This is the notes I took at the seminar Masterclass in Derived Category Methods in Ring Theory, Aarhus University, 13-16 August 2024. Some of the notations have been altered for my own preference.
Category of Chain Complexes
Let \(R\) be a ring acting on the left, and is a \(k\)-algebra (e.g. \(k=\mathbb{Z}\)).
Definition 1.
An \(R\)-complex is a graded \(R\)-module with an endomorphism \(d\) of degree \(-1\), such that the \(d\circ d = 0\), \[\dots M_{v+1}\xrightarrow{d_{v+1}} M_v\xrightarrow{d_v} M_{v-1}\xrightarrow{d_{v-1}}\dots\]
We use the notation \(M^\sharp\) to mean the underlying graded \(R\)-module which is also identified with the complex with zero differential.
A homomorphism of \(R\)-complexes \(M\to N\) is a graded homomorphism \(M^\sharp\to N^\sharp\), while a chain map is a homomorphism of complexes, we allow chain map with non-zero degree \(|\alpha|\), that satisfy \[d_N \circ \alpha = (-1)^{|\alpha|} \alpha \circ d_M\]
A morphism \(M\to N\) is a chain map of degree \(0\).
Denote \(C(R)\) the category of \(R\)-complexes and morphisms of complexes, \(C({}_{R}\mathbf{Mod}_{S})=C(R\otimes_k S^o)\) the category of \(R-S\)-bicomplexes and morphisms of bicomplexes.
Given a complex, we look at certain subcomplexes that are actually graded modules
Definition 2. \[Z(M): Z_v(M) = \ker d_v\] \[B(M): B_v(M) = \mathrm{Im}d_{v+1}\] \[C(M): C_v(M) = \mathrm{coker}d_{v+1} = M_v/B_v(M)\] \[H(M): H_v(M) = Z_v(M)/B_v(M)\] Note that the differentials restricted to \(Z(M)\) and \(B(M)\) are automatically zero.
Remarks 1. A morphism at complexes \(M\to N\) restricts to \(Z(M)\to Z(N)\) and \(B(M)\to B(N)\), and induces a morphism \(H(M)\to H(N)\).
Definition 3. An \(R\)-complex \(M\) is called acyclic if \(H(M)=0\).
Definition 4. A chain map \(M\xrightarrow{\alpha} N\) is called null-homotopic if there exists a degree \(|\alpha|+1\) chain map \(h: M\to N\) such that \(\alpha = d^N\circ h + (-1)^{|\alpha|}h\circ d^M\).
Definition 5. Two chain maps \(M\xrightarrow{\alpha} N\) and \(N\xrightarrow{\beta} M\) are called homotopic if \(\alpha-\beta\) is null-homotopic.
Proposition 1. \(\alpha,\beta:M\to N\) are homotopic \(\Rightarrow\) \(H(\alpha)=H(\beta)\).
The total Hom complex
Given \(R\)-complexes \(M\) and \(N\), the total hom \[\mathrm{Hom}_R(M,N)^\sharp = \mathrm{Hom}_R(M^\sharp,N^\sharp)\] i.e. \[\mathrm{Hom}_R(M,N)_v = \prod_{i\in \mathbb{Z}} \mathrm{Hom}_R(M_i,N_{i+v})\] with differential given by \[d^{\mathrm{Hom}(M,N)} \alpha = d^N\circ \alpha - (-1)^{|\alpha|} \alpha \circ d^M\]
Proposition 2. Let \(M\), \(N\) be \(R\)-complexes, \[Z(\mathrm{Hom}(M,N)) = \{ \text{chain maps } M\to N \}\] \[B(\mathrm{Hom}(M,N)) = \{ \text{null-homotopic chain maps } M\to N \}\] \[Z_0(\mathrm{Hom}(M,N)) = \mathrm{Hom}_{C(R)}(M,N)\]
Definition 6. Given two morphisms \(\alpha:M'\to M, \beta: N\to N'\), there is a functor map \[\mathrm{Hom}_R(M,N)\xrightarrow{\mathrm{Hom}(\alpha,\beta)} \mathrm{Hom}_R(M',N')\] mapping \(\theta\mapsto (-1)^{|\alpha|\cdot(|\beta|+|\theta|)}\beta\circ\theta\circ\alpha\). We denote \(\mathrm{Hom}(M,\beta)=\mathrm{Hom}(\mathrm{id}_M,\beta)\) and \(\mathrm{Hom}(\alpha,N)=\mathrm{Hom}(\alpha,\mathrm{id}_N)\).
Theorem 1. Hom is a functor \(C(R)^{op}\times C(R)\to C(k)\). If we have more structures on the complexes, we can have for example \[C(R-Q^{op})\times C(R-S^{op})\to C(Q-S^{op})\]
Definition 7 (Shift). Let \(M\) be an \(R\)-complex, define the suspension \[(\Sigma^s M)_v = M_{v-s}, \quad (\partial^{\Sigma^s M})_v = (-1)^s \partial_{v-s}^M.\]
Proposition 3. \[\mathrm{Hom}(\Sigma^s M,N) = \Sigma^{-s}\mathrm{Hom}(M,N)\] \[\mathrm{Hom}(M,\Sigma^s N) = \Sigma^s\mathrm{Hom}(M,N)\]
Total Tensor Product
Given \(R^o\)-complex \(M\) and \(R\)-complex \(N\), the total tensor product is \[(M\otimes_R N)^\sharp = M^\sharp\otimes_k N^\sharp\] where in degree \(v\), \[(M\otimes_R N)_v = \coprod_{i\in \mathbb{Z}} M_i\otimes_R N_{v-i}\] with differential given by \[d^{M\otimes_R N}(m\otimes n) = d^M(m)\otimes n + (-1)^{|m|}m\otimes d^N(n).\]
Definition 8. Given \(R\)-complex morphism \(\alpha:M\to M'\) and \(\beta:N\to N'\), there is a morphism \[\alpha\otimes \beta: M\otimes_R N\to M'\otimes_R N', \quad m\otimes n\mapsto (-1)^{|\beta||m|} \alpha(m)\otimes \beta(n).\]
Theorem 2. The tensor product is a functor \[-\otimes_R-: C(R^o)\times C(R)\to C(k).\] if we have more structures on the complexes, we can have for example \[-\otimes_R-: C({}_{Q}\mathbf{Mod}_{R^o})\times C({}_{R}\mathbf{Mod}_{S^o})\to C({}_{Q}\mathbf{Mod}_{S^o}).\]
Its behavior with shifting is
Proposition 4. \[\Sigma^s(M\otimes_R N) = \Sigma^s M\otimes_R N = M\otimes_R \Sigma^s N.\]
Definition 9. Let \(M\) be an \(R\)-complex, define \[\sup M = \sup\{ i\in \mathbb{Z}: H_iM\neq 0 \}\] \[\inf M = \inf\{ i\in \mathbb{Z}: H_iM\neq 0 \}\] define \[\mathrm{amp}\,M = \sup M - \inf M\]
Definition 10 (Hard Truncations). \[(M_{\le n})_v := \begin{cases} M_v & v\le n \\ 0 & v>n \end{cases} \quad (d^{M_{\le n}})_v = \begin{cases} d^M_v & v\le n \\ 0 & v>n \end{cases}\]
\[(M_{\ge n})_v := \begin{cases} M_v & v\ge n \\ 0 & v<n \end{cases} \quad (d^{M_{\ge n}})_v = \begin{cases} d^M_v & v\ge n \\ 0 & v<n \end{cases}\]
Definition 11 (Soft Truncations). \[(M_{\supset n})_v := \begin{cases} M_v & v>n \\ Z_v(M) & v=n\\ 0 & v<n \end{cases}\]
\[(M_{\subset n})_v := \begin{cases} 0 & v>n \\ C_v(M) & v=n\\ M_v & v<n \end{cases}\]
Special Morphisms and Natural transformations
Mapping Cone
A mapping cone of a morphism \(\alpha:M\to N\) in \(C(R)\) is a complex \(C(\alpha)\) defined as \[C(\alpha)^\sharp = N^\sharp \oplus \Sigma M^\sharp\] with differentials given by \[d^{C(\alpha)} = \begin{pmatrix} d^N & \alpha_{v-1} \\ 0 & -d_{v-1}^M \end{pmatrix}.\]
Theorem 3. The sequence \[0\to N\to C(\alpha)\to \Sigma M\to 0\] is split in \(C(R)\) iff \(\alpha\) is null-homotopic.
