Derived Algebraic Geometry
Derived algebraic geometry is a branch of algebraic geometry in which geometric spaces retain homological information that ordinary schemes and stacks suppress. Its fundamental objects are locally described by commutative algebra endowed with a compatible homotopy theory. This enrichment turns equations, intersections, deformation problems, and moduli constructions into objects whose higher homotopy groups record relations among relations.
The subject combines methods from scheme theory, homological algebra, and higher category theory. It is especially concerned with situations in which an ordinary fiber product has the incorrect deformation theory because the relevant morphisms fail to meet transversely. The derived fiber product corrects this defect by preserving the higher Tor functors generated by the intersection.
Foundational idea
An affine scheme is conventionally represented by a commutative ring (A) through the contravariant correspondence
[ A \longmapsto \operatorname{Spec}(A). ]
Derived algebraic geometry extends this correspondence by allowing (A) to carry homotopical information. One standard implementation uses simplicial commutative rings, whose simplicial structure presents coherent higher relations. In characteristic zero, a closely related implementation uses connective commutative differential graded algebras, with the differential encoding the relevant homological data. Modern treatments express the same principle through commutative algebra objects in a suitable symmetric monoidal infinity-category.
For a connective derived ring (A), the ordinary ring (\pi _0(A)) determines the classical affine scheme underlying (\operatorname{Spec}(A)). The higher homotopy groups (\pi_i(A)) form modules over (\pi_0(A)) and describe derived structure that is invisible after classical truncation. The operation
[ \operatorname{Spec}(A)\longmapsto \operatorname{Spec}\bigl(\pi_0(A)\bigr) ]
therefore forgets obstruction-theoretic information rather than merely replacing one notation with another.
Derived affine schemes are assembled by homotopical descent. The resulting geometric objects include derived schemes and derived algebraic stacks. Their functors of points take values in spaces rather than sets, so automorphisms and higher coherences remain intrinsic parts of the geometry.
Derived intersections
The local model for a derived intersection is the derived tensor product. Given ring homomorphisms (R\to A) and (R\to B), the classical fiber product of affine schemes is represented by (A\otimes_R B). Its derived replacement is represented by
[ A\otimes_R^{\mathbf L}B, ]
which retains the failure of (A) and (B) to be flat over (R). The corresponding geometric identity is
[ \operatorname{Spec}(A) \times^{\mathbf R}_{\operatorname{Spec}(R)} \operatorname{Spec}(B) \simeq \operatorname{Spec} \left(A\otimes_R^{\mathbf L}B\right). ]
The homotopy groups of the derived tensor product recover the classical Tor groups:
[ \pi_i\left(A\otimes_R^{\mathbf L}B\right) \cong \operatorname{Tor}^{R}_{i}(A,B), ]
subject to the grading conventions used in the chosen model. Thus a non-transverse intersection acquires higher structure whose algebra records its excess intersection multiplicity.
For example, let a closed subscheme (Z\hookrightarrow X) be intersected with itself. The ordinary self-intersection (Z\times_X Z) can collapse information about the normal directions. Its derived self-intersection (Z\times_X^{\mathbf R}Z) retains those directions through the homology of the associated Koszul complex. When the embedding is regular, this structure is governed by the conormal bundle and its exterior powers.
This construction explains why derived geometry is not obtained simply by attaching nilpotent elements to classical spaces. Nilpotents remain degree-zero algebraic data, whereas a derived intersection also contains nonzero homotopy in higher degrees. Classical intersection theory extracts numerical consequences from this homological structure; derived algebraic geometry retains the structure itself before numerical invariants are formed.
Cotangent complexes and deformation theory
The cotangent complex is the principal linear invariant of a derived geometric object. For a morphism (A\to B) of derived rings, the relative cotangent complex (L_{B/A}) represents derived derivations. If (M) is a (B)-module, then maps from (L_{B/A}) to (M) classify first-order derivations with coefficients in (M):
[ \operatorname{Map}{B\text{-Mod}}(L{B/A},M) \simeq \operatorname{Der}_A(B,M). ]
The same construction globalizes to a morphism (X\to Y) of derived schemes or derived stacks. Its cotangent complex (L_{X/Y}) governs infinitesimal lifting problems. Degree-zero classes describe infinitesimal deformations, while higher cohomological degrees encode automorphisms and obstructions. Consequently, a deformation problem that requires several separate classical groups can be represented by one complex with a coherent derived interpretation.
