Derived Algebraic Geometry

⚡ Key Takeaways

Derived Algebraic Geometry (DAG) enriches classical algebraic geometry by replacing commutative rings with derived/homotopical commutative rings (simplicial commutative rings, connective $E_\infty$ -rings, or commutative differential graded algebras) and schemes with derived schemes and derived stacks. Key result in words: intersections, quotients, and mapping spaces that are "badly behaved" classically (non-transverse intersections, non-flat fibers, infinite-dimensional moduli) acquire correct, homotopy-invariant structure carrying higher cohomological data, so that constructions like the cotangent complex, virtual fundamental classes, and deformation theory become functorial and exact. It applies to enumerative geometry, intersection theory, geometric representation theory, mathematical physics (TQFT, BV quantization), and the geometric Langlands program. Developed chiefly by Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi (2004–2014), building on ideas of Grothendieck, Quillen, Illusie, Deligne, Kontsevich, and Kapranov–Ciocan-Fontanine.

💡 The Core Insight

Classical algebraic geometry is built on the 0-truncated world: a commutative ring $A$ is just a set with operations, and $\mathrm{Spec}\,A$ remembers only $\pi_0$ . But the natural operations of geometry — tensor product (= intersection), invariants (= quotient), limits and colimits — are not exact. The derived fix is to work in a setting where these operations are automatically derived: tensor product becomes derived tensor product, fixed points become homotopy fixed points, fiber products become homotopy pullbacks. A derived scheme is a pair $(\mathcal{X}, \mathcal{O}_\mathcal{X})$ where $\mathcal{O}_\mathcal{X}$ is a sheaf of derived rings whose higher homotopy groups $\pi_i\mathcal{O}_\mathcal{X}$ are coherent $\pi_0\mathcal{O}_\mathcal{X}$ -modules. The slogan: "geometry over derived rings, done $\infty$ -categorically, makes every functor exact."

DefinitionSimplicial Commutative Rings and Connective

E_\infty

-Rings

A simplicial commutative ring is a simplicial object $A_\bullet$ in the category $\mathrm{CRing}$ of commutative rings, i.e. a functor $A_\bullet : \Delta^{\mathrm{op}} \to \mathrm{CRing}$ . Its homotopy groups $\pi_i A := \pi_i(A_\bullet)$ (taking $A_\bullet$ as a simplicial set) form a graded-commutative ring with $\pi_0 A$ an ordinary commutative ring and each $\pi_i A$ a $\pi_0 A$ -module. The $\infty$ -category $\mathrm{sCAlg}$ of simplicial commutative rings is obtained by inverting weak equivalences. In characteristic zero there is an equivalence

1

\quad \mathrm{sCAlg}_{\mathbb{Q}} \;\simeq\; \mathrm{cdga}^{\leq 0}_{\mathbb{Q}},

where $\mathrm{cdga}^{\leq 0}_{\mathbb{Q}}$ denotes connective (non-positively graded, cohomological convention) commutative differential graded $\mathbb{Q}$ -algebras, via the Dold–Kan normalization. Derivation hint: The Dold–Kan correspondence equates simplicial $\mathbb{Q}$ -modules with non-positively graded chain complexes; promoting it to the commutative-algebra level requires characteristic zero because the symmetric powers used to define the multiplication on $N(A_\bullet)$ involve dividing by symmetric-group orders. A connective $E_\infty$ -ring $R$ is a commutative algebra object in the symmetric monoidal $\infty$ -category of connective spectra; its homotopy ring $\pi_*R = \bigoplus_{n\geq 0}\pi_n R$ is graded-commutative. There is a forgetful functor $\mathrm{sCAlg}\to \mathrm{CAlg}^{\mathrm{cn}}$ which is an equivalence rationally but not integrally.

