D-modules is the study of differential equations using algebraic methods. The main example to consider is the followinf differential equation \[\left(\sum_{i=1}^m a_i(z) \partial^i\right) f = 0.\] Here \(a_i\) are polynomials and \(\partial = \frac{\partial }{\partial z}\) is a differential operator. It is evident that the solutions of the equation forms a linear space, but on different open sets, the solution sets might be very different. An observation of Cauchy is that, on a small ball \(B(z,\varepsilon)\) which does not have any zeros of \(a_m(z)\), the equation admits \(m\) solutions.
When talking about differential equations, there will be both \(\mathcal{O}\)-modules and \(\mathbb{C}\)-sheaves. For algebraic variety \(X\), let \(\mathcal{O}_X\) be the sheaf of regular functions in Zariski topology and
there are natural multiplication actions \[\Gamma(O_X)\to \mathrm{End}_\mathbb{C}(\mathcal{O}_X) \quad \mathcal{O}\to {\mathcal{H}om}_\mathbb{C}(\mathcal{O}, \mathcal{O}).\]
A derivation of \(\mathcal{O}_X\) is a \(\mathbb{C}\)-linear morphism \(\delta: \mathcal{O}_X\to \mathcal{O}_X\) satisfying \[\delta_U(s_U t_U) = \delta_U(s_U) t_U + s_U \delta_U(t_U).\]
So derivations have to be \(k\)-linear rather than \(\mathcal{O}_X\)-linear, so they eventually live in the ring \(\mathrm{End}_k(\mathcal{O}_X)\). But this ring also lives \(\Gamma(\mathcal{O}_X)\), which are multiplications by scalars.
One defines
The sheaf of derivations \(\Theta_X\) or \(\mathrm{SDer}_X\subset \mathrm{SEnd}_k(\mathcal{O}_X)\) given by \[\Theta_X(U) = \mathrm{Der}(\mathcal{O}_U).\]
The ring of differential operators \(\mathscr{D}(X)\subset \mathrm{End}_k(\mathcal{O}_X)\) generated by elements \(\Gamma(\mathcal{O}_X)\) and \(\mathrm{Der}(\mathcal{O}_X)\).
The sheaf of differential operators \(\mathscr{D}_X\subset \mathrm{SEnd}_k(\mathcal{O}_X)\) given by \(\mathscr{D}_X(U)=\mathscr{D}(U) \subset \mathrm{End}_k(\mathcal{O}_U)\).
We can talk about the category of \(\mathscr{D}_X\)-modules, which equipps with both \(\mathcal{O}_X\) action and derivation, fixing the problem of not being able to talk about \(\mathcal{O}_X\)-linearity and \(\mathcal{O}_X\)-modules when dealing with derivations.
The simplest example of sheaf of differential operators is \(\mathscr{D}_{\mathbb{A}^1}\). In this case we have \[\mathrm{Der}(\mathcal{O}_{\mathbb{A}^1}) = k[x]\frac{\partial }{\partial x}, \quad \Gamma(\mathcal{O}_{\mathbb{A}^1})=k[x],\] and therefore \[\mathscr{D}(\mathbb{A}^1) = k[x,\partial_x]/([\partial_x, x]=1).\]
Let \(\mathcal{M}\) be a coherent \(\mathcal{O}_X\)-module, an integrable connection on \(\mathcal{M}\) is a morphism of \(k\)-sheaves \[\nabla : \Theta_X \to \mathrm{SEnd}_k(\mathcal{M})\] such that \[\nabla_{f_U\theta_U} s_U = f_U \nabla_{\theta_U} s_U,\] \[\nabla_{\theta_U} (fs) = (\theta f) s + f \nabla_{\theta} s,\] \[\nabla_{[S,T]} = [\nabla_S, \nabla_T].\] Here the third condition is telling you that curvature is zero, thus being an ‘integrable connection’. Without the third condition it is a connection in general sense.
Let \(\mathcal{M}\) be coherent \(\mathcal{O}_X\)-module, an integrable connection on \(\mathcal{M}\) is equivalent to given a structure of \(\mathscr{D}_X\)-module on \(\mathcal{M}\). Moreover, if admits an integral connection, it is locally free (vector bundles).
