Derived Categories
Derived categories¶
Example: Dualizing complex of a projective variety¶
Let \(X = \operatorname{Proj}(k[x,y,z,w]/(xy, xz)) \subset \mathbb{P}^3\) be the union of the plane \(V(x) \cong \mathbb{P}^2\) and the line \(V(y,z) \cong \mathbb{P}^1\). We compute the dualizing complex \(\omega_X^\bullet = \mathbf{R}\mathcal{H}om_{\mathcal{O}_{\mathbb{P}^3}}(\mathcal{O}_X, \omega_{\mathbb{P}^3}[3])\) and show that \(X\) is not Cohen--Macaulay.
Minimal free resolution. The ideal \((xy, xz) = (x) \cap (y,z)\) has two generators with a single syzygy \(z \cdot (xy) - y \cdot (xz) = 0\), giving the minimal resolution
Exactness at the middle term: the kernel of \(\begin{bmatrix} xy & xz \end{bmatrix}\) consists of pairs \((a,b)\) with \(x(ay + bz) = 0\). Since \(x\) is not a zero divisor in \(k[x,y,z,w]\), we need \(ay + bz = 0\), whose syzygies in \(k[y,z]\) are generated by \((z, -y)\).
Dualizing. Since \(\omega_{\mathbb{P}^3} = \mathcal{O}(-4)\) and the resolution \(\mathcal{L}^\bullet \xrightarrow{\sim} \mathcal{O}_X\) consists of locally free sheaves, we compute \(\mathbf{R}\mathcal{H}om(\mathcal{O}_X, \mathcal{O}(-4)[3])\) by applying \(\mathcal{H}om(-, \mathcal{O}(-4))\) to \(\mathcal{L}^\bullet\) and shifting by \([3]\). Dualizing reverses arrows and transposes matrices:
placed in cohomological degrees \(-3, -2, -1\) (after the shift by \([3]\), the term \(\mathcal{O}(-4) = \mathcal{H}om(\mathcal{O}, \mathcal{O}(-4))\) lands in degree \(-3\)).
Non-Cohen--Macaulay detection. The dualizing complex \(\omega_X^\bullet\) is a genuine complex spanning three cohomological degrees, rather than a single sheaf concentrated in one degree. For a Cohen--Macaulay scheme of pure dimension \(d\), the dualizing complex would be a single sheaf \(\omega_X\) in degree \(-d\). Here \(X\) has components of dimensions 2 and 1, so it is not equidimensional, hence not Cohen--Macaulay. Each component individually is Cohen--Macaulay (\(\mathbb{P}^2\) and \(\mathbb{P}^1\) are smooth), but their union fails the condition.
Example: Complexes with the same homology but not quasi-isomorphic¶
Over the polynomial ring \(R = \mathbb{C}[x,y]\), the two complexes
with rightmost term in degree 0, have isomorphic cohomology in every degree yet represent distinct objects in \(D(\operatorname{Mod}_R)\).
Cohomology. For \(\mathcal{L}^\bullet\): the cokernel of \(\begin{bmatrix} x & y \end{bmatrix}\) is \(R/(x,y) \cong \mathbb{C}\), so \(H^0(\mathcal{L}^\bullet) = \mathbb{C}\). The kernel consists of syzygies \((a,b)\) with \(ax + by = 0\); the syzygy module of \((x,y)\) in \(R\) is generated by \((y,-x)\), giving \(H^{-1}(\mathcal{L}^\bullet) \cong R\). For \(\mathcal{M}^\bullet\): the zero differential gives \(H^0(\mathcal{M}^\bullet) = \mathbb{C}\) and \(H^{-1}(\mathcal{M}^\bullet) = R\).
Non-quasi-isomorphism via splitting. The complex \(\mathcal{M}^\bullet\) has zero differential, so it splits in the derived category as a direct sum \(\mathcal{M}^\bullet \cong R[-1] \oplus \mathbb{C}[0]\). If \(\mathcal{L}^\bullet\) were quasi-isomorphic to \(\mathcal{M}^\bullet\), it would also split as \(R[-1] \oplus \mathbb{C}[0]\) in \(D(R)\). Such a splitting would correspond to a section \(\mathbb{C} \to \mathcal{L}^\bullet\) in the derived category, or equivalently, the vanishing of the extension class in \(\operatorname{Ext}_R^1(\mathbb{C}, R)\) that \(\mathcal{L}^\bullet\) represents.