⚠️ Common mistake

Simplicial commutative rings and connective $E_\infty$ -rings coincide rationally but differ over $\mathbb{Z}$ and in positive characteristic. The simplicial theory is "polynomial" — generated under colimits by polynomial rings $\mathbb{Z}[x_1,\dots,x_n]$ — and its free algebras involve divided symmetric powers $\mathrm{Sym}$ ; the $E_\infty$ theory uses genuine homotopy-coherent commutativity and its free algebra on a spectrum involves the extended powers $(X^{\wedge n})_{h\Sigma_n}$ . Lurie's foundational text Higher Algebra / Spectral Algebraic Geometry primarily uses $E_\infty$ -rings; Toën–Vezzosi's Homotopical Algebraic Geometry uses simplicial commutative rings. They are different theories — "derived" vs "spectral" algebraic geometry.

DefinitionDerived Scheme

A derived scheme is a pair $\mathcal{X} = (X, \mathcal{O}_\mathcal{X})$ where:

$X$ is a topological space (or, more precisely, $\mathcal{X}$ is an $\infty$ -topos, or a sheaf on the site of derived affines),
$\mathcal{O}_\mathcal{X}$ is a sheaf of derived rings on $X$ ,

such that: 1. The truncation $t_0\mathcal{X} := (X, \pi_0\mathcal{O}_\mathcal{X})$ is an ordinary scheme; 2. Each homotopy sheaf $\pi_i\mathcal{O}_\mathcal{X}$ is a quasi-coherent $\pi_0\mathcal{O}_\mathcal{X}$ -module. Equivalently, a derived scheme is a functor

2

\quad \mathcal{X} : \mathrm{sCAlg} \longrightarrow \mathcal{S}, \qquad A \longmapsto \mathcal{X}(A),

from simplicial commutative rings to the $\infty$ -category $\mathcal{S}$ of spaces (animae), satisfying étale descent and admitting a cover by derived affine schemes $\mathrm{Spec}\,A := \mathrm{Map}_{\mathrm{sCAlg}}(A,-)$ . Derivation hint: This is the functor-of-points perspective. A scheme is determined by what it does to test rings; pass from $\mathrm{CRing}$ to $\mathrm{sCAlg}$ and from $\mathrm{Set}$ to $\mathcal{S}$ to obtain its derived analog. Descent encodes the gluing axiom in the homotopy-coherent sense.

TheoremDerived Fiber Products Compute Tor

Let $A \to C \leftarrow B$ be maps of ordinary commutative rings. The derived tensor product $A \otimes^{\mathbb{L}}_C B \in \mathrm{sCAlg}$ has homotopy groups

3

\quad \pi_n\!\left(A \otimes^{\mathbb{L}}_C B\right) \;\cong\; \mathrm{Tor}^C_n(A,B).

Consequently the derived fiber product of affine schemes is

4

\quad \mathrm{Spec}\,A \times^{\mathbf{h}}_{\mathrm{Spec}\,C} \mathrm{Spec}\,B \;=\; \mathrm{Spec}\!\left(A \otimes^{\mathbb{L}}_C B\right),

whose classical truncation is the ordinary fiber product $\mathrm{Spec}(A\otimes_C B)$ , while the higher $\pi_n$ measure the failure of transversality / Tor-independence. Derivation hint: Resolve $A$ (or $B$ ) by a flat/free simplicial $C$ -algebra resolution $P_\bullet \xrightarrow{\sim} A$ ; then $A\otimes^{\mathbb{L}}_C B := P_\bullet\otimes_C B$ and its homotopy groups are by definition the derived-functor Tor. The truncation collapses to $\pi_0 = A\otimes_C B$ .