This is a useful tool and a key ingredient in the triangulated structures on the homotopy category \(K(R)\).
Special Morphisms
Definition 12. A morphism \(\alpha:M\to N\) in \(C(R)\) is a quasi-isomorphism if \(H(\alpha)\) is an isomorphism.
Theorem 4. \(\alpha:M\to N\) is quasi-isomorphism \(\Rightarrow\) \(C(\alpha)\) is acyclic.
Example 1. If \(n\ge \sup M\), then \(M\to M_{\subset n}\) is a quasi-isomorphism, similarly if \(n\le \inf M\), then \(M_{\supset n}\to M\) is a quasi-isomorphism.
Remarks 2.
If \(H(M)\cong H(N)\), there need not exist a quasi-isomorphism \(M\to N\) or \(N\to M\) o.o! In fact we can have \(\mathrm{Hom}_{C(R)}(M,N) = 0 = \mathrm{Hom}_{C(R)}(N,M)\).
If there exists quasi-isomorphism \(M\to N\), there need not exist a quasi-isomorphism in the reversed direction.
Homotopy Equivalences
Definition 13.
A morphism \(\alpha:M\to N\in C(R)\) is called a homotopy equivalence if there exists a morphism in the reversed direction \(\beta:N\to M\) such that \(\beta\alpha\sim \mathrm{id}_M\) and \(\alpha\beta\sim \mathrm{id}_N\). Any such beta is called a homotopy inverse of \(\alpha\).
\(M\in C(R)\) is called contractible if \(\mathrm{id}_M\sim 0\), or iff \(0\to N\) is a homotopy equivalence.
Theorem 5. \(\alpha:M\to N\) is a homotopy equivalence \(\Leftrightarrow\) \(C(\alpha)\) is contractible.
Remarks 3. If \(\alpha\) is a homotopy equivalence, then \(\alpha\) is also a quasi-isomorphism. If a complex \(M\) is contractible, then \(M\) is acyclic (exact).
Example 2. A short exact sequence \(\eta : 0\to M'\to M\to M''\to 0\) of \(R\)-modules can be viewed as a complex \(\eta\in C(R)\) which is acyclic. In this way, \(\eta\) is contractible iff \(\eta\) is split exact.
Remarks 4. The homotopy category \(K(R)=C(R)/\sim\) is the category whose objects are the complexes and morphisms are homotopy classes of morphisms in \(C(R)\). It could also be defined as a localization with respect to homotopy equivalences, \(C(R)[\sim^{-1}]\).
Theorem 6. For a functor \(F:C(R)\to C(S)\), the following are equivalent
If \(\alpha\sim \beta\), then \(F(\alpha)\sim F(\beta)\).
If \(\alpha\) is a homotopy equivalence, then \(F(\alpha)\) is a homotopy equivalence.
In this case we say \(F\) preserves homotopy.
Example 3. \(M\in C(R)\), we have \(\mathrm{Hom}_R(M,-)\) and \(M\otimes_R-\) preserve homotopy.
Special Natural Transformations
There are some standard isomorphisms and evaluation morphisms.
Definition 14.
Commutativity \[M\otimes_R N\to N\otimes_{R^{op}} M, \quad m\otimes n\mapsto (-1)^{|m||n|}n\otimes m\]
Associativity \[(M\otimes_R N)\otimes_S P\to M\otimes_R (N\otimes_S P), \quad (m\otimes n)\otimes p\mapsto m\otimes (n\otimes p)\]
Swap \[\mathrm{Hom}_R(M,\mathrm{Hom}_{S^{op}}(N,X)) \xrightarrow{S^{MXN}} \mathrm{Hom}_{S^{op}}(N,\mathrm{Hom}_R(M,X))\] where there is a sign change \((-1)^{|m||n|}\).
Adjunction \[\mathrm{Hom}_R(X\otimes_S N,M)\to \mathrm{Hom}_S(N,\mathrm{Hom}_R(X,M))\] where there is a sign change \((-1)^{|x||n|}\).
Evaluation Morphisms
\[M\xrightarrow{\delta^M_X} \mathrm{Hom}_{S^{op}}(\mathrm{Hom}_R(M,X),X),\quad \delta_X^M(m)(\psi) = (-1)^{|m|}\psi(m)\]
Remarks 5. A derived version of bi-duality is important in at least two cases
\(R=S\), \(X = {}_R D_R\), dualizing complex, we have Grothendieck duality.
\(R=S\), \(X = {}_R R_R\), derived reflexive complexes.
Homothety evaluation morphism
Definition 15. Homothety evaluation morphism \[R\mapsto \mathrm{Hom}_{S^{op}}(X,X)\]
Tensor Evaluation
Exercise.
Homomorphism Evaluation
Homomorphism evaluation \[N\otimes_S \mathrm{Hom}_R(X,M)\xrightarrow{\eta^{MXN}} \mathrm{Hom}_R(\mathrm{Hom}_{S^{op}}(N,X),M)\] \[n\otimes\psi \mapsto (\varphi \mapsto (-1)^{|n|(|\psi|+|\varphi|)}\psi(\varphi(n))\] There are some conditions under which \(\eta^{MXN}\) is an isomorphism. For example if \(N\) is a bounded complex of finitely presented \(S^{op}\)-modules and \(M\) is a complex of injective \(R\)-modules.
Sample Application of Homomorphism Evaluation
Recall that a projective module is flat.
Theorem 7. If \(F\) is a finitely presented flat \(R\)-module, then \(F\) is projective.
This is exercise 1.4.17.
Proof. Let \(M\xrightarrow{\alpha} N\) be a surjection, we need to show that \(\mathrm{Hom}_R(F, \alpha)\) is surjective. As \(\mathbb{Q}/\mathbb{Z}\) is faithfully injective (divisible) abelian group, we have \(\mathrm{Hom}_R(F,\alpha)\) is surjective \(\Leftrightarrow\) \(\mathrm{Hom}_\mathbb{Z}(\mathrm{Hom}_R(F,\alpha),\mathbb{Q}/\mathbb{Z})\) is injective. By hom evaluation and commutativity, the latter is \[\simeq \mathrm{Hom}_\mathbb{Z}(\alpha, \mathbb{Q}/\mathbb{Z})\otimes F)\] ◻
Homological Algebra
There is a fact, an \(R\)-complex is projective iff it is a contractible complex of projective \(R\)-modules.
Definition 16. An \(R\)-complex \(P\) is semi-projective (dg-projective) if the total Hom functor \(\mathrm{Hom}_R(P,-)\) preserves surjective quasi-isomorphisms.
We are not insisting on preserving all surjective morphisms, we only need it to send surjective q-isoms to surjective q-isoms.
The two main examples are
Example 4. A contractible complex of projective modules is semi-projective.
Example 5. A bounded below complex of projective \(R\)-modules \[\dots \to P_v\to P_{v-1}\to \dots \to P_{r}\to 0\] is semi-projective.
There are many ways to characterize these complexes,
Theorem 8. For an \(R\)-complex \(P\), the following are equivalent
\(P\) is semi-projective.
\(\mathrm{Hom}(P, -)\) is exact and preserve q-isoms
For any chain map \(P\to N\) and surjective quasi-isomorphism \(M\to N\), we can lift the map to \(P\to M\).