Introduction to D-modules
Let \(X\) be a smooth variety, \(\mathcal{O}_X, \Theta_X\) be the sheaf of regular functions and sheaf of vector fields (derivations). For any point of \(X\) we can take an affine neighborhood \(U\) and a local coordinate system \(\{x_i, \partial_i\}_{1\le i\le n}\), this is because \(\mathcal{O}_{X,x}\) is regular, \(\Omega^1_{X,x}\) is locally free and we can take local coordinates \(x_i\) and its dual basis, satisfying \[\Theta_U = \bigoplus_{i=1}^n \mathcal{O}_U \partial_i, \quad [\partial_i,\partial_j]=0, \quad [\partial_i, x_j]=\delta_{ij}.\] \[D_U = D_X|_U = \bigoplus_{\alpha\in \mathbb{N}^n} \mathcal{O}_U \partial^\alpha\]
Algebraic Structure of \(D_X\)
The following example is a good way to understand the abstract structure of \(D_X\).
Example.
Let \(U\) be affine, then \(D(U)\) is abstractly the algebra generated by abstract symbols \(\{\tilde{f}, \tilde{\theta} : f\in \mathcal{O}(U), \theta\in \Theta(U)\}\) subject to relations
\(\tilde{f} \tilde{g} = \widetilde{fg}\),
\(\tilde{f}+\tilde{g} = \widetilde{f+g}\),
\(\tilde{\theta_1}+\tilde{\theta_2} = \widetilde{\theta_1+\theta_2}\),
\([\tilde{\theta_1}, \tilde{\theta_2}] = \widetilde{[\theta_1,\theta_2]}\),
\(\tilde{f}\tilde{\theta} = \widetilde{f\theta}\),
\([\tilde{\theta}, \tilde{f}] = \widetilde{\theta(f)}\).
i.e. \(\widetilde{(\cdot)}\) preserves the algebraic structure of \(\mathcal{O}_U\), the Lie algebra structure of \(\Theta_U\), the linear action of \(\mathcal{O}\) on \(\Theta\) and the Leibniz rule.
In summary, we can say in terms of algebraic structures, \[D_X = \mathrm{Ring}(\mathcal{O}_X)+\mathrm{Lie}(\Theta_X)+ \mathcal{O}_X\mathrel{\circlearrowright}\Theta_X+ \mathrm{Leibniz}.\]
Example.
In terms of local coordinates of any element \(P = \sum_\alpha a_\alpha \partial^\alpha\in D(U)\), the multiplication of polynomials is obvioius. By the Leibniz rule \(\theta f - f \theta = \theta(f)\), the multiplication of any derivation \(\partial\) (on the left) is given by the following linear maps \[\partial = M_\partial + C_\partial\] where \(M_\partial : a_\alpha \partial^\alpha \mapsto a_\alpha \partial^\alpha \partial\) multiplies \(\partial\) on the multiple derivations and \(C_\partial : a_\alpha \partial^\alpha \mapsto \partial(a_\alpha) \partial^\alpha\) applies the derivation on the coefficients. Clearly \(M_\partial\) and \(C_\partial\) are \(\mathbb{C}\)-linear maps and they commute with each other. This easily give us the following Leibniz rule \[\partial^\alpha = \prod_i (M_{\partial_i}+C_{\partial_i})^{\alpha_i} = \sum_{\beta\le \alpha} \binom{\alpha}{\beta} M_\beta C_{\alpha-\beta}.\] As a result \[\begin{aligned} \partial^\alpha &= \sum_{\beta\le \alpha} \binom{\alpha}{\beta} M_\beta C_{\alpha-\beta}\\ &= \sum_{\beta} \frac{1}{\beta!} \frac{\alpha!}{(\alpha-\beta)!} M_{\alpha-\beta} C_{\beta} \\ \end{aligned}\] Extending linearly we have by total symbol \(\sigma: D(U)\to \mathcal{O}(U)[\xi]\), \[\sigma(P\cdot Q) = \sum_{\beta} \frac{1}{\beta!} \partial_\xi^\beta\sigma(P)(x,\xi) \cdot \partial_x^\beta \sigma(Q)(x,\xi).\]
Example.
If we write everything reversely as \(P=\sum (-\partial)^\alpha a_\alpha\) and consider the action of any derivation multiplied on the right, we have similarly \[-\partial = M_{(-\partial)} + C_\partial\] since \((-\partial^\beta b) (-\partial) = (-\partial)^\beta(-\partial b + \partial(b))\). Therefore \[(-\partial)^\alpha = \sum_\beta \binom{\alpha}{\beta} M_{(-\partial)^\beta} C_{\partial^{\alpha-\beta}}\]
Order Filtration
There is a ascending filtration called the order filtration on \(D_U\) given by local coordinates on affine \(U\) \[F_lD_U = \sum_{|\alpha|\le l} \mathcal{O}_U \partial^\alpha.\] This filtration is however not dependent on the choice of local coordinates.
