1} = A[P_n]\) for \(n \geq -1\text{.}\)
It should be stressed that while the standard resolution is a “natural” (and functorial) way to construct simplicial resolutions, the resulting resolutions are not preferred in any mathematical sense. In particular, if one starts performing operations one quickly ends up with simplicial resolutions that are not the standard ones but are homotopy equivalent, and the distinction will carry no value (if anything it is more of a hindrance).
Subsection16.4Nonabelian derived functors
We now ready to define an analogue of derived functors for algebras over a given ring. For this, the following definition will be useful.
Definition16.4.1.
Given covariant functors \(F\colon \calC_1 \to \calC_2, G\colon \calC_1 \to \calC_3\text{,}\) the left Kan extension of \(G\) along \(F\) consists of a covariant functor \(L\colon \calC_2 \to \calC_3\) and a natural transformation \(\alpha\colon G \to L \circ F\) which are universal for this property: that is, if \(M\colon \calC_2 \to \calC_3\) is another functor and \(\beta\colon G \to M \circ F\) is a natural transformation, then there is a unique natural transformation \(\sigma\colon L \to M\) making the second diagram in Figure 16.4.2 commute.
Figure16.4.2.
Remark16.4.3.
Note that in Definition 16.4.1, both the commutativity of the second diagram in Figure 16.4.2 and the uniqueness of \(\sigma\) are well-posed because a natural transformation is specified by a collection of morphisms between prescribed sources and targets, so the comparison of these is set-theoretic and not category-theoretic.
As usual with a definition via a universal property, the use of the definite article is justified by the observation that any two objects satisfying the definition are uniquely isomorphic. However, \(\alpha\) is not itself guaranteed to be an isomorphism of functors; that is, \(G\) is not necessarily isomorphic to the restriction of \(L\) along \(F\text{.}\)
Here is the motivating example.
Example16.4.4.
Let \(G\colon \Mod_A \to \calA\) be a right exact covariant functor to an abelian category. Let \(\calC\) be the subcategory of \(K^-(A)\) consisting of complexes of projective modules. Using the fact that simplicial resolutions by projective modules give rise to projective resolutions (Example 16.3.3), we may check that the usual left derived functor of \(G\) is the left Kan extension of \(G\colon \calC \to K^-(\calA)\) along the inclusion \(\calC \to K^-(A)\text{.}\) The point is that the formation of projective resolutions corresponds to replacing general objects of \(K^-(A)\) by cofibrant objects.
With Example 16.4.4 in mind, it is now clear how to proceed with modules replaced by rings.
Definition16.4.5.
Let \(\Poly_A\) be the full subcategory of \(\Ring_A\) consisting of polynomial rings over \(A\) in finitely many variables (i.e., the essential image of the restriction to finite sets of the left adjoint of the forgetful functor from \(\Ring_A\) to sets). Note that objects in \(\Poly_A\) do not come with a specified choice of polynomial generators, and so morphisms in \(\Poly_A\) are not required to respect these generators.
Proposition16.4.6.
For \(A \in \Ring\) and \(F\colon \Poly_A \to D(\Ab)\) a covariant functor, the functor \(F\) admits a left Kan extension \(LF\colon \Ring_A \to D(\Ab)\) along the inclusion \(\Poly_A \to \Ring_A\text{,}\) which moreover has the following properties. (We call \(LF\) the left derived functor of \(F\text{.}\))
The natural transformation \(\alpha\) is an equivalence: that is, \(LF\) restricts to \(F\) on \(\Poly_A\text{.}\)
\(LF\) commutes with filtered colimits. In particular, if \(A[S]\) is a polynomial algebra on a possibly infinite set \(S\text{,}\) we can compute \(LF(A[S])\) as the colimit of \(F(A[T])\) over all finite subsets \(T\) of \(S\text{.}\)
Given a simplicial resolution \(P_\bullet \to B\) of an object \(B \in \Ring_A\text{,}\)\(LF(B)\) is the colimit of \(LF(P^\bullet)\) (see Remark 16.4.7). (For example, this means we can evaluate \(LF\) using the standard resolution, as per Example 16.3.4.)