Any exact sequence of complexes \[0\to M'\to M\to P\to 0\] where \(M'\) is acyclic, splits.
\(P\) is a complex of projective modules and the functor \(\mathrm{Hom}(P,)\) preserves acyclic complexes.
Proposition 5. The semi-projectivity satisfy two out of three property. If in an exact sequence \[0\to P'\to P\to P''\to 0\] of complexes of projective modules, two out of the three complexes are semi-projective, then the third is also semi-projective.
Definition 17. An \(R\)-complex \(L\) is semi-free if it is graded free on a basis \(E\) of homogeneous elements with \(E=\sqcup_{n\ge 0} E^n\) such that \[E^0\subset Z(L), \quad \partial^L E^n\subset R\left\langle \bigcup_{i=0}^{n-1} E^i \right\rangle\]
Example 6. A bounded below complex of free \(R\)-modules is semi-free. \[\dots L_v\to L_{v-1}\to\dots \to L_0\to 0\] with \(E^v\) a basis for \(L_v\).
Here we see \(E^0\) are cycles and indeed \(\partial^L E^n\subset R\left\langle \bigcup_{i=0}^{n-1} E^i \right\rangle\).
Proposition 6. A semi-free \(R\)-complex is semi-projective.
Definition 18. A semi-projective resolution of an \(R\)-complex \(M\) is a q-isom from semi-projective complex \(P\) to \(M\), \(P\xrightarrow{\sim} M\).
Sketch of the Construction
Here is a sketch of the construction, given an \(R\)-complex \(M\), then we take \(Z^0\) to be a set of homogeneous generators of \(H(M)\), take \(L^0\) to be a graded free \(R\)-module that surjects on a basis \(E^0\) that surjects onto \(Z^0\), \(\pi^0:L^0\rightarrow Z(M)\), \(\partial^{L^0} = 0\).
Do the same to \(H(\ker \pi^0)\), adjust \(\partial, \pi\) accordingly.
Theorem 9. Let \(M\) be an \(R\)-complex, there exists a semi-projective resolution \(\pi: P\to M\), moreover
One can choose \(\pi\) surjective and \(P\) with \(P_v=0\) for \(v<\inf M^\sharp\).
One can choose \(P\) with \(P_v=0\) for \(v<\inf M^\sharp\).
If \(R\) is left noetherian and \(H(M)\) is bounded below and degree-wise finitely generated, then \(P\) can be chosen degree-wise finitely generated with \(P_v=0\) for \(v<\inf M^\sharp\).
Definition 19. An \(R\)-complex \(I\) is semi-injective if \(\mathrm{Hom}_R(,I)\) is exact and map surjective quasi-isomorphisms to injective quasi-isomorphisms.
Example 7. A contractible complex of injective \(R\)-modules is semi-injective.
Example 8. Bounded above complex of injective \(R\)-modules is semi-injective.
Proposition 7. If \(P\) is semi-projective then for a faithfully injective \(k\)-module \(\mathbb{E}\), \(\mathrm{Hom}_k(P,\mathbb{E})\) is semi-injective.
Definition 20. A semi-injective resolution of an \(R\)-complex \(M\) is a q-isom from \(M\) to a semi-injective complex \(I\), \(M\xrightarrow{q\simeq} I\).
Theorem 10. For every \(R\)-complex \(M\), there exists a semi-injective resolution \(M\xrightarrow{q\simeq} I\). Moreover
One can choose the map \(M\to I\) to be injective and \(I\) with \(I_v=0\) for \(v>\sup M^\sharp\).
One can choose \(I\) with \(I_v=0\) for \(v>\sup M^\sharp\).
Definition 21. An \(R\)-complex \(F\) is semi-flat if \(\otimes_R F\) preserves injective q-isoms.
Remarks 6. This is equivalent to asking that it preserves injective q-isoms and also preserve exactness.
Example 9. A contractible complex of flat \(R\)-modules is semi-flat.
Example 10. A bounded below complex of flat \(R\)-modules is semi-flat.
There are some similar characterizations.
Proposition 8. For an \(R\)-complex \(F\), the following are equivalent
\(F\) is semi-flat.
\(\otimes_R F\) is exact and preserves q-isoms.
The character complex \(\mathrm{Hom}(F,\mathbb{E})\) is semi-injective, where \(\mathbb{E}\) is a faithfully injective \(k\)-module.
\(F\) is a complex of exact modules and the functor \(\otimes_R F\) preserves acyclic complexes.
Corollary 1. A semi-projective complex is semi-flat.
Homotopy Categories
Definition 22. The homotopy category \(K(R)\) has the same object as the category of complexes \(C(R)\), and the morphisms are homotopy classes of morphisms in \(C(R)\), i.e. \[K(R)(M,N) = Z_0(\mathrm{Hom}(M,N))/B_0(\mathrm{Hom}(M,N)) = H_0(\mathrm{Hom}(M,N))\]
There is a canonical quotient functor \(Q_R:C(R)\to K(R)\) which maps \(M\mapsto M\) and \(\alpha\mapsto [\alpha]\). The connection
Proposition 9. Let \(\alpha\) be a morphism in \(C(R)\),
\([\alpha]\) is zero iff \(\alpha\) is null-homotopic.
\([\alpha]\) is an isomorphism iff \(\alpha\) is a homotopy equivalence.
Proposition 10. A complex \(M\in C(R)\) is zero in \(K(R)\) iff \(M\) is contractible.
Theorem 11. The homotopy category \(K(R)\) with \(\Sigma\) is a triangulated category with the distinguished triangles being those isomorphic to \[M\to N \xrightarrow{(1,0)^t} \mathrm{Cone}(\alpha) \xrightarrow{(0, 1^{\Sigma M})} \Sigma M\]
The homotopy category has a universal property whose special case is useful.
Theorem 12. Let \(F: C(R)\to C(S)\) be a functor that preserves homotopy, then there is a unique functor \(K(F): K(R)\to K(S)\) that makes the following diagram commute
There is an important consequence of splitting,
Proposition 11. Let \(P\) be semi-projective, \(\beta:M\to P\) a q-isom. There exists a \(\gamma:P\to M\) with \(1^p\sim \beta\gamma\).
Corollary 2. A q-isom of semi-projective complexes is a homotopy equivalence.
The Derived Category and Derived Functors
The derived category \(D(R)\) is the localization of the homotopy category \(K(R)\) with respect to the class of quasi-isomorphisms. It is the category whose objects are the complexes and morphisms are homotopy classes of morphisms in \(C(R)\).
There exists a universal functor \[V_R: K(R)\to D(R),\quad \alpha\mapsto\alpha 1^{-1}\] which maps q-isoms to isoms.
The universality of the functor gives, for every functor \(F: K(R)\to \mathcal{U}\) that maps q-isoms to isoms, there is a unique functor \(F'\) that makes the following diagram commute
Moreover,
\(D(R)\) and \(V_R\) are triangulated.
If \(\mathcal{U}\) and \(F\) are triangulated, then so is \(F'\).
Construction of Derived Category (Sketch)
The objects of \(D(R)\) are \[\mathrm{Obj}\, D(R) = \mathrm{Obj}\, K(R)\] \[\mathrm{Hom}_{D(R)}(M,N) = \{ \text{Fractions } \alpha\varphi^{-1}\}\] where we invert quasi-isomorphisms, the morphisms (right fractions here) look like the diagrams \[M\xleftarrow{\varphi} U\xrightarrow{\alpha} N\] one can also use left fractions.
Some creepy notations used in the book:
\(\sim\) homotopy equivalence relation in \(C(R)\),
\(\simeq\) quasi-isomorphism relation in \(C(R)\),
\(\cong\) isomorphism relation in \(C(R)\).
\(\approxeq\) homotopy equivalence relation in \(C(R)\), isoms in \(K(R)\).