DefinitionThe Cotangent Complex

For a map of derived rings $A \to B$ , the cotangent complex $L_{B/A} \in \mathrm{Mod}_B$ is the value at $B$ of the left-derived functor of Kähler differentials, characterized by the universal property that for any $B$ -module $M$ ,

5

\quad \mathrm{Map}_{\mathrm{Mod}_B}\!\left(L_{B/A}, M\right) \;\simeq\; \mathrm{Der}_A(B, M) \;\simeq\; \mathrm{fib}\!\left(\mathrm{Map}_{\mathrm{CAlg}_{A}/B}(B, B\oplus M) \to \ast\right),

where $B\oplus M$ is the square-zero extension and $\mathrm{Der}_A(B,M)$ denotes $A$ -linear derivations. It satisfies $\pi_0 L_{B/A} = \Omega^1_{\pi_0 B/\pi_0 A}$ and, for a complete intersection $B = A[x_1,\dots,x_n]/(f_1,\dots,f_r)$ ,

6

\quad L_{B/A} \;\simeq\; \Big[\, \textstyle\bigoplus_{j} B\cdot dg_j \;\xrightarrow{\,(\partial f_i/\partial x_j)\,}\; \bigoplus_i B\cdot dx_i \,\Big]

concentrated in cohomological degrees $-1$ and $0$ . Derivation hint: $L_{B/A}$ is computed by taking a simplicial polynomial resolution $P_\bullet\to B$ and forming $\Omega^1_{P_\bullet/A}\otimes_{P_\bullet}B$ . For a regular sequence the resolution is the Koszul complex, giving the two-term presentation; transversality is the statement that $H^{-1}=0$ .

ProofWhy Derived Intersection Carries the Right Dimension

We show that the derived self-intersection of a smooth subvariety has the expected virtual dimension, with corrections recorded in higher homotopy.

Step 1 (Setup). Let $Z \hookrightarrow X$ be a smooth closed subvariety of codimension $c$ in a smooth variety $X$ , with ideal sheaf $\mathcal{I}_Z$ . Classically, $Z\cap Z = Z$ as sets, but the scheme-theoretic self-intersection $Z\times_X Z$ is non-reduced and "wrong-dimensional."

Step 2 (Derived pullback). Form the derived fiber product $Z\times^{\mathbf{h}}_X Z$ . By equation [4] this is $\mathrm{Spec}$ of $\mathcal{O}_Z \otimes^{\mathbb{L}}_{\mathcal{O}_X}\mathcal{O}_Z$ .

Step 3 (Compute Tor via the Koszul complex). Since $Z\hookrightarrow X$ is a regular embedding, $\mathcal{O}_Z$ admits a Koszul resolution over $\mathcal{O}_X$ by $\wedge^\bullet N^\vee$ where $N = N_{Z/X}$ is the normal bundle. Tensoring,

7

\quad \pi_n\!\left(\mathcal{O}_Z\otimes^{\mathbb{L}}_{\mathcal{O}_X}\mathcal{O}_Z\right) \;\cong\; \mathrm{Tor}^{\mathcal{O}_X}_n(\mathcal{O}_Z,\mathcal{O}_Z) \;\cong\; \wedge^n N_{Z/X}.

Derivation hint: This is the HKR-type computation for a regular embedding; the Tor algebra of a regular embedding is the exterior algebra on the (shifted) normal bundle, a classical result generalizing the structure of the Koszul complex.

Step 4 (Interpretation). The derived self-intersection has classical truncation $Z$ but carries higher homotopy $\wedge^n N$ in degree $n$ . Its virtual dimension is

8

\quad \dim^{\mathrm{vir}}\!\left(Z\times^{\mathbf{h}}_X Z\right) \;=\; \dim Z - c \;=\; 2\dim Z - \dim X,

the expected dimension of an intersection of two copies of $Z$ in $X$ , even though the classical intersection has dimension $\dim Z$ . The discrepancy is precisely captured by the higher $\pi_n$ . $\blacksquare$ Derivation hint: Virtual dimension is the Euler characteristic of the cotangent complex restricted to a point, $\dim^{\mathrm{vir}} = \chi(L_{\mathcal{X}}^\vee)$ ; for the derived intersection the cotangent complex has the normal bundle appearing in degree $-1$ , lowering the virtual dimension by $c$ .