Proposition.
(\(X\) smooth)
\(F_lD_X\) is locally free.
\(F_0D_X = \mathcal{O}_X\), \((F_lD_X) (F_mD_X) \subset F_{l+m}D_X\).
\([F_lD_X, F_mD_X ] \subset F_{l+m-1}D_X\).
Recall that \(D_X\) is non-commutative, but since we have a filtration we can derive associated graded ring (commutative) given by (with the convention that \(F_{-1}D_X=0\)) \[\mathrm{gr}D_X = \bigoplus_{l\ge 0} F_lD_X/F_{l-1}D_X= \bigoplus_{l\ge 0} \mathrm{gr}_l D_X\]
Proposition.
Let \(U\) be an affine chart with coordinates \(\{x_i, \partial_i\}_{1\le i\le n}\), then If we take \(\xi_i\) to be the image of \(\partial_i\mapsto \mathrm{gr}_1 D_U\), then
\(\mathrm{gr}_l D_U = \bigoplus_{|\alpha|=l} \mathcal{O}_U \xi^\alpha \),
\(\mathrm{gr}D_U = \mathcal{O}_U[\xi_1,\cdots, \xi_n]\).
The associated polynomial \(\sigma_l : F_l \to \mathcal{O}[\xi]\) of \(P\) is called the principal symbol \(\sigma_l(P)\).
D-modules and Connections
Giving a \(D_X\)-module structure is the same as giving a \(\mathcal{O}_X\) module \(M\) and a (flat) connection \[\nabla: \Theta_X \to \mathrm{SEnd}_k(M) \in \mathrm{Mor}({\bf Ab}_k)\] such that
\(\nabla_{f\theta} s = f\nabla_\theta s\),
\(\nabla_\theta(fs) = \theta(f)s + f\nabla_\theta s\),
(flatness/integrability) \(\nabla_{[\theta_1,\theta_2]} = [\nabla_{\theta_1}, \nabla_{\theta_2}]\).
We have
\[\{D_X\text{-modules}\} \leftrightarrow \{\text{Flat connections on }\mathcal{O}_X\text{-modules}\}\]
\[M\mapsto (M,\nabla_\theta s := \theta s), \quad (M, \theta m = \nabla_\theta m)\leftrightarrow (M, \nabla)\] Note that connection only requires the first two conditions, the third condition is called the flatness or integrability condition.
This can be seen as the \(D_X\) module relations are defined by the algebraic structure of \(\mathcal{O}_X\), Lie algebra structure of \(\Theta_X\), linear action of \(\mathcal{O}\mathrel{\circlearrowright}\Theta\) and the Leibniz rule. To be precise,
The first condition reflects \(\mathcal{O}\mathrel{\circlearrowright}\Theta\)
The second condition reflects the Leibniz rule \([\theta, f] = \theta(f)\).
The flatness is derived from the Lie algebra structure of \(\Theta\).
As the algebraic structure of \(\mathcal{O}\), it is automatically encoded in the \(\mathcal{O}_X\)-module structure of \(M\).
Remarks on multiplication
As a little warning, the \(f\theta\) in \(\nabla_{f\theta}\) is the \(\mathcal{O}\mathrel{\circlearrowright}\Theta\) structure and not the multiplication in \(D_X\). In \(D_X\) the multiplications \(f\theta\) is different from \(\theta f\), on ly the former equals the action of \(\mathcal{O}\mathrel{\circlearrowright}\Theta\), the latter is the action of multiplying \(f\) followed by \(\theta\).
Remarks on terminologies
Although the term integrable connections and flat connections are the same, in the theory of \(D_X\)-modules people usually use integrable connections to mean something stronger, requiring an extra condition of being locally free and finitely generated. The category of such integrable connections are denoted by \(\mathrm{Conn}(X)\). They are the most important examples of left \(D_X\)-modules. You can also say they are vector bundles with flat connections.
Left and Right D-modules
Recall that given a left \(G\)-module \(M\), we can use the anti-homomorphism \((\cdot)^{-1}:G^{op}\to G\) to obtain a right \(G\)-module \(M\) by acting \(mg = g^{-1}m\). On a general non-commutative ring \(R\) however, there is no obvious anti-homomorphism and thus no obvious way to get a right \(R\)-module from a left \(R\)-module. However, for \(D_X\)-modules, this is possible due to the following adjoint anti-involution.