We compute this Ext group via the Koszul resolution of \(\mathbb{C} = R/(x,y)\):
Applying \(\operatorname{Hom}_R(-, R)\): the complex \(R \xleftarrow{\begin{bmatrix} x \\ y \end{bmatrix}} R^{\oplus 2} \xleftarrow{\begin{bmatrix} -y & x \end{bmatrix}} R\) has \(H^1 = \operatorname{Ext}_R^1(\mathbb{C}, R) \cong R/(x,y) \cong \mathbb{C} \neq 0\). (More generally, by local duality, \(\operatorname{Ext}_R^i(\mathbb{C}, R) = 0\) for \(i < 2\) and \(\operatorname{Ext}_R^2(\mathbb{C}, R) \cong \mathbb{C}\) when \(R\) has dimension 2. But here we need \(\operatorname{Ext}^1\) in the derived category sense between \(H^0 = \mathbb{C}\) and \(H^{-1}[1] = R\), which is \(\operatorname{Ext}_R^1(\mathbb{C}, R)\).)
In fact, \(\operatorname{Hom}_{D(R)}(\mathcal{L}^\bullet, \mathcal{L}^\bullet) \not\cong \operatorname{Hom}_{D(R)}(\mathcal{M}^\bullet, \mathcal{M}^\bullet)\) since the latter decomposes (via the splitting) into four summands \(\operatorname{Ext}^*(R, R) \oplus \operatorname{Ext}^*(\mathbb{C}, \mathbb{C}) \oplus \operatorname{Ext}^*(\mathbb{C}, R) \oplus \operatorname{Ext}^*(R, \mathbb{C})\), while \(\mathcal{L}^\bullet\) carries a nontrivial extension class preventing such a decomposition. The derived category remembers not just cohomology objects but also the extension data connecting them.
Derived pushforward and higher direct images¶
Remark: Semismall maps and perverse sheaves¶
Consider the simplest Springer resolution \(f: X = T^*\mathbb{P}^1 \to Y = \operatorname{Spec}(k[x,y,z]/(xy - z^2))\), where \(Y\) is the \(A_1\)-surface singularity (the quadric cone). We verify that \(f\) is semismall and explain why \(\mathbf{R}f_*(\mathbb{Q}_X[2])\) is a perverse sheaf but does not decompose as the direct sum of its cohomology sheaves.
Semismallness. A proper morphism \(f: X \to Y\) between irreducible varieties is semismall if \(2k + \dim Y_k \leq \dim X\) for all \(k \geq 0\), where \(Y_k = \{y \in Y \mid \dim f^{-1}(y) \geq k\}\). It is small if strict inequality holds for \(k \geq 1\). For the Springer resolution: \(\dim X = 2\), the generic fibre is a point (\(k = 0\), \(Y_0 = Y\), \(\dim Y_0 = 2\), check: \(0 + 2 \leq 2\)), and the fibre over the singular point \(0 \in Y\) is \(\mathbb{P}^1\) (\(k = 1\), \(Y_1 = \{0\}\), \(\dim Y_1 = 0\), check: \(2 + 0 = 2 \leq 2\)). Since equality holds at \(k = 1\), the map \(f\) is semismall but not small.
Perversity. For a semismall map, \(\mathbf{R}f_*(\mathbb{Q}_X[\dim X])\) is a perverse sheaf. This is a much stronger constraint than being merely an object of \(D^b_c(Y)\): perverse sheaves form an abelian subcategory, and their simple objects (intersection cohomology complexes) carry deep topological information.