Fact, \[\mathrm{Hom}_{D(R)}(M,N) = \mathrm{Hom}_{K(R)}(P,N)\] where \(P\xrightarrow{q\simeq} M\) is a semi-projective resolution of \(M\). This tells us that the hom set is always a set.
Composition in \(D(R)\)
We define the composition rule as follows \[\mathrm{Hom}_{D(R)}(M,N)\times \mathrm{Hom}_{D(R)}(L,M)\to \mathrm{Hom}_{D(R)}(L,N)\] as sending \[(\alpha\varphi^{-1}, \beta\psi^{-1}) \mapsto (\alpha\gamma)(\psi\chi)^{-1}\]
Triangulated Structure on \(D(R)\)
Definition 23. The distinguished triangles in \(D(R)\) are defined as the set of isomorphism closure in \(D(R)\) of \(V_R(\text{distinguished triangles in } K(R))\).
The semi-projective resolution functor
Definition 24.
Every \(R\)-complex \(M\) has a semi-projective resolution \(\pi^M: P(M)\xrightarrow{q\simeq} M\).
The lifting properties of semi-projective complexes make \(P(-)\) into an endofunctor of the homotopy category \(K(R)\), and \(\pi:P(-)\to \mathrm{id}_{K(R)}\) is a natural transformation.
Derived Functors
Let \(F:K(R)\to K(S)\) be a functor, we can derive a functor by the universal property.
Definition 25. The total left derived functor \(LF\) is the unique functor that makes the diagram commute
Similarly we may define the right derived functor \(RF\) as \[RF(M) = F I(M)\]
Remarks 7. \(P: K(R)\to K(R)\) maps q-isoms to isoms.
where if \(\alpha\) is a q-isom, this will force \(P(\alpha)\) to be a q-isom.
Abstract Definition to Derived Functors
\[(LF)\circ V_R = V_S\circ F\circ P \xrightarrow{\lambda = V_SF\pi} V_S F\] \((LF,\lambda)\) is object of a category \[\mathcal{L}_F = \{ (F', \lambda') : F':D(R)\to D(S), \lambda': F'\circ V_R \to V_S\circ F \}\] where the morphisms are natural transformations.
Theorem 13. \((LF,\lambda)\) is the terminal object in \(\mathcal{L}_F\).
Remarks 8.
One can also derive contravariant functors by switching the roles of \(P\) and \(I\).
One can also derive functors in multiple variables.
Now we turn to the most important example, tensor and hom.
Derived Tensor and Hom
Recall \(R\) is a \(k\)-algebra.
Definition 26.
The Hom functor \(\mathrm{Hom}_R(-,-): C(R)^{op}\times C(R)\to C(k)\) preserves homotopy and induces a functor \[\mathrm{Hom}_R(-,-): K(R)^{op}\times K(R)\to K(k)\] We can right derive this functor to get the derived Hom functor \[R\mathrm{Hom}_R(-,-): D(R)^{op}\times D(R)\to D(k)\] To compute its value \[\begin{aligned} R\mathrm{Hom}_R(M,N) &= \mathrm{Hom}_R(P(M), I(N)) \\ &\simeq \mathrm{Hom}_R(P(M),N) \simeq \mathrm{Hom}_R(M,I(N)). \end{aligned}\]
Similarly we can derive the total tensor functor \[-\otimes_R-: K(R)\times K(R)\to K(k)\] we write its derived version as \[-\otimes^L_R-: D(R)\times D(R)\to D(k)\] \[\begin{aligned} M\otimes^L_R N &= P(M)\otimes_R P(N) \\ &\simeq P(M)\otimes_R N\simeq M\otimes_R P(N) \end{aligned}\]
Remarks 9. In fact it suffices to let \(P(M)\) or \(P(N)\) to be a semi-flat resolution.
Definition 27. Let \(M\) and \(N\) be \(R\)-complexes, \[\mathrm{Ext}^i_R(M,N) := H_{-i}(R\mathrm{Hom}_R(N,M))\] note that we used homology notation, in cohomology notation it will be denoted as \(H^i(R\mathrm{Hom}_R(N,M))\) instead. Similarly \[\mathrm{Tor}^R_i(M,N) := H_i(M\otimes^L_R N)\]
Standard Isomorphisms in \(D(R)\)
Recall that we have commutativity, associativity, swap, and adjunction isomorphisms in \(C(R)\).
Proposition 12. These natural isomorphisms exists in derived versions in \(D(R)\).
Derived Associativity
The ordinary associativity isomorphism is \[(M\otimes_R X)\otimes_S N \xrightarrow{\simeq} M\otimes_R (X\otimes_S N)\] is a natural isomorphism of functors, and it lives naturally in \(K(R)\), \[C(R^o)\times C(R\text{-}S^o)\times C(S)\to C(k) \Rightarrow K(R^o)\times K(R\text{-}S^o)\times K(S)\to K(k)\]
Now we look at
This is a natural isom of functors \[D(R^o)\times D(R\text{-}S^o)\times D(S)\to D(k)\]
Boundedness and Finiteness Conditions
We want to know where do we have homology in the \(R\mathrm{Hom}\) and \(\otimes^L\) functors. There are general lowerbounds for non-vanishing of \(\mathrm{Ext}\) and \(\mathrm{Tor}\),
For the derived Hom functor, we have \[-\sup R\mathrm{Hom}_R(M,N) = \inf \{i: \mathrm{Ext}^i_R(M,N)\neq 0\}\ge \inf M - \sup N.\]
For the derived tensor product, we have \[\inf(M\otimes^L_R N) = \inf\{i:\mathrm{Tor}_i(M,N)\neq 0\} \ge \inf M + \inf N.\]
Remarks 10. Upper bounds on non-vanishing of \(\mathrm{Ext}\) and \(\mathrm{Tor}\) are closely related to homological dimensions of complexes, which will be discussed in later sections.
\[R\mathrm{Hom}_R(-,-): D_\sqsupset(R)^{op}\times D_{\sqsubset}(R)\to D_\sqsubset(k)\] \[-\otimes^L_R-: D_\sqsupset(R^o)\times D_\sqsupset(R)\to D_\sqsupset(k)\]
In some cases one even has \(f\) attached to \(D^f\) to each of the \(D\) in the above, which means all the homology module are finitely generated.
Homological Dimensions
Remarks 11. In order to avoid the confusion caused by using sup and inf in homology and cohomology, I will introduce the following notations, the left bound, \[\mathrm{lb}\,M = \begin{cases} \sup M & \text{for homology index} \\ -\inf M & \text{for cohomology index} \end{cases}\] the right bound, \[\mathrm{rb}\,M = \begin{cases} \inf M & \text{for homology index} \\ -\sup M & \text{for cohomology index} \end{cases}\] Note that their definition varies depending on what index convention you use. And we have the co-left bound, co-right bounds \[\mathrm{colb}\,M = -\mathrm{lb}\,M, \quad \mathrm{corb}\,M = -\mathrm{rb}\,M.\]
Definition 28. \(M\) an \(R\)-complex, define \[\begin{aligned} \mathrm{pd}_R M &= \inf\{i: \text{There is a semi-projective res } P \to M \text{ with } P_v = 0 \text{ for } v>i\} \\ &= \inf_{P\to M} \sup P\\ &= \inf_{P\to M} \mathrm{lb}\,P, \end{aligned}\] which lies in \(\mathbb{Z}\cup \{\pm \infty\}\).
Remarks 12.
\(M\) can be acyclic (exact) complex \(0\xrightarrow{\simeq} M\), \(\mathrm{pd}_R M = -\infty\)
If \(M\) does not have a semi-projective \(P\) with \(P_v=0\) for \(v\gg 0\), then the set is empty then by definition \(\mathrm{pd}_R M = \infty\).
This definition is sensitive on shift, i.e. \[\mathrm{pd}_R \Sigma^i M = \mathrm{pd}_R M + i\] \[\mathrm{pd}_R M \ge \mathrm{lb}\,M\] note that \(\Sigma\) is always left shift whatever index convention you use.