Adjoint Anti-involution.
Let \(U\) be affine with coordinates \(\{x_i, \partial_i\}_{1\le i\le n}\), then the anti-involution \((\cdot)^\dagger : D_U\to D_U\) given by \[(a_\alpha \partial^\alpha)^\dagger = (-\partial)^\alpha a_\alpha \] defines a anti-involution on \(D_X\), i.e. \((PQ)^\dagger = Q^\dagger P^\dagger\).
Note that this involution map depends on the choice of coordinate.
Proof. It suffices to verify \((a\partial^\alpha b\partial^\beta)^\dagger = (b\partial^\beta)^\dagger (a\partial^\alpha)^\dagger = (-\partial)^\beta b (-\partial)^\alpha a\). This is a direct computation. \[(a\partial^\alpha b\partial^\beta)^\dagger = \left(a\sum_{\gamma} \binom{\alpha}{\gamma} \partial^\gamma(b) \partial^{\alpha-\gamma+\beta}\right)^\dagger = \sum_\gamma \binom{\alpha}{\gamma} (-\partial)^{\alpha-\gamma+\beta} a\partial^\gamma (b)\] and by the formulas in previous examples we have \[(-\partial)^\beta b (-\partial)^\alpha a = \sum_{\gamma} \binom{\alpha}{\gamma} \left(M_{(-\partial)^{\alpha-\gamma}} C_{\partial^\gamma} (-\partial)^\beta b\right) \cdot a. = \sum_{\gamma} \binom{\alpha}{\gamma} (-\partial)^{\alpha-\gamma+\beta} \partial^\gamma (b) a.\] ◻
Corollary.
Any left \(D_X\)-module \(M\) can be viewed as a \(D_X^{op}\)-module or equivalently a right \(D_X\)-module by \[s P := P^\dagger s.\]
Recall that left \(D_X\)-modules are \(\mathcal{O}_X\)-modules with a flat-connection. However right \(D_X\)-modules are [not]{.underline} connections. To understand what it does, we can examine the relations
\(s(f\theta) = (sf)\theta = (fs)\theta \quad\Leftrightarrow \nabla_{f\theta}s = \nabla_\theta(fs)\),
\(s(\theta f) = s(f\theta+\theta(f)) = (s\theta)f \quad\Leftrightarrow \nabla_{f\theta}s + \theta(f)s = f\nabla_\theta s\).
\(s[\theta_1, \theta_2] = s(\theta_1\theta_2 - \theta_2\theta_1) \quad \Leftrightarrow \nabla_{[\theta_1,\theta_2]}s = [\nabla_{\theta_2}, \nabla_{\theta_1}]s\).
These relations seem a bit weird, in general we use \(\nabla'=-\nabla\) to simplify them as
\(\nabla'_{f\theta}s = \nabla'_\theta (f s)\),
\(\nabla'_{f\theta}s = f\nabla'_\theta s + \theta(f)s \quad\Leftrightarrow \nabla'_\theta (fs) = \theta (f)s + f\nabla'_\theta s\),
\(\nabla'_{[\theta_1,\theta_2]}s = [\nabla'_{\theta_1}, \nabla'_{\theta_2}]s\).
So most relations are similar to that of a connection except the first one. This is infact a Lie derivative. On the top differential \(\Omega_X = \wedge \Omega^1_X\) there is a natural action of \(\Theta_X\) by Lie derivative, \[(\mathcal{L}_\theta \omega) (\theta_{1\dots n}) = \theta (\omega(\theta_{1\dots n})) - \sum_i \omega(\dots,[\theta,\theta_i],\dots),\] which satisfy
\(\mathcal{L}_{f\theta}\omega = \mathcal{L}_\theta (f\omega)\),
\(\mathcal{L}_\theta (f\omega) = \theta(f)\omega + f\mathcal{L}_\theta \omega\),
\(\mathcal{L}_{[\theta_1,\theta_2]}\omega = [\mathcal{L}_{\theta_1}, \mathcal{L}_{\theta_2}]\omega\).