Non-splitting. We claim \(\mathbf{R}f_*(\mathbb{Q}_X[2])\) does not decompose as \(\bigoplus_i R^i f_*(\mathbb{Q}_X)[2-i]\) (the naive direct sum of shifted higher direct images). The higher direct images are: \(R^0 f_*(\mathbb{Q}_X) = \mathbb{Q}_Y\) (since \(f\) is birational with connected fibres), and \(R^2 f_*(\mathbb{Q}_X) = \mathbb{Q}_{\{0\}}\) (the skyscraper at the singular point, recording \(H^2(\mathbb{P}^1) = \mathbb{Q}\)). If the decomposition held, \(\mathbf{R}f_*(\mathbb{Q}_X[2])\) would be \(\mathbb{Q}_Y[2] \oplus \mathbb{Q}_{\{0\}}\), which is not a perverse sheaf (the summand \(\mathbb{Q}_Y[2]\) is not perverse on the singular space \(Y\)). The actual perverse sheaf \(\mathbf{R}f_*(\mathbb{Q}_X[2]) = IC(Y, \mathbb{Q})\) is the intersection cohomology complex, an indecomposable object encoding the nontrivial extension between the regular and singular strata.
Fourier--Mukai transforms on elliptic curves¶
Let \(E\) be an elliptic curve over \(\mathbb{C}\) with origin \(p_0\). The normalized Poincare bundle on \(E \times E\) is
satisfying \(\mathscr{P}|_{p \times E} = \mathcal{O}_E(p - p_0)\) and \(\mathscr{P}|_{E \times p} = \mathcal{O}_E(p - p_0)\). The Fourier--Mukai transform is
Theorem: Mukai¶
The Fourier--Mukai transform satisfies \(\Phi \circ \Phi = \iota^* \circ [-1]\), where \(\iota: E \to E\) is the negation map \(p \mapsto -p\) and \([-1]\) is the homological shift. In particular, \(\Phi\) is an autoequivalence of \(D^b(E)\) with \(\Phi^4 = [-2]\).
The proof computes the convolution kernel \(\mathscr{P} \circ \mathscr{P} = \mathbf{R}\pi_{13,*}(\pi_{12}^*\mathscr{P} \otimes \pi_{23}^*\mathscr{P})\) on \(E \times E \times E\) in four steps.
Step 1: Reducing the convolution. By Lemma ecag-0220, \(\pi_{12}^*\mathscr{P} \otimes \pi_{23}^*\mathscr{P} \cong f^*\mathscr{P}\) where \(f: E \times E \times E \to E \times E\) is \((x,y,z) \mapsto (x+z, y)\).
Step 2: Flat base change. The diagram
is cartesian, where \(m(x,z) = x + z\). Since \(\pi_1\) is flat (a projection), flat base change gives \(\mathbf{R}\pi_{13,*}(f^*\mathscr{P}) = m^*(\mathbf{R}\pi_{1,*}\mathscr{P})\).
Step 3: Computing \(\mathbf{R}\pi_{1,*}\mathscr{P}\). For \(p \neq p_0\), the restriction \(\mathscr{P}|_{p \times E} = \mathcal{O}_E(p - p_0)\) is a nontrivial degree-zero line bundle, so \(H^0 = H^1 = 0\) by Riemann--Roch (\(\chi = 0\) and \(H^0 = 0\) for a nontrivial degree-0 bundle on a genus-1 curve forces \(H^1 = 0\)). At \(p = p_0\), \(\mathscr{P}|_{p_0 \times E} = \mathcal{O}_E\) has \(H^0 = H^1 = k\). Cohomology and base change (applicable since \(\mathscr{P}\) is flat over \(E\) via \(\pi_1\)) gives:
Therefore \(\mathbf{R}\pi_{1,*}\mathscr{P} = \mathcal{O}_{p_0}[-1]\).
Step 4: Pulling back. The fibre \(m^{-1}(p_0) = \{(x,z) \mid x + z = p_0\} = \{(x, -x)\} = \Gamma_\iota\) is the graph of the negation map. So
The Fourier--Mukai transform with kernel \(\mathcal{O}_{\Gamma_\iota}\) is \(\iota^*\), so \(\Phi^2 = \iota^* \circ [-1]\).