If \(P\) is a semi-projective complex, then there is a quasi-isom \(P\xrightarrow{q\simeq} M\Leftrightarrow\) there exists some isom \(P\sim M\) in \(D(R)\). so \(\mathrm{pd}_R M\) only depends on the isomorphism. Class of \(M\) in \(D(R)\).
If \(M\) is an \(R\)-module, viewed as a complex \(0\to M\to 0\), then \(\mathrm{pd}_RM\) is the usual projective dimension of a module.
Remarks 13. It may be tempting to define the projective dimension as the projective dimension of the complex \(M\in C(R)\) viewed as an element in the abelian category \(C(R)\), since the category \(C(R)\) is abelian with enough projective. But \(R\) is not a projective object in the abelian category of complexes as it is not acyclic, in fact if we use this as the definition, we have \[\mathrm{pd}_R R = \infty.\] And it should be warned that this definition is not what we are doing here.
Example 11. \(M\) be an \(R\)-module with projective resolution. \[\dots \to P_1\to P_0\to M\to 0\] Let \(x\in R\) be a central element, define \(K(x,M)\) to be the complex \[0\to M\xrightarrow{x} M\to 0\] then \(\mathrm{pd}_R K(x,M) = 1\).
Here we have a complex
\(x=1\), \[K(x,M) = 0\to M\to M\to 0 \sim 0 \in D(R)\]
\(x=0\) \[K(x,M) = M\oplus \Sigma M, \mathrm{pd}K(x,M) = \mathrm{pd}M + 1\]
\(R=\mathbb{Z}\), \(M= \mathbb{Q}\oplus\mathbb{Z}\), \(x=2\) \[K(x,M) \cong \mathbb{Z}/2\mathbb{Z}\in D(\mathbb{Z})\] \[\mathrm{pd}_\mathbb{Z}K(x,M) = 1 = \mathrm{pd}_\mathbb{Z}M\]
Theorem 14. For \(M\in D(R)\) and \(n\in \mathbb{Z}\), the following are equivalent
\(\mathrm{pd}_R M \le n\)
\(-\inf R\mathrm{Hom}_R(M,N) \le n - \inf N\) for all \(R\)-complex \(N\)
\(n\ge \sup M\) and \(\mathrm{Ext}^{n+1}_R(M,N) = 0\) for all \(N\)
For some (equivalently every) semi-projective resolution \(P\to M\), the \(R\)-module \(C_n(P) = \mathrm{coker}\partial_{n+1}^P\) is projective.
Furthermore, \[\begin{aligned} \mathrm{pd}_R M &= \sup \{ -\inf R\mathrm{Hom}_R(M,N) : \text{module }N \} \\ &= \sup \{ i : \mathrm{Ext}^i_R(M,N) \neq 0, \exists \text{ module } N \} \end{aligned}\]
Remarks 14. \(M,N\in D(R)\), we have an inequality \[\begin{aligned} \inf M - \sup N &\le - \sup R\mathrm{Hom}_R(M,N) = \inf \{ i : \mathrm{Ext}^i(M,N)\neq 0 \} \\ &\le -\inf R\mathrm{Hom}_R(M,N) = \sup \{i : \mathrm{Ext}^i_R(M,N)\neq 0\} \\ &\le \mathrm{pd}_R M - \inf N. \end{aligned}\]
Theorem 15. Let \(R\) be left Noetherian and \(M\in D_\sqsupset(R)\), \[\mathrm{pd}_R M = \sup \{ -\inf R\mathrm{Hom}_R(M,R/I) : I\subset R \text{ left ideal} \}\] Furthermore, if \(\mathrm{pd}_R M <\infty\), then \[\mathrm{pd}_R M = - \inf R\mathrm{Hom}_R(M,R)\]
Similarly we can define injective dimension \[\begin{aligned} \mathrm{id}_R M &= \inf \{i: \text{There is a semi-injective res } M\to I \text{ with } I_{-v} = 0 \text{ for } v>i\}\\ &= \inf_{M\to I} (-\inf I)\\ &= \inf_{M\to I} \mathrm{corb} I \end{aligned}\]
Minimal semi-injective resolution
Theorem 16. For a semi-injective \(R\)-complex \(I\), the following are equivalent
Every q-isom \(I\to I\) is an isom in \(C(R)\).
Every q-isom \(I\to M\) has a left inverse in \(C(R)\).
The only acyclic sub-complex \(A\subset I\) is \(A=0\).
For each \(n\), \(Z_n(I)=\ker d_n^I\) is an essential subset of \(I_n\).
If any of the above holds, then \(I\) is a minimal semi-injective \(R\)-complex. A minimal semi-injective resolution of \(M\) is a q-isom \(M\to I\) with \(I\) minimal semi-injective.
If you have a minimal semi-injective resolution, you can read off the injective dimension.
Theorem 17. If \(M\to I\) is a minimal semi-injective resolution, then the injective dimension \[\mathrm{id}_R M = -\inf I^\sharp = \mathrm{corb}\,I^\sharp\]
Example 12. For \(M=\mathbb{Z}\in C(\mathbb{Z})\), \[I = 0\to \mathbb{Q}\to \mathbb{Q}/\mathbb{Z}\to 0\] \(I\) is a minimal semi-injective resolution of \(M\), \[\mathrm{id}_\mathbb{Z}\mathbb{Z}= 1\]
Flat Dimension
Definition 29. The flat dimension of an \(R\)-complex \(M\) is
\(\mathrm{fd}_R M\) is defined using semi-flat resolution of \(M\).
\(\mathrm{fd}_R M\) controls the homological supremum of \((-)\otimes_R M\).
Theorem 18. \(M\in D(R)\), we always have \[\mathrm{fd}_R M \le \mathrm{pd}_R M\] and equality holds if \(R\) is left Noetherian and \(M\in D_\sqsupset(R)\).
Flat injective duality
\(M\in D(R)\), \(E\) faithfully injective \(k\)-module, now \(\mathrm{Hom}_R(M,E)\in D(R^o)\).
Injective dimension \(\mathrm{id}_{R^o} \mathrm{Hom}_k(M,E) = \mathrm{fd}_R M\).
\[\mathrm{fd}_{R^o} \mathrm{Hom}_k(M,E) = \mathrm{id}_R M\] provided that \(R\) is left Noetherian and \(M\in D_\sqsubset(R)\).
Evaluation Morphisms In Derived Category
Recall that there were the biduality, homothety, tensor evaluation, homomorphism evaluation morphisms. The point is that, these natural morphisms exists in derived versions.
Derived Tensor Evaluation
let \(M\in D(R)\), \(X\in D(R\text{-}S^o)\), \(N\in D(S)\) \[R\mathrm{Hom}_R(M,X)\otimes_S^L N \xrightarrow{\theta^{MXN}} R\mathrm{Hom}_R(M,X\otimes^L_SN)\]
Theorem 19. The crazy thing is that, \(\theta^{MXN}\) is usually an isomorphism o.o! For example for if \(R\) is left Noetherian and one of the following holds
\(M\in D_\square^f(R)\) with \(\mathrm{pd}_R M<\infty\), ( \(M\) is a perfect \(R\)-complex ).
\(M\in D^f_\sqsupset(R), X\in D_\sqsubset(R\text{-}S^o), \mathrm{fd}_S N<\infty\).
Derived Homomorphism Evaluation
\[N\otimes_S R\mathrm{Hom}_R(X,M)\xrightarrow{\eta^{MXN}} R\mathrm{Hom}_R(R\mathrm{Hom}_{S^o}(N,X),M)\]
Theorem 20. \(\eta^{MXN}\) is an isom in \(D(k)\) if for example \(S\) is right Noetherian and one of the following holds
\(N\in D_\square^f(S^o), \mathrm{pd}_{S^o} N<\infty\) ( \(N\) is a perfect \(S^o\)-complex ).