Given such \(\nabla'\) or \(\mathcal{L}\), the right \(D_X\)-module structure is given by \[s\theta := -\nabla'_\theta s.\] i.e. \[\{D_X^{op}\text{-modules}\} \leftrightarrow \{\text{Lie derivatives on }\mathcal{O}_X\text{-modules}\}\] \[M\mapsto (M, \nabla'_\theta s := -s\theta), \quad M\leftrightarrow (M, \nabla')\]
Right Module Structure on Top Differential \(\Omega_X\)
In terms of local coordinates, the right module structure on \(\Omega_X\) is given by \[f \,\mathrm{d}x_1\wedge \cdots \wedge \,\mathrm{d}x_n P = (P^\dagger f) \,\mathrm{d}x_1\wedge \cdots \wedge \,\mathrm{d}x_n.\]
The right \(D_X\) module structure on \(\Omega_X\) gives a map \(D^{op}_X\to \mathrm{SEnd}_\mathbb{C}(\Omega_X)\) of \(\mathbb{C}_X\) algebras, by the locally-freeness of \(\Omega_X\) we have an isomorphism of sheaves of rings \[\mathrm{SEnd}_{\mathbb{C}}(\Omega_X)\cong \Omega_X\otimes_{\mathcal{O}_X} \mathrm{SEnd}_\mathbb{C}(\mathcal{O}_X) \otimes_{\mathcal{O}_X} \Omega_X^{\vee}.\] which gives a canonical isomorphism of \(\mathbb{C}_X\)-algebras \[D_X^{op}\cong \Omega_X \otimes_{\mathcal{O}_X} D_X \otimes_{\mathcal{O}_X} \Omega_X^{\vee}.\] This is seen by computing the image of \(D_X^{op}\) by the previous results on the right actions of \(D_X^{op}\) on local coordinates of the top differentials.
Remarks On Why Right Module Structure
One may wonder why \(\Omega_X\) is a right \(D_X\)-module and not a left \(D_X\)-module. One explanation is that the right module structure is naturally given by the Lie derivative. To see why using \[f\,\mathrm{d}x_1\wedge \cdots \wedge \,\mathrm{d}x_n \mapsto P f \,\mathrm{d}x_1\wedge \cdots \wedge \,\mathrm{d}x_n\] will not work, we can see how coordinate transformation laws determine the right module structure and not the left in the example of \(\Omega_X^1\) in the \(1\)-dimensional case. Consider two different coordinates \(x,y\), and assume \[\omega = a \,\mathrm{d}x = b \,\mathrm{d}y, \quad a = b y'\] and we compute the action of \[P = \frac{\partial }{\partial x} = y'\frac{\partial }{\partial y}.\]
If \(\Omega_X^1\) is a left \(D_X\)-module, we would expect \[P\omega = \frac{\partial a}{\partial x} \,\mathrm{d}x = y'\frac{\partial b}{\partial y} \,\mathrm{d}y, \quad \frac{\partial a}{\partial x} = (y')^2 \frac{\partial b}{\partial y}\] which is in general not true.
For the right action, we should expect \[\omega P = -\frac{\partial a}{\partial x} \,\mathrm{d}x = -\frac{\partial }{\partial y} ( y' b ) \,\mathrm{d}y, \quad \frac{\partial a}{\partial x} = y' \frac{\partial }{\partial y} (y'b),\] this is exactly \(\frac{\partial a}{\partial x} = y'\frac{\partial a}{\partial y}\).
Another curious question is, why isn’t \(\Omega^1\) a \(D\) or \(D^{op}\) module? Why is only top differential \(\Omega_X\) a right module? The above example might provide an intuitive explanation, only in the top differential, we can divide two differentials and get a function. When \(\dim X > 1\), the quotient \(\,\mathrm{d}y / \,\mathrm{d}x\) no longer make sense since \(\mathfrak{m}/\mathfrak{m}^2\) is not \(1\)-dimensional. But \(\bigwedge^n \mathfrak{m}/\mathfrak{m}^2\) is.
Tensor and Hom Over \(\mathcal{O}\)
Given left \(D_X\) modules \(M,N\), right \(D_X\)-modules \(M',N'\), we have the following module structures
These results might be a bit surprising, but remember that we are tensoring over \(\mathcal{O}\) and so there are some subtleties. It is automatically \(\mathcal{O}\)-module, to become \(D\)-module three extra conditions need to satisfy. To check the module structure quickly we can test the relation of \(f\theta\) acting on left or right to see if \((f\theta) m = f(\theta m)\) or \(m(f\theta) = (m f)\theta\), this works a lot of times.
For example we can see why \(M\otimes_\mathcal{O}N'\) with the left \(D\)-structure given by \[\theta(m\otimes n) = \theta m\otimes n - m\otimes n\theta\] does not work by observing that \[(f\theta)(m\otimes n) = f\theta m\otimes n - m\otimes nf\theta\] while \[f(\theta(m\otimes n)) = f\theta m\otimes n - m\otimes n\theta f.\] they differ by a term \(\theta(f)m\otimes n\).
