[117], tag 05QI or [125], Chapter 10; we are skipping a lot of details. We do not develop the point of view of triangulated categories very thoroughly; for more on that point of view, see [84], Chapter II.
We pick up from Section 9 (retaining notation) and introduce the derived category associated to an abelian category. This amounts to checking that we can perform the localization of the homotopy category \(K(\calA)\) at the family of quasi-isomorphisms.
Here we take a “classical” point of view on derived categories; however, it is better in the long run to express the construction in the language of \(\infty\)-categories. See Subsection 16.5 and Remark 13.4.1 for further remarks in this vein.
Subsection10.1Localization in a category
Remark10.1.1.
Recall from Remark 9.5.2 that we are in the situation of having to construct one category from another by “formally inverting” some morphisms. We are familiar with processes of these type from algebra, such as the group completion of a monoid (e.g., passage from positive integers to arbitrary integers) or the localization of a ring at a multiplicative subset (e.g., passage from integers to rational numbers). The category-theoretic situation is similar but rather fraught with arrows, and somewhat complicated by the fact that composition of morphisms is not commutative. Similar (but a bit less fraught) considerations apply to localization in a noncommutative ring.
To isolate a key difficulty, imagine trying to define a morphism in the localization category as a formal composition \(g^{-1} \circ f\) where \(f\) is a morphism and \(g^{-1}\) is the “formal inverse” of another morphism. Then the composition of two such morphisms would have the form \(g_1^{-1} \circ f_1 \circ g_2^{-1} \circ f_2\) and we would then need to rewrite the inner composition \(f_1 \circ g_2^{-1}\) as a composition \(g_3^{-1} \circ f_3\) in the opposite order. Then the total composition would become
We give only a brief summary of the formalism needed to make this idea work. See [117], tag 04VB for further details.
Definition10.1.2.
Let \(\calC\) be a category (not necessarily abelian or even additive). Let \(S\) be a collection of morphisms in \(\calC\text{.}\) We say that \(S\) is a left multiplicative system if the following conditions hold.
The collection \(S\) contains all identity morphisms and is closed under composition (of composable pairs).
Given the solid arrows as in Figure 10.1.3 with \(t \in S\text{,}\) for some choice of \(Y'\) there exist dashed arrows with \(s \in S\) forming a commutative square.
Figure10.1.3.
You should think of this as saying that the “formal composition” \(g \circ t^{-1}\colon Z \to Y\) can be refactored as \(s^{-1} \circ f\text{,}\) with the formal inverse moved from the right to the left.
For every pair of morphisms \(f,g\colon X \to Y\) and every \(t \in S\) with target \(X\) such that \(f \circ t = g \circ t\text{,}\) there exists a morphism \(s \in S\) with source \(Y\) (and unspecified target) such that \(s \circ f = s \circ g\text{.}\) (In this case, the morphisms \(f\) and \(g\) are going to be conflated in the localization, and we want that to make sense with respect to composition on both sides.)
If \(\calC\) is an additive category, it is equivalent to require that for every morphism \(f\colon X \to Y\) and every \(t \in S\) with target \(X\) such that \(f \circ t = 0\text{,}\) there exists a morphism \(s \in S\) with source \(Y\) (and unspecified target) such that \(s \circ f = 0\text{.}\)
Similarly, a right multiplicative system is a collection of morphisms of \(\calC\) that constitutes a left multiplicative system in the opposite category. A multiplicative system is a collection of morphisms of \(\calC\) which is simultaneously a left multiplicative system and a right multiplicative system.
We say that a multiplicative system is saturated if for any three composable morphisms \(f,g,h\) with \(f \circ g, g \circ h \in S\text{,}\) we also have \(g \in S\text{.}\) For example, the collection of all isomorphisms has this property.
Definition10.1.4.
Let \(\calC\) be a category and let \(S\) be a multiplicative system. We define the category \(S^{-1} \calC\) as follows. (There are some steps to verify that this is a well-posed definition of a category; see [117], tag 04VD.)