\(N\in D_\sqsupset(S^o), X\in D_\sqsubset(R\text{-}S^o), \mathrm{id}_R M<\infty\).
Dualizing Complexes and Iwanaga-Gorenstein Categories
Derived biduality
Here \(M\in D(R), X\ni D(R\text{-}S^o)\) \[\delta_X^M : M \to R\mathrm{Hom}_R(R\mathrm{Hom}_R(M,X),X)\]
To simplify things, we make the blanket assumption in this section: \(R\) noetherian, and projective as a \(k\)-algebra (module).
Theorem 21. The biduality morphism \[\delta_R^M : M\to R\mathrm{Hom}_{R^o}(R\mathrm{Hom}_R(M,R),R)\] is an isomorphism in \(D(R)\) if \(M\in D^f_\square(R)\) and \(\mathrm{pd}_R M < \infty\).
Definition 30. An \(R\)-\(R^o\) complex \(D\) is dualizing for \(R\) if
\(H(D)\) is degree-wise finitely generated over \(R\) and \(R^o\).
The injective dimensions \(\mathrm{id}_R D < \infty\), \(\mathrm{id}_{R^o} D < \infty\). (because the boundedness
\(\chi':R\to R\mathrm{Hom}_R(D,D)\) isom in \(D(R\text{-}R^o)\).
\(\chi: R\to R\mathrm{Hom}_{R^o}(D,D)\) isom in \(D(R\text{-}R^o)\).
Example 13. If \(k\) is a field, then \(k\) is dualizing for \(k\). Let \(R\) be a finite dimensional \(k\)-algebra, \(D=\mathrm{Hom}_k(R,k)\) injective \(R\)-module, look at \[\mathrm{Hom}_R(-,D)=\mathrm{Hom}_R(-,\mathrm{Hom}_k(R,k)) \cong \mathrm{Hom}_k(R\otimes_k -, k) \cong \mathrm{Hom}_k(-,k)\]
\[R\xrightarrow{\chi^0} \mathrm{Hom}_R(D,D) = \mathrm{Hom}_R(\mathrm{Hom}_k(R,k), \mathrm{Hom}_k(R,k)) \xrightarrow{swap} \mathrm{Hom}_R(R,\mathrm{Hom}_k(\mathrm{Hom}_k(R,k),k))\]
Theorem 22. Let \(D\) be a dualizing complex for \(R\). For \(M\in D^f(R)\) \[\delta_D^M: M\to R\mathrm{Hom}_{R^o}(R\mathrm{Hom}_R(M,D),D)\] is an isomorphism in \(D(R)\).
Theorem 23 (Grothendieck Duality). There is an adjoint equivalence of triangulated categories \[D^f(R^o) \xrightarrow{R\mathrm{Hom}_{R^o}(-,D)} D^f(R)^{op}\] \[D^f(R^o) \xleftarrow{R\mathrm{Hom}_R(-,D)} D^f(R)^{op}\] unit and counit are biduality. It restricts to \[D^f_\sqsubset(R^o) \leftrightarrow D^f_\sqsupset(R)^{op}\quad D^f_\sqsupset(R^o) \leftrightarrow D^f_\sqsubset(R)^{op}\] \[D^f_\square(R^o) \leftrightarrow D^f_\square(R)^{op}\quad I^f(R^o) \leftrightarrow P^f(R)^{op}\]
Definition 31. We introduce the following notations for the categories of complexes \[P(R) = \{M\in D_\square(R) : \mathrm{pd}_R M < \infty \}\] \[I(R) = \{M\in D_\square(R) : \mathrm{id}_R M <\infty\}\] \[F(R) = \{M\in D_\square(R) : \mathrm{fd}_R M <\infty\}\] \[I^f(R) = I(R)\cap D^f(R)\]
Definition 32. An \(R\)-complex \(M\) is derived reflexive if
The complex itself has to be \(\in D^f_\square(R)\).
\(R\mathrm{Hom}_R(M,R)\in D^f_\square(R^o)\)
Bi-duality \(\delta_R^M : M\xrightarrow{q\simeq} R\mathrm{Hom}_R(R\mathrm{Hom}_R(M,R),R)\).
Denote \[\mathcal{M}(R)=\{M\in D(R) : M \text{ is derived reflexive} \}.\]
Theorem 24. There is an adjoint equivalence of triangulated categories, \[\mathcal{R}(R^o) \xrightarrow{R\mathrm{Hom}_{R^o}(-,R)} \mathcal{R}(R)^{op}\] \[\mathcal{R}(R^o) \xleftarrow{R\mathrm{Hom}_R(-,R)} \mathcal{R}(R)^{op}\] where \(\mathcal{P}^f(R^o)\) and \(\mathcal{P}^f(R)^{op}\) sits inside the arrows.
Definition 33. \(R\) is Iwanaga-Gorenstein if \(\mathrm{id}_R R <\infty\) and \(\mathrm{id}_{R^o} R<\infty\), i.e. injective dimension of \(R\) is finite as both a left and right module.
Theorem 25. The following are equivalent
\(R\) is Iwanaga-Gorenstein.
Every flat \(R\) and \(R^o\)-module has finite injective dimension.
Every injective \(R\) and \(R^o\)-module has finite flat dimension.
Theorem 26. The following are equivalent
\(R\) is Iwanaga-Gorenstein.
\(R\) is a dualizing complex for \(R\).
\(R\) has a dualizing complex \(D\) with projective dimension finite over both \(R\) and \(R^o\), i.e. \(\mathrm{pd}_R D <\infty\) and \(\mathrm{pd}_{R^o} D <\infty\).
Now let \(D\) be a dualizing complex, look at the adjoint functors \[D(R)\xrightarrow{D\otimes_R^L-} D(R),\quad D(R)\xleftarrow{R\mathrm{Hom}_R(D,-)} D(R)\] The unit \(\alpha^N\) is an isom if \(\theta^{DDN}\) is an isom, and the co-unit \(\beta^M\) is an isom if \(\eta^{MDD}\) is an isom.
We introduce the following notations
\(\hat{A}(R) = \{ N\in D(R) : \alpha^N \text{ is an isom} \}\)
\(A(R) = \{ N\in \hat{A}(R) : N, D\otimes_R^L N \in D_\square(R) \}\)
\(\hat{B}(R) = \{ M\in D(R) : \beta^M \text{ is an isom} \}\)
\(B(R) = \{ M\in \hat{B}(R) : M, R\mathrm{Hom}_R(M,D) \in D_\square(R) \}\)
note that \(\mathcal{F}(R)\subset A(R)\), \(I(R)\subset B(R)\).
Theorem 27 (Foxby-Sharp Equivalence). There is an adjoint equivalence of triangulated categories
Proposition 13. Let \(D\) be a dualizing complex for \(R\), then \[A^f(R) = \mathcal{R}(R)\]
Theorem 28. The following are equivalent
\(R\) is Iwanaga-Gorenstein
\(\mathcal{F}(R) = I(R)\) and \(\mathcal{F}(R^o) = I(R^o)\)
\(R\) has a dualizing complex and \(\mathcal{R}(R)=D^f_\square(R)\) and \(\mathcal{R}(R^o)=D^f_\square(R^o)\)
\(R\) has a dualizing complex and \(A(R)=D_\square(R)\)
\(R\) has a dualizing complex and \(B(R)=D_\square(R)\)
Theorem 29. If \(D\) is a dualizing complex for \(R\), and \(M\) an \(R\)-complex of finite flat dimension, then \[\mathrm{pd}_R M\le \max(\sup M, \mathrm{id}_R D + \sup(D\otimes_R^L M)) < \infty\]
Homological Invariants in Commutative Algebra
Definition 34. For \(\mathfrak{p}\in \mathrm{Spec}R\), \(k(\mathfrak{p}) = R_\mathfrak{p}/\mathfrak{p}R_\mathfrak{p}\) is the residue field of \(R\) at \(\mathfrak{p}\). For \(M\in D(R)\), \[\mathrm{supp}_R M = \{\mathfrak{p}\in \mathrm{Spec}R : k(\mathfrak{p})\otimes_R^L M\neq 0 \in D(R)\}.\] \[\mathrm{cosupp}_R M = \{\mathfrak{p}\in \mathrm{Spec}R : R\mathrm{Hom}_R(k(\mathfrak{p}),M)\neq 0 \in D(R)\}.\]
Remarks 15. For an \(R\)-module \(M\) one has the classical support \[\mathrm{Supp}_R M = \{\mathfrak{p}\in \mathrm{Spec}R : m_\mathfrak{p}\neq 0\}\] One always have \[\mathrm{supp}_R M \subset \mathrm{Supp}_R M\] with equality if \(M\) is finitely generated.
