In mathematics, a diffiety (/dəˈfaɪəˌtiː/) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.^{[1]}
In algebraic geometry the main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of polynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic ideal generated by the initial set of polynomials.
When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to differentiate the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with a differential ideal.
An elementary diffiety will consist therefore of the infinite prolongation of a differential equation , together with an extra structure provided by a special distribution. Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations. Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (nonelementary) diffiety as an object that locally looks like an elementary diffiety.
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
Let be an dimensional smooth manifold. Two dimensional submanifolds , of are tangent up to order at the point if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of , whose derivatives at agree up to order . One can show that being tangent up to order is a coordinateinvariant notion and an equivalence relation.^{[2]} One says also that and have same th order jet at , and denotes their equivalence class by or . The jet space of submanifolds of , denoted by , is defined as the set of all jets of dimensional submanifolds of at all points of : As any given jet is locally determined by the derivatives up to order of the functions describing around , one can use such functions to build local coordinates and provide with a natural structure of smooth manifold.^{[2]}

For instance, for one recovers just points in and for one recovers the Grassmannian of dimensional subspaces of . More generally, all the projections are fibre bundles.
As a particular case, when has a structure of fibred manifold over an dimensional manifold , one can consider submanifolds of given by the graphs of local sections of . Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle turns out to be an open and dense subset of .^{[3]}
The jet prolongation of a submanifold is
The map is a smooth embedding and its image , called the prolongation of the submanifold , is a submanifold of diffeomorphic to .
A space of the form , where is any submanifold of whose prolongation contains the point , is called an plane (or jet plane, or Cartan plane) at . The Cartan distribution on the jet space is the distribution defined by
A differential equation of order on the manifold is a submanifold ; a solution is defined to be an dimensional submanifold such that . When is a fibred manifold over , one recovers the notion of partial differential equations on jet bundles and their solutions, which provide a coordinatefree way to describe the analogous notions of mathematical analysis. While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as Lagrangian submanifolds and minimal surfaces.
As in the jet bundle case, the Cartan distribution is important in the algebrogeometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold is a solution if and only if it is an integral manifold for , i.e. for all .
One can also look at the Cartan distribution of a PDE more intrinsically, defining
Given a differential equation of order , its th prolongation is defined as
Below we will assume that the PDE is formally integrable, i.e. all prolongations are smooth manifolds and all projections are smooth surjective submersions. Note that a suitable version of Cartan–Kuranishi prolongation theorem guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then the inverse limit of the sequence extends the definition of prolongation to the case when goes to infinity, and the space has the structure of a profinitedimensional manifold.^{[5]}
An elementary diffiety is a pair where is a th order differential equation on some manifold, its infinite prolongation and its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution is dimensional and involutive. However, due to the infinitedimensionality of the ambient manifold, the Frobenius theorem does not hold, therefore is not integrable
A diffiety is a triple , consisting of
such that is locally of the form , where is an elementary diffiety and denotes the algebra of smooth functions on . Here locally means a suitable localisation with respect to the Zariski topology corresponding to the algebra .
The dimension of is called dimension of the diffiety and its denoted by , with a capital D (to distinguish it from the dimension of as a manifold).
A morphism between two diffieties and consists of a smooth map whose pushforward preserves the Cartan distribution, i.e. such that, for every point , one has .
Diffieties together with their morphisms define the category of differential equations.^{[3]}
The Vinogradov spectral sequence (or, for short, Vinogradov sequence) is a spectral sequence associated to a diffiety, which can be used to investigate certain properties of the formal solution space of a differential equation by exploiting its Cartan distribution .^{[6]}
Given a diffiety , consider the algebra of differential forms over
and the corresponding de Rham complex:
Its cohomology groups contain some structural information about the PDE; however, due to the Poincaré Lemma, they all vanish locally. In order to extract much more and even local information, one thus needs to take the Cartan distribution into account and introduce a more sophisticated sequence. To this end, let
be the submodule of differential forms over whose restriction to the distribution vanishes, i.e.
Note that is actually a differential ideal since it is stable w.r.t. to the de Rham differential, i.e. .
Now let be its th power, i.e. the linear subspace of generated by . Then one obtains a filtration
and since all ideals are stable, this filtration completely determines the following spectral sequence:
The filtration above is finite in each degree, i.e. for every
so that the spectral sequence converges to the de Rham cohomology of the diffiety. One can therefore analyse the terms of the spectral sequence order by order to recover information on the original PDE. For instance:^{[7]}
Many higherorder terms do not have an interpretation yet.
As a particular case, starting with a fibred manifold and its jet bundle instead of the jet space , instead of the spectral sequence one obtains the slightly less general variational bicomplex. More precisely, any bicomplex determines two spectral sequences: one of the two spectral sequences determined by the variational bicomplex is exactly the Vinogradov spectral sequence. However, the variational bicomplex was developed independently from the Vinogradov sequence.^{[8]}^{[9]}
Similarly to the terms of the spectral sequence, many terms of the variational bicomplex can be given a physical interpretation in classical field theory: for example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges, etc.^{[10]}
Vinogradov developed a theory, known as secondary calculus, to formalise in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs (i.e. the space of integral manifolds of a given diffiety).^{[11]}^{[12]}^{[13]}^{[3]}
In other words, secondary calculus provides substitutes for functions, vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way on the space of solution. Furthermore, the space of these new objects are naturally endowed with the same algebraic structures of the space of the original objects.^{[14]}
More precisely, consider the horizontal De Rham complex of a diffiety, which can be seen as the leafwise de Rham complex of the involutive distribution or, equivalently, the Lie algebroid complex of the Lie algebroid . Then the complex becomes naturally a commutative DG algebra together with a suitable differential . Then, possibly tensoring with the normal bundle , its cohomology is used to define the following "secondary objects":
Secondary calculus can also be related to the covariant Phase Space, i.e. the solution space of the EulerLagrange equations associated to a Lagrangian field theory.^{[15]}