The objects of \(S^{-1} \calC\) are the objects of \(\calC\text{.}\)
For \(X,Y \in \calC\) two objects, the morphisms \(X \to Y\) in \(\calC\) are given by pairs \((f\colon X \to Y', s\colon Y \to Y')\) where \(Y' \in \calC\) is a third object modulo the following equivalence relation: two pairs
\begin{equation*}
(f_i\colon X \to Y_i, s_i\colon Y \to Y_i)
\end{equation*}
for \(i=1,2\) are equivalent if there is a third pair with \(i=3\) fitting into a diagram as in Figure 10.1.5 for some morphisms \(Y_i \to Y_3\) in \(\calC\) (not necessarily in \(S\)).
Figure10.1.5.
You should think of a pair \((f,s)\) as corresponding to the formal composition \(s^{-1} \circ f\text{.}\)
The composition of a pair \((f\colon X \to Y', s\colon Y \to Y')\) with a pair \((g\colon Y \to Z', t\colon Z \to Z')\) is defined to be the equivalence class of a pair \((h \circ f\colon X \to Z'', u \circ t\colon Z \to Z'')\) where \(h\) and \(u \in S\) are chosen (using the definition of a left multiplicative system) to fill in the commutative square Figure 10.1.6.
Figure10.1.6.
The identity morphism on \(X\) is the class of \((\id_X, \id_X)\text{.}\)
One can similarly form the localization of the opposite category, then take the opposite category of the result (using the definition of a right multiplicative system). This gives the same answer; see [117], tag 04VL.
The morphisms of \(\calC\) which become isomorphisms in \(S^{-1} \calC\) also form a multiplicative system; in fact, this is the smallest saturated multiplicative system containing \(S\) ([117], tag 04VB), and so equals \(S\) if and only if \(S\) is itself saturated.
Remark10.1.7.
In Definition 10.1.4, we have glossed over a serious set-theoretic difficulty; since the definition of a morphism is quantified over an unspecified third object \(Y'\) of \(\calC\text{,}\) it is not clear that the collection of morphisms between two fixed objects is a set, as is required in the definition of a category.
One way to avoid this issue is to only consider localizations of categories which are small, meaning that there is a set of objects which meets every isomorphism class. Then one can instead quantify \(Y'\) over this set of representatives without losing anything.
A more robust mechanism is to use the Gabriel-Zisman theorem which gives a criterion for constructing localizations even when the ambient category is not small. See [125], Theorem 10.3.7.
Remark10.1.8.
There is a way to interpret ring-theoretic localization as a special case of localization of categories. See [117], tag 0BM1.
Subsection10.2Distinguished triangles
Recall that a short exact sequence of complexes gives rise to a long exact sequence of cohomology groups. This serves as inspiration for the following discussion of triangles in the homotopy category.
Definition10.2.1.
A triangle in \(\Comp(\calA)\) is a tuple \((A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta)\) coming from a diagram of the form
We may as well start with the triangle associated to a mapping cone. In this case, the morphism \(\delta\) coincides with the family of connecting homomorphisms coming from the short exact sequence of complexes
and so the sequence in question is just the long exact sequence in cohomology. Alternatively, we can first prove Lemma 10.2.4 and then use this to reduce to checking exactness at \(H^i(C^\bullet)\text{.}\)
Lemma10.2.4.
The set of distinguished triangles is preserved by forward and backward rotation.
The set of distinguished triangles is preserved by the shift operators, so it will be enough to check preservation by forward rotation. That is, given a triangle of the form
we must produce a commutative diagram in \(K(\calA)\) of the form of Figure 10.2.5 in which the dashed arrow is an isomorphism in \(K(\calA)\text{.}\)
Figure10.2.5.
We construct the arrow \(K[1]^\bullet \to \Cone(g)^\bullet = \Cone(f)^\bullet \oplus L[1]^\bullet\) so that the first factor is the injection \(K^{n+1} \to L^n \oplus K^{n+1}\) and the second factor is \(-f^{n+1}\text{.}\) We construct the arrow \(\Cone(f)^\bullet \oplus L[1]^\bullet = \Cone(g)^\bullet \to K[1]^\bullet\) as the projection onto \(\Cone(f)^n \oplus L^{n+1} \to \Cone(f)^n\) followed by \(h^n\text{.}\) One may check as in [117], tag 014I that these maps are inverses in \(K(\calC)\text{.}\)
Corollary10.2.6.