Verification. Since \(\iota^2 = \operatorname{id}\) and \([-1]\) commutes with pullback, \(\Phi^4 = (\iota^* \circ [-1])^2 = [-2]\). A quasi-inverse is \(\Phi^{-1} = \iota^* \circ [1] \circ \Phi\).
Alternatively, \(\Phi\) can be shown to be an equivalence by verifying that it sends the "orthonormal basis" \(\{\mathcal{O}_p\}_{p \in E}\) of skyscraper sheaves to the family \(\{\mathcal{O}_E(p - p_0)\}\) of line bundles, preserving Ext groups (\(\operatorname{Ext}^i(\mathcal{O}_E(p'-p_0), \mathcal{O}_E(p-p_0)) = H^i(\mathcal{O}_E(p-p')) = 0\) for \(p \neq p'\) and \(i \neq 0,1\)) and commuting with the Serre functor (\(\mathscr{P}_x \otimes \omega_E = \mathscr{P}_x\) since \(\omega_E = \mathcal{O}_E\)).
Lemma: Divisor class identity on \(E \times E\)¶
For any closed point \(p \in E\), let \(D_p = \{(x,y) \in E \times E \mid x + y = p\}\) be the fibre of the addition map \(m: E \times E \to E\) over \(p\). Then
We verify this by the see-saw principle. On a product \(A \times B\) of complete varieties, two line bundles are isomorphic if their restrictions agree on all fibres \(\{a\} \times B\) and on at least one fibre \(A \times \{b_0\}\).
Restriction to \(\{q\} \times E\). The right side restricts to \(\mathcal{O}_E(p - p_0)\). For the left side: \(D_p \cap (\{q\} \times E) = \{(q, p-q)\}\) and \(D_{p_0} \cap (\{q\} \times E) = \{(q, p_0 - q)\}\), so
Restriction to \(E \times \{q\}\). Both sides restrict to \(\mathcal{O}_E(p - p_0)\) by the same computation with roles exchanged.
The see-saw principle gives the desired isomorphism.
Lemma: Convolution identity for the Poincare bundle¶
Let \(f: E \times E \times E \to E \times E\) be \((x,y,z) \mapsto (x+z, y)\). Then \(\pi_{12}^*\mathscr{P} \otimes \pi_{23}^*\mathscr{P} \cong f^*\mathscr{P}\) on \(E \times E \times E\).
We apply the see-saw principle on \(E \times E \times E\) viewed as a family over the second factor.
Restriction to \(E \times \{p\} \times E\). The bundle \(\pi_{12}^*\mathscr{P}\) restricts to \(\mathscr{P}|_{E \times p}\) pulled back via the first coordinate, which is \(\pi_1^*\mathcal{O}_E(p - p_0)\). Similarly \(\pi_{23}^*\mathscr{P}\) restricts to \(\pi_2^*\mathcal{O}_E(p - p_0)\) (where \(\pi_1, \pi_2\) now denote projections on the slice \(E \times E\)). The product is \(\pi_1^*\mathcal{O}_E(p-p_0) \otimes \pi_2^*\mathcal{O}_E(p-p_0)\).
For \(f^*\mathscr{P}\): the restricted map sends \((x,z) \mapsto (x+z, p)\), so
The key identifications are: \(f^*\mathcal{O}(\Delta)|_{E \times \{p\} \times E} = \mathcal{O}(D_p)\) (since \(\Delta = \{a = b\}\) pulls back under \((x,z) \mapsto (x+z, p)\) to \(\{x+z = p\}\)), and \(f^*\pi_1^*\mathcal{O}_E(-p_0)|_{E \times \{p\} \times E} = \mathcal{O}(-D_{p_0})\) (via \(\pi_1 \circ f = m\)). The factor \(f^*\pi_2^*\mathcal{O}_E(-p_0)\) restricts to \(\mathcal{O}_E(-p_0)\) pulled back from the constant map to \(p\), contributing trivially to the slice. Therefore \(f^*\mathscr{P}|_{E \times \{p\} \times E} = \mathcal{O}(D_p - D_{p_0})\), which equals \(\pi_1^*\mathcal{O}_E(p-p_0) \otimes \pi_2^*\mathcal{O}_E(p-p_0)\) by Lemma ecag-0219.