There is another interesting fact, if \(X\) is a smooth curve with genus \(g\), we have a \(D_X\)-module \(\deg\mathcal{O}= 0\) and a right \(D_X\)-module \(\deg\Omega_X = 2g-2\). In fact it is proved that a line bundle is equippable with \(D_X\)-module structure iff \(\deg\mathcal{L}=0\), and with right \(D_X\)-module structure iff \(\deg\mathcal{L}= 2g-2\).
Remarks on dual
Note that for a left \(D_X\)-module \(M\), the \(\mathcal{O}\)-dual \(M^* = Hom_\mathcal{O}(M,\mathcal{O})\) is still a left \(D_X\)-module!
This might sound weird, but remember that we are taking dual over \(\mathcal{O}\). If you take the \(D_X\)-dual \(M^\vee = \mathrm{Hom}_{D_X}(M,D_X)\), then this is a right \(D_X\)-module.
2+1 Mixed Associativity
Let \(M, N\) and \(M'\) be two left and one right \(D_X\)-modules. There is a canonical isomorphism of \(\mathbb{C}_X\)-modules \[(M'\otimes_{\mathcal{O}_X} N)\otimes_{D_X} M \cong M'\otimes_{D_X} (M\otimes_{\mathcal{O}_X} N) \cong (M'\otimes_{D_X} M)\otimes_{\mathcal{O}_X} N.\] Note that all three of them are of the form \(V_D\otimes_D {}_DW\).
Category of left and right D-modules are equivalent
There is an equivalence of categories \[\{D_X\text{-modules}\} \leftrightarrow \{D_X^{op}\text{-modules}\}.\] given by the two functors \[\Omega_X\otimes_{\mathcal{O}_X}(\cdot) : {}_{D_X}\mathbf{Mod} \to {}_{}\mathbf{Mod}_{D_X}, \quad {\mathcal{H}om}_{\mathcal{O}_X}(\Omega_X, \cdot) : {}_{}\mathbf{Mod}_{D_X} \to {}_{D_X}\mathbf{Mod}.\] In fact here \(\Omega^{\vee}_X\otimes_{\mathcal{O}_X}(\cdot) = {\mathcal{H}om}_{\mathcal{O}_X}( \Omega_X, \cdot)\).
Inverse Images
Given a morphism \(f:X\to Y\) of smooth varieties, we can pushforward tangent vectors and pullback cotangent vectors naturally. Therefore we have the tangent map \[T(f) : \Theta_X \to f^*\Theta_Y = \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\Theta_Y\] and the cotangent map \[T^*(f) : \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\Omega^1_Y = f^*\Omega^1_Y \to \Omega^1_X\] given by \(a(x,y)\,\mathrm{d}y\mapsto a(x,f(x))\,\mathrm{d}f(x)\) naturally. In fact the tangent map is the dual of the cotangent map, \[\begin{aligned} {\mathcal{H}om}_{\mathcal{O}_X}(\Omega^1_X, \mathcal{O}_X) &\to {\mathcal{H}om}_{\mathcal{O}_X}(f^*\Omega^1_Y,\mathcal{O}_X) \\ &= f^*{\mathcal{H}om}_{\mathcal{O}_Y}(\Omega^1_Y, \mathcal{O}_Y). \end{aligned}\] The latter equality comes from the canonical map \[f^*{\mathcal{H}om}_{\mathcal{O}_Y}(\mathcal{F},\mathcal{G}) \to {\mathcal{H}om}_{\mathcal{O}_X}(f^*\mathcal{F},f^*\mathcal{G})\] which is an isomorphism when \(\mathcal{F}\) is locally free and finitely generated, satisfied by \(\Omega^1_Y\).