Any morphism in \(K(\calC)\) can be included into a distinguished triangle (in any position).
Any morphism can be included as the first morphism of a distinguished triangle using the mapping cone. For the other positions, apply Lemma 10.2.4.
Lemma10.2.7.
Given any collection of solid arrows in Figure 10.2.8 forming a commutative diagram in \(K(\calA)\text{,}\) in which the rows form distinguished triangles, there exists a dashed morphism such that the vertical arrows form a morphism of triangles in \(K(\calA)\text{.}\)
We may assume at once that \(C = \Cone(f), C' = \Cone(f')\text{.}\) In this case, commutativity of the square in \(K(\calA)\) implies the existence of a homotopy \(h\) for the map \(b \circ f - f' \circ a\text{.}\) We may then write down a morphism \(c\colon \Cone(f) \to \Cone(f')\) by the formula
and verify that this yields a morphism of triangles. (Compare [117], tag 014F.)
The following result is akin to the universal property of kernels and cokernels.
Corollary10.2.9.
Let \(f\colon X^\bullet \to Y^\bullet\) be a morphism in \(K(\calC)\text{.}\) Then every morphism \(h\colon Y^\bullet \to Z^\bullet\) such that \(h \circ f = 0\) in \(K(\calC)\) can be factored through \(Y^\bullet \to \Cone(f)^\bullet\text{.}\)
We return to our unfinished business from Remark 9.5.2, namely modifying the homotopy category so as to force every quasi-isomorphism to acquire an inverse. Thanks to the cone construction, we can relate quasi-isomorphisms to acyclic objects, which are easier to handle.
Definition10.3.1.
An object \(K^\bullet\) of \(\Comp(\calA)\) is acyclic if \(H^n(K^\bullet) = 0\) for all \(n \in \ZZ\text{;}\) this property is preserved under isomorphisms in \(K(\calA)\text{.}\) By Lemma 10.2.3, if two of the three complexes in a distinguished triangle are acyclic, then so is the third. From this (and the preservation of the acyclic property under shifts) we may deduce that the full subcategory of \(K(\calA)\) consisting of acyclic objects is also a triangulated category.
Lemma10.3.2.
A morphism \(f\colon K^\bullet \to L^\bullet\) in \(K(\calA)\) is a quasi-isomorphism if and only if there exists a distinguished triangle \((K^\bullet, L^\bullet, M^\bullet, f, g, h)\) in which \(M^\bullet\) is acyclic.
It suffices to check the conditions for a left multiplicative system, as the symmetric argument will imply the conditions for a right multiplicative system. The first condition in Definition 10.1.2 is evidently satisfied: every identity morphism is a quasi-isomorphism, and any composition of quasi-isomorphisms is a quasi-isomorphism.
To check the second condition, apply Corollary 10.2.6 to fit \(g\) into a distinguished triangle \((X, Y, Z, g, h, i)\text{,}\) then set \(Y' = \Cone(i[-1])\text{;}\) we obtain the map \(s\) by filling the diagram Figure 10.3.4 using Lemma 10.2.4 (to rotate) and Lemma 10.2.7. (We deduce from Lemma 10.2.3 that \(s\) is a quasi-isomorphism.)
Figure10.3.4.
To check the third condition, start with a morphism \(f\colon X \to Y\) and a quasi-isomorphism \(t\colon Z \to X\) such that \(f \circ t = 0\text{,}\) Apply Corollary 10.2.6 to fit \(t\) into a distinguished triangle \((Z, X, Q, t, d, h)\text{.}\) By Corollary 10.2.9, we can choose a morphism \(i\colon Q \to Y\) such that \(i \circ d = f\text{.}\) Apply Corollary 10.2.6 again to fit \(i\) into a distinguished triangle \((Q, Y, W, i, j, k)\text{;}\) then \(j \circ f = j \circ i \circ d = 0 \circ d = 0\text{.}\) By Lemma 10.3.2, \(t\) being a quasi-isomorphism implies that \(Q\) is acyclic, which in turn implies that \(j\) is a quasi-isomorphism. (Compare [117], 05RG.)
Figure10.3.5.
Definition10.3.6.