Example 14.
The \(\mathrm{supp}_R k(\mathfrak{p}) = \{\mathfrak{p}\} = \mathrm{cosupp}_R k(\mathfrak{p})\).
The injective hall \(E=E_R(R/\mathfrak{p})\) of \(R/\mathfrak{p}\), which is an injective module and \(R/\mathfrak{p}\) is an essential submodule of \(E\), \[\begin{aligned} \mathrm{supp}_R E_R(R/\mathfrak{p}) &= \{\mathfrak{p}\} \\ \mathrm{cosupp}_R E_R(R/\mathfrak{p}) &= \{\mathfrak{q}: \mathfrak{q}\subset \mathfrak{p}\} \end{aligned}\]
Theorem 30. For \(M\in D(R)\), one has \(\mathrm{cosupp}_R M = \varnothing \Leftrightarrow\mathrm{supp}_R M=\varnothing \Leftrightarrow M = 0 \in D(R)\) (i.e. \(M\) acyclic, exact).
Support Formula
Theorem 31. For \(M,N\in D(R)\), one has \[\mathrm{supp}_R (M\otimes_R^L N) = \mathrm{supp}_R M \cap \mathrm{supp}_R N.\] i.e. \(M\otimes_R^L N = 0\) iff \(\mathrm{supp}_R M \cap \mathrm{supp}_R N = \varnothing\).
Example 15.
\(R=\mathbb{Z}\), \(M= \mathbb{Z}/2\mathbb{Z}\), \(N=\mathbb{Q}\). \[M\otimes_R^L N = \mathbb{Z}/2\mathbb{Z}\otimes_\mathbb{Z}\mathbb{Q}= 0\]
\(\mathrm{Supp}_\mathbb{Z}(0)=\varnothing\)
\(\mathrm{Supp}_\mathbb{Z}(\mathbb{Z}/2\mathbb{Z})\cap \mathrm{Supp}_\mathbb{Z}(\mathbb{Q}) = \{(2)\} \cap \mathrm{Spec}(\mathbb{Z}) = \{(2)\}\)
Cosupport Formula
Theorem 32. For \(M,N\in D(R)\) one has \[\mathrm{cosupp}_R R\mathrm{Hom}_R(M,N) = \mathrm{supp}_R M \cap \mathrm{cosupp}_R N.\] In particular, \(R\mathrm{Hom}_R(M, N) = 0\) in \(D(R)\) iff \(\mathrm{supp}_R M \cap \mathrm{cosupp}_R N = \varnothing\).
Koszul Complex
\(x\in R\) be an element of a ring, the Koszul complex is \[K(x) = 0 \to R \xrightarrow{x} R \to 0\] where since we are using homology indexing, the degrees are \(1, 0\), for cohomology index, it should be \(0, 1\) instead and the definitions related with it will need to be changed. For a sequence of elements \(x = (x_1,\dots,x_n)\), the Koszul complex is \[K(x) = K(x_1)\otimes_R \cdots \otimes_R K(x_n)\]
Example 16. \(K(x_1,x_2)\) looks like \[0\to R\xrightarrow{(-x_2,x_1)^t} R^2 \xrightarrow{(x_1,x_2)} R\to 0\]
Depth and width
Let \(\mathfrak{a}= (x_1,\dots,x_n)\) be an ideal of \(R\), \(M\in D(R)\), the \(\mathfrak{a}\)-depth of \(M\) is defined as \[\begin{aligned} \mathfrak{a}\text{-}\mathrm{depth}_R M &= n + \mathrm{colb}\,( K(x) \otimes M ) \\ &= \mathrm{colb}\,\mathrm{Hom}_R(K(x),M)\\ &= \mathrm{colb}\,R\mathrm{Hom}_R(R/\mathfrak{a}, M)\\ &= \mathrm{colb}\,R\Gamma_\mathfrak{a}M \end{aligned}\]
One always have an inequality \[\mathfrak{a}\text{-}\mathrm{depth}_R M \ge \mathrm{colb}\,M\] and equality holds in some cases.
The \(\mathfrak{a}\)-width is defined as \[\begin{aligned} \mathfrak{a}\text{-}\mathrm{width}_R M &= \mathrm{rb}\,(K(x)\otimes_R M)\\ &= n+ \mathrm{rb}\,R\mathrm{Hom}_R(K(x),M)\\ &= \mathrm{rb}\,(R/\mathfrak{a}\otimes^L_R M). \end{aligned}\] Similarly we have \[\mathfrak{a}\text{-}\mathrm{width}_R M \ge \mathrm{rb}\,M\] and equality holds for example if \(\mathfrak{a}\subset J(R)\) and \(M\in D^f(R)\). Thus for a finitely generated \(R\)-module \(M\neq 0\), the \(\mathfrak{a}\)-width of \(M\) is \(0\).
Remarks 16. Say \(\mathfrak{a}\subset J(R)\), \(M\neq 0\) is a finitely generated \(R\)-module, \(\mathfrak{a}\text{-}\mathrm{depth}_R M =\) the maximal length of an \(M\)-regular sequence contained in the ideal \(\mathfrak{a}\), which is finite.
Example 17. Let \(R\in \mathbb{Z}, M = \mathbb{Q}, x\neq 0\in \mathbb{Z}\). \[K(x)\otimes_\mathbb{Z}\mathbb{Q}= 0\to \mathbb{Q}\xrightarrow{x} \mathbb{Q}\to 0\] which \(= 0\) in \(D(R)\). The \((x)\)-depth of \(\mathbb{Q}\) is \(\infty\).
Depth and flat dimension
Definition 35. For a local ring \((R,\mathfrak{m}, k)\) and \(M\in D(R)\), we define \[\mathrm{depth}_R M = \mathfrak{m}\text{-}\mathrm{depth}_R M\] and width to be \[\mathrm{width}_R M = \mathfrak{m}\text{-}\mathrm{width}_R M.\]
How does depth and width interact with \(\otimes^L\) and \(R\mathrm{Hom}\)?
You can always compute \[\mathrm{depth}_R R\mathrm{Hom}_R(M,N) = \mathrm{width}_R M + \mathrm{depth}_R N\]
\[\mathrm{depth}_R (M\otimes^L_R N) \ge \mathrm{depth}_R M + \mathrm{depth}_R N - \mathrm{depth}R\] provided that \(\mathrm{fd}_R M< \infty\) and \(N\in D_\sqsubset(R)\).