These natural operations allow us to define a \(D_X\)-module structure on a pullback \(f^*M\) of a \(D_Y\)-module \(M\) by \[f^*: {}_{D_Y}\mathbf{Mod} \to {}_{D_X}\mathbf{Mod}, \quad M\mapsto f^*M,\] \[\nabla_\theta (\psi \cdot s\circ f) = \theta\psi + (\nabla_{T_f\theta} s)\circ f.\] To be precise, let \(\psi\otimes s\in \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}M\), then for \(\theta\in \Theta_X\) we define \[\theta(\psi\otimes s) = \theta\psi \otimes s + \psi\otimes T_f\theta s.\] Given local good coordinate system \(\{y_i,\partial_{y_i}\}\) on \(Y\), the tangent map can be computed as \[T_f(\theta) = \sum_i \theta(y_i\circ f) \partial_{y_i},\] so \[\theta(\psi\otimes s) = \theta\psi\otimes s + \sum_i \psi \theta(y_i\circ f)\otimes \partial_{y_i} s.\]
As such, any \(M\in {}_{D_Y}\mathbf{Mod}\) gives a \(D_X\)-module \(f^*M = \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}M\). The pullback \[f^*D_Y= \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}D_Y\] is therefore a left \(D_X\) module. But at the same time it is a right \(f^{-1}D_Y\) module, and the two module structures are compatible (note that the left module structure acts on both components, but the right module structure only acts on the right component). We denote this \((D_X,f^{-1}D_Y)\)-module by \(D_{X\to Y}\).
Proposition.
Given a morphism \(f:X\to Y\) of smooth varieties, the pullback functor \[f^*: {}_{D_Y}\mathbf{Mod} \to {}_{D_X}\mathbf{Mod}, \quad M\mapsto f^*M\] also given by \(M\mapsto D_{X\to Y}\otimes_{f^{-1}D_Y} f^{-1}M\) by the associativity of tensor product.
Proof.
\[\begin{aligned} D_{X\to Y}\otimes_{f^{-1}D_Y} f^{-1}M &= (\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}D_Y)\otimes_{f^{-1}D_Y} f^{-1}M \\ &= \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}M. \end{aligned}\]◻
Direct Images
Left tensoring \(D_{X\to Y} = \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}D_Y\) allows us to pullback \(D_Y\) modules to \(D_X\)-modules. What about tensoring \(D_{X\to Y}\) on the right? Intuitively this could pushforward a right \(D_X\) module \(M\) into a right \(f^{-1}D_Y\)-module \(M\otimes_{D_X} D_{X\to Y}\). Pushing forward again we reach a \(D_Y\) module \[{}_{}\mathbf{Mod}_{D_X}\to {}_{}\mathbf{Mod}_{D_Y}: M\mapsto f_*(M\otimes_{D_X}D_{X\to Y}).\] But this is not very homologically friendly since it involves both a left exact functor \(f_*\) and a right exact functor \(\otimes\). We will give the right answer in terms of derived categories later, here we explain the above process can be used to pushforward left \(D_X\)-modules as well.
It suffices to use the categorical equivalence of left and right \(D\)-modules given by \(\Omega_X\) and \(\Omega_Y\) i.e. we map \[M\mapsto \Omega_Y^{\vee}\otimes_{\mathcal{O}_Y} f_*((\Omega_X\otimes_{\mathcal{O}_X}M)\otimes_{D_X} D_{X\to Y}).\] From the isomorphism \[(\Omega_X \otimes_{\mathcal{O}_X} M)\otimes_{D_X} D_{X\to Y} \cong (\Omega_X\otimes_{\mathcal{O}_X} D_{X\to Y})\otimes_{D_X} M\] note that the right \(D_X\)-module structure and right \(f^{-1}D_Y\)-module structure on \(\Omega_X\otimes_{\mathcal{O}_X}D_{X\to Y}\) are compatible, the above isomorphism is actually an isomorphism of \(f^{-1}D_Y\)-modules.
Definition.
Define the following \((f^{-1}D_Y; D_X)\)-bimodule \[D_{Y\leftarrow X} = (\Omega_X\otimes_\mathcal{O}D_{X\to Y}) \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\Omega_Y^\vee.\] Intuitively, this is just changing the direction of two \(D\)-module structures by tensoring with corresponding functors.
This gives a clean definition of the pushforward functor for left \(D_X\)-modules \[f_*: {}_{D_X}\mathbf{Mod} \to {}_{D_Y}\mathbf{Mod}, \quad M\mapsto f_*(D_{Y\leftarrow X}\otimes_{D_X} M).\]
Some Categories of \(D\)-Modules
Our module \(D_X\) is locally free and thus quasi-coherent over \(\mathcal{O}_X\). These quasi-coherent sheaves are fundamental in algebraic geometry, and we shall mainly deal with \(D_X\)-modules that are quasi-coherent.
The category of quasi-coherent \(\mathcal{O}_X\)-modules is denoted by \({}_{}\mathbf{Mod}_{qc}(\mathcal{O}_X)\). For the special case of affine varieties,
the global sections functor \[\Gamma(X,) : {}_{}\mathbf{Mod}_{qc}(\mathcal{O}_X) \to {}_{}\mathbf{Mod}(\Gamma(X,\mathcal{O}_X))\] is exact.