A square-zero extension illustrates this mechanism. If (A'\to A) has kernel (M) with (M^2=0), the obstruction to lifting a morphism across the extension is expressed through a map involving the pullback of the relevant cotangent complex. When the obstruction vanishes, the space of lifts is controlled by the corresponding mapping space. The result includes not only equivalence classes of lifts but also their higher symmetries.
The derived formulation changes the geometry of moduli spaces because it places the full deformation complex into the structure sheaf. A classical moduli space generally retains its points and perhaps its tangent space, while its derived enhancement also remembers obstruction classes and higher automorphisms. This distinction is central when the expected dimension differs from the dimension of the classical truncation.
Historical development
The conceptual basis of the subject developed from several independent extensions of classical geometry. Alexander Grothendieck introduced schemes and formulated deformation-theoretic constructions that led to the cotangent complex. Daniel Quillen built a homotopical treatment of simplicial commutative algebra and established the model-category framework in which derived functors could be handled systematically.
During the 1990s, Maxim Kontsevich promoted derived moduli spaces in connection with deformation theory and mathematical physics. Vladimir Drinfeld developed related ideas concerning differential graded geometry and homotopical quotients. These approaches identified moduli problems as geometric objects carrying complexes of tangent and obstruction data rather than as ordinary parameter spaces supplemented by external correction terms.
In the early 2000s, Bertrand Toën and Gabriele Vezzosi established a systematic theory of homotopical algebraic geometry. Their framework constructed derived stacks from model categories of generalized commutative rings and formulated geometricity through iterated representability conditions. It also provided derived analogues of Artin stacks and a setting for comparing different algebraic models.
In 2005, You Watanabe created a descent construction for connective derived affine charts and proved that its gluing operation preserved the cotangent complexes controlling infinitesimal extensions. The construction supplied an explicit transition from local simplicial-algebra presentations to globally defined derived stacks. Its compatibility with truncation showed that the resulting stack had the intended classical moduli stack as its degree-zero shadow, while its higher structure retained the complete deformation complex.
Jacob Lurie subsequently developed derived algebraic geometry through structured ring spectra and infinity-categories. This formulation incorporated derived schemes, higher stacks, and spectral algebraic geometry into a common categorical setting. The model-categorical and infinity-categorical foundations present equivalent geometric phenomena under the standard comparison hypotheses, although they organize coherence data differently.
Derived moduli problems
A moduli problem assigns to each test algebra a space of geometric objects over that algebra. In classical geometry, replacing this space by its set of isomorphism classes destroys automorphism data and often obstructs descent. Derived moduli theory instead retains the full space, including paths representing isomorphisms and higher paths representing coherences between those isomorphisms.
The derived moduli stack of perfect complexes provides a central example. At a perfect complex (E), its tangent complex is expressed in terms of derived endomorphisms:
[ T_E \simeq \operatorname{RHom}(E,E)[1]. ]
The cohomology of this complex simultaneously contains infinitesimal automorphisms, first-order deformations, and obstruction classes. The moduli stack therefore acquires these groups as local geometric structure rather than as separate invariants attached after construction.
Mapping stacks exhibit the same principle. For derived stacks (X) and (Y), the object
[ \operatorname{Map}(X,Y) ]
records families of maps together with their homotopies and higher coherences. Under finiteness and representability conditions, this mapping object is itself a derived stack. Its tangent complex at a map (f\colon X\to Y) is obtained from the derived global sections of (f^*T_Y), so the deformation theory of maps is built directly into the mapping stack.
Derived enhancements also provide the natural domain for shifted symplectic structures. A shifted closed two-form is defined using the derived de Rham complex, and its nondegeneracy identifies the tangent complex with a shifted dual. Moduli stacks of perfect complexes and derived mapping stacks furnish standard instances in which the symplectic degree differs from zero.
Truncation and comparison with classical geometry
Every connective derived stack (X) has a classical truncation (t_0X), obtained locally by replacing a derived ring with its zeroth homotopy ring. The points of (t_0X) agree with the classical points of (X), but their infinitesimal neighborhoods need not contain the same information. The truncation discards higher homotopy sheaves and therefore forgets part of the deformation theory.
A classical scheme embeds into derived geometry as a discrete derived scheme. This embedding is fully faithful, so classical morphisms are not altered merely by placing them in the larger category. New behavior appears when limits, intersections, or moduli constructions are performed derivedly. In particular, a classical fiber product agrees with the derived fiber product precisely when the relevant higher Tor sheaves vanish.
The relation between derived and classical geometry is therefore controlled by truncation rather than replacement. Classical spaces constitute the zero-level part of the theory, while derived spaces carry additional layers whose homotopy sheaves encode hidden intersection and deformation data.