∎

mathematician · Jacob Lurie | b. 1977

Jacob Lurie reformulated the foundations of homotopy theory and algebraic geometry through his treatises Higher Topos Theory (2009), Higher Algebra, and Spectral Algebraic Geometry. His framework of $\infty$ -categories (quasi-categories, after Joyal) gave the technical machinery — a robust theory of limits, colimits, presentable $\infty$ -categories, and stable $\infty$ -categories — needed to make "everything derived" precise. His representability theorems generalize Artin's, providing criteria for when a moduli functor on $E_\infty$ -rings is a derived (spectral) stack, and his work on derived deformation theory established the equivalence between formal moduli problems and differential graded Lie algebras (jointly developed with the school of Hinich, Pridham, and Lurie's own DAG X). Lurie received the MacArthur Fellowship (2014) and the Breakthrough Prize in Mathematics (2014). His parallel co-developers, Bertrand Toën and Gabriele Vezzosi, built the simplicial-commutative-ring side under the name Homotopical Algebraic Geometry (HAG), providing a model-categorical and then $\infty$ -categorical foundation in their two-part series (2005, 2008).

🧠 Intuition

Think of an ordinary scheme as a silhouette — it records the set of solutions to equations but forgets how those solutions came to coincide. When two curves in a plane meet tangentially, the classical intersection number is computed correctly (via multiplicities), but the scheme you literally write down, $\mathrm{Spec}(\mathbb{C}[x,y]/(f,g))$ , "doesn't know" it should have been a transverse, dimension-zero intersection — it just looks fat at the tangency. The derived intersection adds a ghostly higher structure: in addition to the visible fat point, it records the "missing" relations as elements of $\pi_1, \pi_2, \dots$ of the structure sheaf. These ghosts are exactly what you need so that the Euler characteristic (virtual count) is invariant under deformation. A second mental picture: classical geometry is like measuring a function only at points; derived geometry is like measuring the function together with all its derivatives in a homotopy-coherent way. The cotangent complex $L_\mathcal{X}$ plays the role of the "first derivative" — for smooth $\mathcal{X}$ it is just the cotangent bundle in degree 0, but its degree $-1$ part detects how badly $\mathcal{X}$ fails to be smooth (obstructions), and its degree $+1$ part detects higher stacky / automorphism structure. The whole of DAG can be summarized as: do classical geometry, but every time you take a quotient or an intersection, keep the chain complex instead of just its homology.

Figure

Classical vs. derived intersection: the derived structure restores the expected virtual dimension by recording higher homotopy on the same underlying space.

ExampleDerived Intersection of a Conic and Tangent Line

Given. In $\mathbb{A}^2 = \mathrm{Spec}\,\mathbb{C}[x,y]$ , take the parabola $C: y - x^2 = 0$ and its tangent line at the origin $L: y = 0$ . We compute the derived intersection $C\times^{\mathbf{h}}_{\mathbb{A}^2} L$ .

Step 1. $\mathcal{O}_C = \mathbb{C}[x,y]/(y-x^2)$ , $\mathcal{O}_L = \mathbb{C}[x,y]/(y)$ . The classical intersection ring is

\mathbb{C}[x,y]/(y-x^2,\,y) = \mathbb{C}[x]/(x^2),

a length-2 fat point — the intersection multiplicity is 2 (Bézout, consistent with degree $2\times1$ ).

Step 2. Compute $\mathcal{O}_C\otimes^{\mathbb{L}}_{\mathbb{C}[x,y]}\mathcal{O}_L$ . Resolve $\mathcal{O}_C$ by the Koszul complex on the regular element $y-x^2$ :

9

\quad \big[\,\mathbb{C}[x,y]\xrightarrow{\,y-x^2\,}\mathbb{C}[x,y]\,\big] \xrightarrow{\ \sim\ } \mathcal{O}_C.