Suppose that \(\calA\) is a small abelian category. By Proposition 10.3.3, we may apply Definition 10.1.4 to construct the localization of \(K(\calA)\) at the saturated multiplicative system of quasi-isomorphisms. The result is called the derived category of \(\calA\text{,}\) denoted \(D(\calA)\text{.}\) Similarly, we may define the bounded below derived category \(D^+(\calA)\text{,}\) the bounded above derived category \(D^-(\calA)\text{,}\) and the bounded derived category \(D^b(\calA)\text{.}\)
An important case is when \(\calA\) is the category of modules over a ring \(A\text{.}\) This is not a small category, but modulo set-theoretic issues (see Remark 10.3.8) we can still define \(D^*(\calA)\) for \(* \in \{\emptyset, -, +, b\}\text{;}\) we denote this also by \(D^*(A)\text{.}\)
As in the homotopy category, we say that a triangle in \(D(\calA)\) is distinguished if it is isomorphic to the triangle associated to some mapping cone.
The following example shows that Lemma 10.2.3 does not admit a converse.
Example10.3.7.
Let \(f\colon \ZZ \to \ZZ\) be the multiplication-by-\(p\) map for some prime \(p\text{.}\) In \(D(\Ab)\) the cone of \(f\) is isomorphic to \(\ZZ/p\ZZ\) placed in degree \(0\text{,}\) so we obtain a distinguished triangle of the form
but is not distinguished in \(D(\Ab)\text{:}\) otherwise we could apply Lemma 10.2.7 to compare the two triangles, yielding a contradiction.
Remark10.3.8.
To work around the fact that the derived category construction requires a small abelian category as input, one can view the full category \(\Mod_A\) as a 2-colimit of full subcategories consisting of modules of increasingly larger cardinalities. This works because \(\Mod_A\) is not just an abelian category but a Grothendieck abelian category; see [117], tag 09PA. For a more general abelian category \(\calA\text{,}\) however, the set-theoretic difficulty becomes a genuine obstruction; see [117], tag 07JS.
Remark10.3.9.
Just as the properties of the category \(\Mod_A\) are abstracted by the notion of an abelian category, the properties of homotopy categories and derived categories are abstracted by the notion of a triangulated category. A triangulated category is an additive category equipped with a collection of distinguished triangles and shift functors subject to various conditions analogous to some of the properties we have seen above (especially Lemma 10.2.4 and Lemma 10.2.7). See [117], tag 05QI for further discussion.
Proposition10.3.10.
Assume that \(\calA\) has enough injectives and let \(F\colon \calA \to \calA'\) be a left exact functor. Then the right derived functor \(RF\colon K^+(\calA) \to K^+(\calA')\) (see Definition 9.5.1) takes acyclic objects to acyclic objects, and so induces a functor \(RF\colon D^+(\calA) \to D^+(\calA')\text{.}\)
The right derived functor preserves distinguished triangles, so using the criterion of Lemma 10.3.2 it is enough to check that \(RF\) takes acyclic objects to acyclic object. For this, see [117], tag 05TA.
Subsection10.4Truncation
Definition10.4.1.
For \(K^\bullet \in \calC\) and any \(n \in \ZZ\text{,}\) the canonical truncation \(\tau^{\geq n} K^\bullet\) is the complex given by the second row of Figure 10.4.2, equipped with the morphism \(K^\bullet \to \tau^{\geq n} K^\bullet\) defined by the vertical arrows.
Figure10.4.2.
Similarly, the canonical truncation \(\tau^{\leq n} K^\bullet\) is the complex given by the first row of Figure 10.4.3, equipped with the morphism \(\tau^{\leq n} K^\bullet \to K^\bullet\) defined by the vertical arrows.
Figure10.4.3.
Lemma10.4.4.
For any interval \(I\text{,}\) the following conditions on an object \(K^\bullet \in D(A)\) are equivalent.
We have \(H^i(K^\bullet) = 0\) for all \(i \notin I\text{.}\)
There exists an isomorphism \(K^\bullet \to L^\bullet\) in \(D(A)\) such that \(L^i = 0\) for all \(i \notin I\text{.}\)
Suppose for simplicitly that \(I = [0, \infty)\text{,}\) the other cases being similar. In this case, if \(H^i(K^\bullet) = 0\) for all \(i \lt 0\text{,}\) then the morphism \(K^\bullet \to \tau^{\geq 0} K^\bullet\) is a quasi-isomorphism.
