The Deligne-Riemann-Roch theorem is a functorial refinement of the Grothendieck-Riemann-Roch theorem in the case of a family of projective curves. This theorem gives a canonical isomorphism between certain line bundles on a base. I will discuss this theorem, as well as a local Deligne-Riemann-Roch theorem that I obtained. The corresponding local theorem gives an isomorphism between two central extensions of a group ind-scheme, which is the semidirect product of the group of invertible functions on a formal punctured disk defined over the field of rational numbers and the automorphism group of this disk. These сentral extensions are by the multiplicative group scheme. In connection with these central extensions, canonical 2-cocycles arise. But one 2-cocycle is very difficult to calculate, while the other 2-cocycle is written quite explicitly using cup-products of explicitly defined 1-cocycles, then applying the Contou-Carrère symbol to this expression. The Contou-Carrère symbol is a generalization of the usual tame symbol from number theory when a field is replaced by an arbitrary commutative ring.