See the references given in [18], lecture VII, section 1.
Remark16.4.7.
In practice, we will be considering cases in which \(F\) can be lifted to a functor \(\tilde{F}\colon \Poly_A \to \Comp(\Ab)\text{,}\) in which case the colimit in part (2) of Proposition 16.4.6 can be interpreted as the totalization of a double complex made out of the terms \(L\tilde{F}(P^\bullet)\text{.}\) Otherwise, one should replace the derived category \(D(\Ab)\) with its \(\infty\)-categorical analogue and take the colimit there (where it can be reinterpreted as the geometric realization).
To give a concrete example of the effect of the colimit, note that if \(B\) is the coequalizer of two maps \(f_0, f_1\colon P_1 \to P_0\text{,}\) then \(LF(B) = \Cone(f_0 - f_1)\text{.}\)
Remark16.4.8.
An important basic example will be given by the exterior power \(\wedge^i\colon \Mod_A \to \Mod_A\text{,}\) which will give us the derived exterior power \(L\wedge^i\colon \Mod_A \to D(A)\text{.}\) This in turn extends to a functor \(L\wedge^i\colon D^{\leq 0}(A) \to D(A)\text{.}\) (As indicated in [18], Lecture VII, Remark 1.4, this is a point at which we are forced to be a bit sloppy by not working in the language of \(\infty\)-categories, but never mind.)
Remark16.4.9.
In what follows we will frequently use the following construction without explicit comment. Let \(G'\colon \calC_1 \to \calC_3\) be another functor admitting a left Kan extension \((L'\colon \calC_2 \to \calC_3, \alpha\colon G' \to L' \circ F)\text{,}\) and suppose that \(\gamma\colon G \to G'\) is a natural transformation. Then we obtain a natural transformation \(\alpha \circ \gamma\colon G \to L' \circ F\) to which we may apply the universal property of the left Kan extension of \(L\text{,}\) so as to obtain a natural transformation \(\sigma\colon L \to L'\text{.}\) That is, a natural transformation between two functors from \(\calC_2\) to \(\calC_3\) can be uniquely specified by giving its restriction (along \(F\)) to \(\calC_1\text{.}\)
Subsection16.5Under the hood: \(\infty\)-categories
Remark16.5.1.
It was mentioned in passing earlier that the derived category of \(A\)-modules, for some \(A \in \Ring\text{,}\) is more robust to work with in the language of (stable) \(\infty\)-categories. This allows us to be more careful about making identifications “up to homotopy”; rather than simply declaring two morphisms of complexes to be equal if there is a homotopy between them, in the homotopical approach one records the data of the homotopy witness and keeps track of it as one performs further operations.
One reason this is advantageous is that the formation of mapping cones is not functorial in the derived category as we have described it, but it becomes functorial in the stable \(\infty\)-category (because of the retention of the homotopy data). A minimal example is given by the map from \(A \to 0\) to \(0 \to A\text{.}\)
Another reason is that one cannot perform any reasonable descent on the functor \(A \mapsto D(A)\) without the homotopical data: for instance, for a Zariski covering of three or more opens, it is not generally possible to lift descent data from objects in derived categories to chain complexes. Again, recording the homotopy data makes it possible to perform this lifting.
Exercises16.6Exercises
1.
Prove that for any \(n \geq 0\text{,}\) the unique morphism \(\Delta[n] \to \Delta[0]\) is a homotopy equivalence, with a homotopy inverse given by the map \(\Delta[0] \to \Delta[n]\) induced by the map \([0] \to [n]\) taking \(0\) to \(n\text{.}\)
Let \(\calC\) be a category admitting finite nonempty coproducts. Prove that for any simplicial object \(U\) in \(\calC\text{,}\) the maps \(e_0, e_1\colon U \times \Delta[1] \to U\) are homotopy equivalences.
