the relative tangent sheaf of f, defined as the element in . For example, when f is a smooth morphism, is simply a vector bundle, known as the tangent bundle along the fibers of f. Using A1-homotopy theory, the Grothendieck–Riemann–Roch theorem has been extended by to the situation where f is a proper map between two smooth schemes. Generalising and specialising Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination and to the non-proper case by considering cohomology with compact support. The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem to arithmetic schemes. The Hirzebruch–Riemann–Roch theorem is (essentially) the special case where Y is a point and the field is the field of complex numbers. A version of Riemann–Roch theorem for oriented cohomology theories was proven by Ivan Panin and Alexander Smirnov. It is concerned with multiplicative operations between algebraic oriented cohomology theories (such as algebraic cobordism). The Grothendieck-Riemann-Roch is a particular case of this result, and the Chern character comes up naturally in this setting. Examples Vector bundles on a curve A vector bundle of rank and degree (defined as the degree of its determinant; or equivalently the degree of its first Chern class) on a smooth projective curve over a field has a formula similar to Riemann–Roch for line bundles. If we take and a point, then the Grothendieck–Riemann–Roch formula can be read as hence, This formula also holds for coherent sheaves of rank and degree . Smooth proper maps One of the advantages of the Grothendieck–Riemann–Roch formula is it can be interpreted as a relative version of the Hirzebruch–Riemann–Roch formula. For example, a smooth morphism has fibers which are all equi-dimensional (and isomorphic as topological spaces when base changing to ). This fact is useful in moduli-theory when considering a moduli space parameterizing smooth proper spaces. For example, David Mumford used this formula to deduce relationships of the Chow ring on the moduli space of algebraic curves. Moduli of curves For the moduli stack of genus curves (and no marked points) there is a universal curve where (is the moduli stack of curves of genus and one marked point. Then, he defines the tautological classes where and is the relative dualizing sheaf. Note the fiber of over a point this is the dualizing sheaf . He was able to find relations between the and describing the in terms