The Auslander-Buchsbaum Formula
Set \((R,\mathfrak{m}, k)\) be a local ring, \(M\in D_\square^f(R)\). If \(\mathrm{pd}_R M<\infty\), then \[\mathrm{pd}_R M = \sup (k\otimes_R^L M) = \mathrm{depth}R - \mathrm{depth}_R M.\]
Proof. Set \(N=k\) in the depth-width formula for \(M\otimes_R^L N\). There are versions of the equalities in the Auslander-Buchsbaum formula for complexes over non-local rings. ◻
Theorem 33. \(R\) is any commutative noetherian ring, \(M\in D(R)\) is an \(R\)-complex. \[\mathrm{fd}_R M = \sup \{ \sup (k(\mathfrak{p})\otimes_R^L M) : \mathfrak{p}\in \mathrm{Spec}R \}\] If \(\mathrm{fd}_R M <\infty\), then \[\mathrm{fd}_R M = \sup \{ \mathrm{depth}R_\mathfrak{p}- \mathrm{depth}_{R_\mathfrak{p}} M_\mathfrak{p}: \mathfrak{p}\in \mathrm{Spec}(R)\}\]
Width and Injective Dimension
You can always compute \[\mathrm{width}_R R\mathrm{Hom}_R(M,N) = \mathrm{width}_R M + \mathrm{width}_R N\]
\[\mathrm{width}_R (M\otimes^L_R N) \ge \mathrm{depth}_R M + \mathrm{width}_R N - \mathrm{depth}R\] provided that \(\mathrm{id}_R M< \infty\) and \(N\in D_\sqsubset(R)\).
Theorem 34 (The Bass Formula). \(M\in D^f_\square(R)\), if \(\mathrm{id}_R M<\infty\), then \[\mathrm{id}_R M = - \inf R\mathrm{Hom}_R(k,M) = \mathrm{depth}R - \inf M.\] This is Chouinand Formula for injective dimension.
Mathis Duality
Let \((R,\mathfrak{m},k)\) be local, \(E=E_R(k)\) the injective hull of the \(R\)-module \(k\), then
Theorem 35. Let \((R,\mathfrak{m},k)\) be local and complete, \(\hat{R}=R\), for example \(R=k[[ x_1,\dots,x_n]]/I\), then there are equivalences of triangulated categories Here \(D^{f/art/l}(R)\) is the full subcategory of \(D(R)\) whose objects \(M\) satisfy that each homology module \(H_i(M)\) is finitely generated / artinian / has finite length. Moreover, if \(D\) is a normalized dualizing complex for \(R\) (i.e. \(\sup D = \dim R\)), then Grothendieck duality \[D^f(R) \xleftrightarrow{R\mathrm{Hom}_R(-,D)} D^f(R)^{op}\] restricts to Mathis duality, i.e. \[R\mathrm{Hom}_R(M,D) \simeq R\mathrm{Hom}_R(M,E_R(k))\] for \(M\in D^f(R)\).
The Derived Category of A Commutative Noetherian Ring
Let \(\mathfrak{a}\) be an ideal and one can look at an obvious sequence of quotients \[R/\mathfrak{a}^3 \to R/\mathfrak{a}^2\to R/\mathfrak{a}\]
Definition 36. The \(\mathfrak{a}\)-torsion functor \[\Gamma_\mathfrak{a}= \mathrm{colim}_{u\ge 1} \mathrm{Hom}_R(R/\mathfrak{a}^u, -) : C(R)\to C(R)\] it finds all things that gets killed by some powers of \(\mathfrak{a}\).
Derived \(\mathfrak{a}\)-torsion \[R\Gamma_\mathfrak{a}: D(R)\to D(R)\]
Definition 37. \[H_\mathfrak{a}^n(M) = H_{-n}(R\Gamma_\mathfrak{a}(M))\]
Theorem 36. Let \(\mathfrak{a}\) be an ideal and \(M\in D^f(R)\). One can compute \(R\Gamma_\mathfrak{a}(M)\) as \[R\Gamma_\mathfrak{a}(M) = R\mathrm{Hom}_R(R\mathrm{Hom}_R(M,D),R\Gamma_\mathfrak{a}(D)),\] where \(D\) is a dualizing complex for \(R\). Compare biduality \(M\simeq R\mathrm{Hom}_R(R\mathrm{Hom}(M,D),D)\) which is always an isomorphism.
In local algebra, this has a particularly nice interpretation.
Definition 38. The Krull dimension of \(R\) is the dimension of \(\mathrm{Spec}(R)\) as a poset.
Definition 39. A dualizing complex \(D\) for \(R\) is normalized if the supremum of \(D\) is equal to the dimension of \(R\), \(\sup D = \dim R\).
Theorem 37 (Local Duality Theorem). If \((R, \mathfrak{m}, k)\) is local, \(D\) is a normalized dualizing complex for \(R\), then for \(M\in D^f(R)\), \[R\Gamma_m(M) = \mathrm{Hom}_R(R\mathrm{Hom}_R(M,D),E(k))\] \[H_\mathfrak{m}^n(M) = \mathrm{Hom}_R(\mathrm{Ext}_R^{-n}(M,D),E(k))\]
\[D\simeq 0 \to \coprod_{\dim R/\mathfrak{p}=\dim R} E(R/\mathfrak{p}) \to \dots \to E(k) \to 0\]
Proposition 14. \(\mathfrak{a}\subset R\) an ideal and \(M\in D(R)\). \[\begin{aligned} \mathfrak{a}\text{-}\mathrm{depth}_R M &= -\sup R\Gamma_\mathfrak{a}(M)\\ &= \inf \{ n : H^n_\mathfrak{a}(M)\neq 0 \} \end{aligned}\]
Proposition 15. \[M\cong R\Gamma_\mathfrak{a}(M) \Leftrightarrow\mathrm{supp}_R M\subset V(\mathfrak{a}) = \{\mathfrak{p}\in \mathrm{Spec}R : \mathfrak{a}\subset \mathfrak{m}\}\]
Theorem 38. They are commutative up to natural isomorphisms, diagrams of equivalences of triangulated categories
Definition 40. \(R\) is Gorenstein if \(R_\mathfrak{p}\) is Iwanaga-Gorenstein i.e. \(\mathrm{id}_{R_p}R_\mathfrak{p}<\infty\) for each \(\mathfrak{p}\).
Theorem 39. The following are equivalent
\(R\) is Iwanaga-Gorenstein,
\(R\) is Gorenstein with \(\dim R<\infty\),
\(R\) has finite Krull-dimension and \[\mathcal{R}(R) = D^f_\square(R).\]
\(R\) has a dualizing complex and the Auslander category \(A(R)=D_\square(R)\).
\(R\) has a dualizing complex and \(B(R)=D_\square(R)\).
\(R\) has a dualizing complex and \(\hat{A}(R) = D(R)\).
\(R\) has a dualizing complex and \(\hat{B}(R) = D(R)\).
Theorem 40. A complex \(M\in D^f_\square(R)\) is derived reflexive iff \[M\simeq R\mathrm{Hom}_R(R\mathrm{Hom}_R(M, R), R).\]
Theorem 41. The following are equivalent
\(R\) is Gorenstein,
For every acyclic complex \(P\) of projective \(R\)-modules also \(\mathrm{Hom}(P,L)\) is acyclic for every projective \(R\)-module \(L\).
For every acyclic complex \(I\) of injective \(R\)-modules, also \(\mathrm{Hom}_R(E,I)\), is acyclic for every injective \(E\).
For every acyclic complex \(F\) of flat \(R\)-modules, also \(E\otimes_R F\) is acyclic for every injective \(R\)-module \(E\).
Theorem 42. For a local ring \((R,\mathfrak{m},k)\) the following are equivalent
\(R\) is regular,
\(\mathrm{pd}_R k < \infty\),
\(\mathcal{P}^f(R) = D^f_\square(R)\),
Definition 41. \(R\) is regular if \(R_\mathfrak{p}\) is regular for every prime \(\mathfrak{p}\).
Theorem 43. The following are equivalent
\(R\) is regular,
Every acyclic complex of projective modules is contractible,
Every acyclic complex of injective modules is contractible,
Every complex of injective \(R\)-modules is semi-injective,
Every complex of projective \(R\)-modules is semi-projective,
Every complex of finitely generated projective \(R\)-modules is semi-projective.