Derivation hint: $y-x^2$ is a non-zero-divisor, so the two-term Koszul complex is a free resolution of $\mathcal{O}_C$ .

Step 3. Tensor with $\mathcal{O}_L = \mathbb{C}[x]$ (setting $y=0$ ): the map $y-x^2$ becomes multiplication by $-x^2$ :

10

\quad \big[\,\mathbb{C}[x]\xrightarrow{\,-x^2\,}\mathbb{C}[x]\,\big].

Derivation hint: substitute $y=0$ into the Koszul differential.

Step 4. Read off homotopy:

\pi_0 = \mathrm{coker}(-x^2) = \mathbb{C}[x]/(x^2), \qquad \pi_1 = \ker(-x^2) = 0.

Boxed answer.

\boxed{\;C\times^{\mathbf{h}}_{\mathbb{A}^2}L = \mathrm{Spec}\big(\mathbb{C}[x]/(x^2)\big),\quad \pi_1 = 0,\ \text{length } = 2.\;}

Here the regular embedding is Tor-independent (the line and conic meet properly), so no higher homotopy appears — the derived and classical answers agree, and the multiplicity is correctly 2.

ExampleDerived Self-Intersection of the Origin in the Line

Given. Let $\{0\} = \mathrm{Spec}\,\mathbb{C} \hookrightarrow \mathbb{A}^1 = \mathrm{Spec}\,\mathbb{C}[x]$ . Compute the derived self-intersection $\{0\}\times^{\mathbf{h}}_{\mathbb{A}^1}\{0\}$ .

Step 1. $\mathcal{O}_{\{0\}} = \mathbb{C} = \mathbb{C}[x]/(x)$ . Resolve by the Koszul complex on $x$ :

11

\quad \big[\,\mathbb{C}[x]\xrightarrow{\,x\,}\mathbb{C}[x]\,\big]\xrightarrow{\ \sim\ }\mathbb{C}.

Derivation hint: $x$ is a regular element; the Koszul complex on one element resolves the quotient.

Step 2. Tensor with $\mathbb{C} = \mathbb{C}[x]/(x)$ , so $x \mapsto 0$ :

12

\quad \big[\,\mathbb{C}\xrightarrow{\,0\,}\mathbb{C}\,\big].

Derivation hint: the differential $x$ vanishes after setting $x=0$ , leaving the zero map.

Step 3. Homotopy groups:

\pi_0 = \mathbb{C}, \qquad \pi_1 = \mathbb{C}.

The $\pi_1$ is the normal "line" $N_{\{0\}/\mathbb{A}^1}\cong\mathbb{C}$ , in agreement with [7]: $\pi_1 = \wedge^1 N$ .

Boxed answer.

\boxed{\;\{0\}\times^{\mathbf{h}}_{\mathbb{A}^1}\{0\} = \mathrm{Spec}\big(\mathbb{C}\oplus \mathbb{C}[1]\big),\quad \pi_0=\mathbb{C},\ \pi_1 = N_{\{0\}/\mathbb{A}^1}=\mathbb{C}.\;}

The derived structure sheaf is the free divided-power / exterior algebra $\mathbb{C}[\epsilon]$ with $\epsilon$ in homotopy degree 1, $|\epsilon|=1$ . Virtual dimension $= 0 - 1 = -1$ , the expected dimension of intersecting two points in a line. The $\pi_1$ is the ghost that makes deformation invariance hold.

⚠️ Common mistake

"Derived geometry just means working with the derived category of sheaves." No. The derived category $D(X)$ of an ordinary scheme $X$ is a derived linear invariant of $X$ , but the underlying space $X$ is still classical. Derived algebraic geometry derives the space itself — the structure sheaf becomes a sheaf of derived rings, not merely a coefficient system. One can (and does) study quasi-coherent sheaves $\mathrm{QCoh}(\mathcal{X})$ on a derived scheme, which is a further enrichment.

