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368 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe pair .U; A/, together with the section 1 is a decent datum which we denotedd.C; U;L/.The assignment .U; A/ 7! dd.C; U;L/extends to a cartesian functorddW Triv R .X/ ! Desc R .X/:Lemma 4.12. The functors St and dd are mutually quasi-inverse equivalences.4.4 Base change. For an R-linear category C and homomorphism of algebras R ! Swe denote by C ˝R S the category with the same objects as C and morphisms definedby Hom C˝RS.A; B/ D Hom C .A; B/ ˝R S.For an R-algebra A the categories .A ˝R S/ C and A C ˝R S are canonicallyisomorphic.For a prestack C in R-linear categories we denote by C ˝R S the prestack associatedto the fibered category U 7! C.U / ˝R S.For U X, A; B 2 C.U /, there is an isomorphism of sheaves Hom C˝RS .A; B/ DHom C .A; B/ ˝R S.Lemma 4.13. Suppose that A is a sheaf of R-algebras and C is an R-algebroid stack.1. . A C ˝R S/z is an algebroid stack equivalent to E .A ˝R S/ C .2. B C ˝R S is an algebroid stack.Proof. Suppose that A is a sheaf of R-algebras. There is a canonical isomorphismof prestacks .A S/ C Š A C ˝R S which induces the canonical equivalenceE.A ˝R S/ C Š AC D˝R S.The canonical functor A C ! A C induces the functor A C ˝R S ! A C ˝R S,hence the functor AC D˝R S ! . A C ˝R S/z.The map A ! A ˝R S induces the functor AC ! .A E ˝R S/ C which factorsthrough the functor AC˝R S ! .A E ˝R S/ C . From this we obtain the functor. A C ˝R S/z!E .A ˝R C .We leave it to the reader to check that the two constructions are mutually inverseequivalences. It follows that . A C ˝R S/z is an algebroid stack equivalent toE.A ˝R S/ C .Suppose that C is an R-algebroid stack. Let U be a cover such that 0 C.N 0U/ isnonempty. Let L be an object in 0 C.N 0U/; put A WD End 0 C .L/. The equivalenceA C ! 0 C induces the equivalence . A C ˝R S/z!. 0 C ˝R S/z. Since the formeris an algebroid stack so is the latter. There is a canonical equivalence . 0 C ˝R S/zŠ 0 . B C ˝R S/; since the former is an algebroid stack so is the latter. Since the propertyof being an algebroid stack is local, the stack B C ˝R S is an algebroid stack.