If \(\Gamma(X,M)=0\) for \(M\in {}_{}\mathbf{Mod}_{qc}(\mathcal{O}_X)\), then \(M=0\).
The category of quasi-coherent \(D_X\)-modules is denoted by \({}_{}\mathbf{Mod}_{qc}(D_X)\). This is an abelian category. We say \(X\) is \(D\)-affine if
The global sections functor \[\Gamma(X,) : {}_{}\mathbf{Mod}_{qc}(D_X) \to {}_{}\mathbf{Mod}(\Gamma(X,D_X))\] is exact.
If \(\Gamma(X,M)=0\) for \(M\in {}_{}\mathbf{Mod}_{qc}(D_X)\), then \(M=0\).
Let \(X\) be \(D\)-affine, then
\(M\in {}_{}\mathbf{Mod}_{qc}(D_X)\) is generated by global sections.
\(\Gamma(X,): {}_{}\mathbf{Mod}_{qc}(D_X)\simeq {}_{}\mathbf{Mod}(\Gamma(X,D_X))\) is an equivalence of categories.
Inverse Images and Direct Images Using Derived Categories
We shall define several functors on derived categories of \(D\)-modules and study its fundamental properties.
Recall the following fundamental lemma about the category of modules over a sheaves of rings.
Lemma. Let \(\mathcal{O}\) be a sheaf of rings on a topological space \(X\), for the category \({}_{}\mathbf{Mod}(\mathcal{O})\), we have
Any object \(M\) can be embedded into an injective object \(M\to I\).
Any object can be is a quotient of a flat object \(F\to M\).
This means any object \(M^\bullet\in D^+(\mathcal{O})\) is quasi-isomorphic to a complex of injective modules in \(D^+(\mathcal{O})\), and any object in \(D^-(\mathcal{O})\) is quasi-isomorphic to a complex of flat modules in \(D^-(\mathcal{O})\).
The usual push-forward (direct image) functor \(f_*\) and its derived functor \(Rf_*\) in sheaf theory can be displayed and understood in the following diagram
For a morphism of algebraic varieties \(f:X\to Y\) and a sheaf of rings \(R_Y\) on \(Y\), \(Rf_*\) sends \(D^b(f^{-1}R_Y)\to D^b(R_Y)\) and commutes with direct sums. These come from familiar properties about abelian sheaves (in the case \(R=\mathbb{Z}_Y\) which is the second line in the above diagram).
Let \(X\) be smooth algebraic variety. As a note for notations, \(D^\sharp(D_X)\) is the derived category of sheaves of \(D_X\)-modules, while \(D^\sharp_{qc}\) and \(D^\sharp_c\) means the full subcategory of \(D^\sharp\) consisting of complexes with quasi-coherent or coherent cohomology. So that
any object of \(D^b(D_X)\) is represented by a bounded complex of flat \(D_X\)-module,
any object of \(D^b_{qc}(D_X)\) is represented by a bounded complex of locally projective quasi-coherent \(D_X\)-modules.
Theorem (Equivalence Of Quasi-coherent Cohomology)
Let \(X\) be smooth. The inclusion functors \[D^b({}_{}\mathbf{Mod}_{qc}(D_X)) \to D^b_{qc}(D_X)\] \[D^b({}_{}\mathbf{Mod}_{c}(D_X)) \to D^b_{c}(D_X)\] from the complexes of quasi-coherent objects to the complexes having quasi-coherent cohomologies, are categorical equivalences.
Let \(f:X\to Y\) be a morphism of smooth algebraic varieties, we can define the left derived functor \(Lf^*\) using flat resolution, \[Lf^* : D^b(D_Y)\to D^b(D_X)\quad M^\bullet \mapsto D_{X\to Y}\otimes_{f^{-1}D_Y} f^{-1}M^\bullet,\] and it preserves \(D^b_{qc}(D_Y)\to D^b_{qc}(D_X)\). Note that it does not necessarily preserve \(D^b_c\) to \(D^b_c\), since when \(f\) is a closed embedding with \(\dim X<\dim Y\), \(D_{X\to Y}\) can be of infinite rank.
The functor \(Lf^*\) defined above is called the inverse image functor. In practice the shifted inverse image functor is more useful \[f^\dagger := Lf^*[\dim X-\dim Y] : D^b(D_Y)\to D^b(D_X).\] (you can think of the shift as matching their top cohomology).