⚠️ Common mistake

"The truncation $t_0\mathcal{X}$ loses no information." The truncation functor $t_0$ (taking $\pi_0$ ) has a fully faithful right adjoint embedding ordinary schemes into derived schemes, so classical schemes are genuinely a full subcategory. But for a non-classical derived scheme, $t_0\mathcal{X}$ discards all the higher $\pi_i$ — and these are exactly what controls deformation theory, virtual classes, and obstruction theory. Two very different derived schemes can have the same truncation.

CounterexampleQuotients break naive expectations

Consider the action of $\mathbb{G}_m$ on the affine plane $\mathbb{A}^2$ by scaling. The classical GIT quotient or the naive set-theoretic quotient is badly behaved (non-separated, or collapses orbits). The derived/stacky quotient $[\mathbb{A}^2/\mathbb{G}_m]$ is a smooth derived (here, classical Artin) stack whose cotangent complex

13

\quad L_{[\mathbb{A}^2/\mathbb{G}_m]} \;\simeq\; \big[\,\mathcal{O}\cdot\partial_t \to \Omega^1_{\mathbb{A}^2}\,\big]

has a degree $+1$ contribution from the Lie algebra of the stabilizer. Derivation hint: the fiber sequence $\mathfrak{g}\otimes\mathcal{O}\to L_{\mathbb{A}^2}\to L_{[\mathbb{A}^2/\mathbb{G}_m]}$ places the (co)Lie algebra in positive degree, the hallmark of a stacky direction; this is the dual phenomenon to the negative-degree obstructions of derived intersections. The lesson: negative homotopy degrees of $L$ signal derived (intersection/obstruction) structure, while positive degrees signal stacky (quotient/automorphism) structure. A derived Artin stack generically has both.

connection

Derived algebraic geometry sits at a crossroads of several theories. It is built on Infinity Categories and Model Categories for its homotopical foundations, and on Higher Topos Theory for its sheaf theory and descent. Its central computational tool, the cotangent complex, is the subject of Deformation Theory and is governed by the equivalence between Formal Moduli Problems and Differential Graded Lie Algebras (Lurie–Pridham). The theory of Quasi-Coherent Sheaves on derived stacks, together with Ind-Coherent Sheaves and the six-functor formalism, underpins applications to Geometric Langlands Program. The spectral variant — replacing simplicial rings by $E_\infty$ -ring spectra — is Spectral Algebraic Geometry and connects DAG to Chromatic Homotopy Theory and topological modular forms. The enumerative side links to Gromov–Witten Theory and Donaldson–Thomas Theory via virtual fundamental classes, and the symplectic-geometric refinement of DAG, shifted symplectic structures (PTVV), connects to Batalin–Vilkovisky Formalism in mathematical physics.

🔬 Where this appears

⏳ History

- 1961–1967 — Grothendieck, EGA/SGA. The functor-of-points philosophy and the systematic use of fiber products and flatness set the stage; Grothendieck's Pursuing Stacks (1983) and the cotangent-complex circle of ideas anticipate higher structures.

- 1967 — Quillen, Homotopical Algebra. Model categories provide the abstract machinery for "doing homotopy theory" in arbitrary settings, the indispensable foundation for derived rings.

- 1970s — Illusie, Complexe Cotangent et Déformations. André–Quillen homology and the cotangent complex

L_{B/A}

are developed, the first systematic derived construction in commutative algebra.

- 1990s — Kontsevich; Deligne; Drinfeld. Letters and lectures propose that moduli problems should be "derived" and that deformation theory in characteristic zero is controlled by dg-Lie algebras ("derived deformation theory"); virtual fundamental classes appear in Gromov–Witten theory (Li–Tian, Behrend–Fantechi, 1996–97).

- 2001 — Ciocan-Fontanine & Kapranov, Derived Quot schemes / dg-manifolds. Concrete constructions of derived moduli (dg-schemes, dg-manifolds) realizing the Kontsevich–Deligne vision.

- **2004–2008 — Toën & V

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development