Restriction to \(\{p_0\} \times E \times \{p_0\}\). All three bundles \(\pi_{12}^*\mathscr{P}\), \(\pi_{23}^*\mathscr{P}\), and \(f^*\mathscr{P}\) restrict to \(\mathcal{O}_E\), since \(\mathscr{P}|_{p_0 \times E} = \mathscr{P}|_{E \times p_0} = \mathcal{O}_E\) and \(f(p_0, y, p_0) = (p_0, y)\).
The see-saw principle gives \(\pi_{12}^*\mathscr{P} \otimes \pi_{23}^*\mathscr{P} \cong f^*\mathscr{P}\).
Six functors¶
Example: Adjoint functors from poset inclusions¶
The inclusion of posets \(i: (\mathbb{Z}, \leq) \hookrightarrow (\mathbb{R}, \leq)\), viewed as a functor between thin categories, admits both a right adjoint (the floor function) and a left adjoint (the ceiling function).
For the right adjunction \(i \dashv \lfloor \cdot \rfloor\): in a poset category, \(\operatorname{Hom}(a,b)\) is a singleton when \(a \leq b\) and empty otherwise. The adjunction \(\operatorname{Hom}_{\mathbb{R}}(i(n), r) \cong \operatorname{Hom}_{\mathbb{Z}}(n, \lfloor r \rfloor)\) translates to
which is precisely the defining property of the floor function. For the left adjunction \(\lceil \cdot \rceil \dashv i\): the condition \(\operatorname{Hom}_{\mathbb{Z}}(\lceil r \rceil, n) \cong \operatorname{Hom}_{\mathbb{R}}(r, i(n))\) becomes
which defines the ceiling function.
This example exhibits the "best approximation" principle underlying adjunctions: \(\lfloor r \rfloor\) is the best integer approximation to \(r\) from below, while \(\lceil r \rceil\) is the best from above. In the six-functor formalism, the adjunctions \((f^*, f_*)\) and \((f_!, f^!)\) encode analogous approximation relationships between categories of sheaves on different spaces: \(f^*\) is the "closest pullback" to a given pushforward, and \(f^!\) is the "closest exceptional pullback" to a given proper pushforward.
Example: Six functors for an open-closed decomposition¶
Consider the stratification of \(\mathbb{C}\) (analytic topology) into the open complement \(\mathbb{C}^\times\) and the closed point \(\{0\}\):
The six-functor formalism on this complementary pair exhibits fundamental asymmetries between proper and non-proper embeddings.
Properness. The closed embedding \(j\) is proper (every closed immersion of finite type is proper), so \(j_* = j_!\). The open embedding \(i\) is not proper: the map \(\operatorname{Spec}(\mathbb{C}((t))) \to \mathbb{C}^\times\) defined by \(t \mapsto t\) extends to \(\operatorname{Spec}(\mathbb{C}[[t]]) \to \mathbb{C}\) (the closed point maps to \(0 \notin \mathbb{C}^\times\)), violating the valuative criterion. Geometrically, the sequence \(1/n \in \mathbb{C}^\times\) converges to \(0 \notin \mathbb{C}^\times\). For a non-proper open embedding, \(i_* \neq i_!\): the functor \(i_*\) extends by nearby sections while \(i_!\) extends by zero (compact support).
Vanishing identities. For a complementary open-closed pair, the exact triangles \(j_!j^! \to \operatorname{id} \to i_*i^*\) and \(i_!i^! \to \operatorname{id} \to j_*j^*\) yield:
- \(j^*i_! = 0\) and \(j^!i_* = 0\): extending from the open stratum and restricting to the closed stratum gives zero.
- \(i^*j_! = 0\) and \(i^!j_* = 0\): extending from the closed stratum and restricting to the open stratum gives zero.
Etale morphisms. For an open embedding \(i\), we have \(i^* = i^!\): both are restriction to the open subset. This holds because \(i\) is etale, so the relative dualizing sheaf \(\omega_{i}\) is trivial, and \(i^!(-) = i^*(-) \otimes \omega_i = i^*(-)\).