Corollary10.4.5.
An object \(K^\bullet\) of \(D(\calA)\) belongs to \(D^+(\calA), D^-(\calA), D^b(\calA)\) if and only if \(H^i(K^\bullet) = 0\) for respectively \(i \gg 0, i \ll 0, |i| \gg 0\text{.}\)
It also follows from Lemma 10.4.4 that \(D^+(\calA), D^-(\calA), D^b(\calA)\) are all full subcategories of \(D(\calA)\text{.}\) For example, for \(D^+(\calA)\) this holds because if we have two bounded-below complexes, any morphism between them is automatically zero at all sufficiently small indices (because any map between two zero objects is zero).
Remark10.4.7.
By Lemma 10.4.4 applied with \(I = \{0\}\text{,}\) the essential image of the functor \([0]\colon \calA \to D(\calA)\) is precisely the intersection of the essential images of the functors \(\tau^{\geq 0}, \tau^{\leq 0}\colon D(\calA) \to D(\calA)\text{.}\)
In the more general framework of triangulated categories, one can define a t-structure (short for truncation structure) to be a pair of functors \(\tau^{\geq 0}, \tau^{\leq 0}\) satisfying suitable conditions, and then define the heart of the t-structure as the intersection of the essential images of these functors. This gives us a way to start with a triangulated category and realize it as a derived category; in fact, by varying the t-structure we can sometimes realize the same triangulated category as a derived category in multiple ways! (The motivating example of this is the construction of perverse sheaves in connection with the Weil conjectures; see [84].)
Subsection10.5Pseudocoherent and perfect complexes
Let us now specialize to the category of modules over a ring and introduce some additional boundedness conditions.
Definition10.5.1.
For \(A \in \Ring, \calA = \Mod_A\text{,}\) an object \(K^\bullet\) of \(D(A) = D(\calA)\) is pseudocoherent (resp. perfect) if it is isomorphic to a bounded above (resp. bounded) complex of finite projective \(A\)-modules. An object of \(\Mod_A\) is pseudocoherent (resp. perfect) if \(M[0]\) is so as an object of \(D(A)\text{.}\)
Lemma10.5.2.
For \(A \in \Ring\text{,}\) an object \(K^\bullet\) of \(D(A)\) is perfect if and only if it is pseudocoherent and has finite Tor dimension.
If \(A\) is a noetherian ring, then a module is pseudocoherent if and only if it is finitely presented, but such a module need not be perfect.
That said, there do exist many noetherian rings over which every pseudocoherent module is perfect. For example, the ring \(A\) is said to have finite global dimension if there exists an integer \(n\) such that every \(A\)-module admits a resolution by projective \(A\)-modules of length \(n\text{.}\) If \(A\) is of finite projective dimension, then every pseudocoherent \(A\)-module is perfect. (See [117], tag 00O2.)
An important special case is the ring \(A = k[x_1,\dots,x_n]\) where \(k\) is a field. This ring has finite global dimension (bounded by \(n\text{,}\) the number of variables) by the Hilbert syzygy theorem ([117], tag 00OQ).
Exercises10.6Exercises
1.
Let \(A\) be a commutative ring. Show that if any two terms of a distinguished triangle in \(D(A)\) are pseudocoherent (resp. perfect), then so is the third.
Let \(A\) be a commutative ring. Prove that if \(K^\bullet \in D^b(A)\) has the property that \(H^i(K^\bullet)\) is perfect for all \(i\text{,}\) then \(K^\bullet\) is perfect.
Let \(J\) be a finitely generated ideal of \(A \in \Ring\text{.}\) Let \(A \to A'\) be a morphism in \(\Ring\) with finite \(J\)-complete Tor amplitude (see Definition 6.5.1).
Show that there exists some \(c \geq 0\) such that for any \(K \in D^{\geq 0}(A)\text{,}\)\(K \widehat{\otimes}^L_A A' \in D^{\geq -c}(A')\text{.}\)
Show that the derived \(J\)-completed base change functor \(M^\bullet \mapsto M^\bullet \widehat{\otimes}^L_A A'\) commutes with totalizations in \(D^{\geq 0}\text{.}\)