Summary. The identities and non-identities for this pair:
| Property | Holds? | Reason |
|---|---|---|
| \(j_* = j_!\) | Yes | \(j\) is proper |
| \(i_* = i_!\) | No | \(i\) is not proper |
| \(i^* = i^!\) | Yes | \(i\) is etale |
| \(j^*i_! = 0\) | Yes | open-closed vanishing |
| \(i^*j_* = 0\) | No | \(i^*j_* \neq 0\) in general |
The last entry deserves comment: \(i^*j_*\mathcal{F}\) is nonzero whenever the sections of \(j_*\mathcal{F}\) on open subsets meeting \(\{0\}\) restrict nontrivially to \(\mathbb{C}^\times\). The distinction between \(i^*j_! = 0\) and \(i^*j_* \neq 0\) reflects the difference between extension by zero and extension by sections from the closed stratum.
Remark: \(\mathcal{O}\)-modules vs. constructible sheaves under \(j^*i_*\)¶
For the open-closed pair \(\mathbb{C}^\times \xhookrightarrow{i} \mathbb{C} \xhookleftarrow{j} \{0\}\), the behavior of \(j^*i_*\) differs sharply depending on whether we work with \(\mathcal{O}\)-modules or constructible sheaves.
In the algebraic category, \(j^*i_*\mathcal{O}_{\mathbb{C}^\times} = 0\). The computation reduces to a tensor product of \(k[t]\)-modules:
since \(t\) acts as zero in \(k[t]/(t)\) and as a unit in \(k[t, t^{-1}]\): the localization \((k[t]/(t))_t = 0\) because the element \(1 \in k[t]/(t)\) satisfies \(t \cdot 1 = 0\), and inverting \(t\) forces \(1 = 0\).
In contrast, for the constant sheaf \(\mathbb{Q}_{\mathbb{C}^\times}\), the stalk \((i_*\mathbb{Q}_{\mathbb{C}^\times})_0 = H^0(D^\times, \mathbb{Q}) = \mathbb{Q} \neq 0\) (sections on a punctured disk are constant), so \(j^*i_*\mathbb{Q}_{\mathbb{C}^\times} \neq 0\).
Remark: Why \(\mathcal{O}\)-modules and local systems differ¶
The discrepancy between \(j^*i_*\mathcal{O}_{\mathbb{C}^\times} = 0\) and \(j^*i_*\mathbb{Q}_{\mathbb{C}^\times} \neq 0\) reflects a fundamental divide between algebraic and topological information.
Local systems and constructible sheaves are governed by the topology of the underlying space: they encode monodromy, nearby cycles, and the fundamental group. The stalk \((i_*\mathscr{L})_0\) for a local system \(\mathscr{L}\) on \(\mathbb{C}^\times\) involves sections on a small punctured disk, which can be nontrivial because \(\pi_1(\mathbb{C}^\times) \cong \mathbb{Z}\).
The structure sheaf \(\mathcal{O}\) imposes algebraic regularity: \(\mathcal{O}_{\mathbb{C}^\times} = \mathcal{O}_{\mathbb{C}}[t^{-1}]\) is obtained by inverting \(t\), and pulling back to \(\{0\}\) via \(j^*\) tensors with \(k[t]/(t)\), which annihilates every element because \(t\) simultaneously acts as a unit (in \(k[t, t^{-1}]\)) and as zero (in \(k[t]/(t)\)). The \(\mathcal{O}\)-module functor \(j^*i_*\) detects the scheme-theoretic structure of the closed point (the maximal ideal and its powers), while the constructible functor detects the topological complement.
D-modules and the Riemann--Hilbert correspondence bridge these two worlds: a D-module on \(\mathbb{C}\) encodes both the algebraic differential equation structure and the topological monodromy of its solutions, reconciling the vanishing of \(j^*i_*\mathcal{O}_{\mathbb{C}^\times}\) with the nonvanishing of \(j^*i_*\mathbb{Q}_{\mathbb{C}^\times}\).