Morita theory for Hopf algebroids and presheaves of groupoids.

*(English)*Zbl 1033.55002It is well-known that a commutative Hopf algebra \((A,\Gamma)\) over a commutative ring \(A\) corresponds to a representable presheaf of groups \((Spec(A),Spec(\Gamma))\) over \(Spec(A)\). More generally, Hopf algebroids over \(A\) correspond to representable presheaves of groupoids over \(Spec(A)\). Derived functors of categories of comodules over Hopf algebroids are important in algebraic topology, and the main result of the paper under review is that certain change of ring isomorphisms for \(Ext\)-groups of comodules over Hopf algebroids can be derived from sheaf theoretic results about presheaves of groupoids using the above correspondence.

More precisely the author shows the following statements. Given a Hopf algebroid \((A,\Gamma)\) there is an equivalence of categories between the category of comodules over \((A,\Gamma)\) and the category of quasi-coherent sheaves over the sheaf of groupoids \((Spec(A),Spec(\Gamma))\). The author then shows that if a map \(\Phi:(X_0,X_1) \to (Y_0,Y_1)\) of presheaves of groupoids over an affine site is an internal equivalence in the sense of [A. Joyal and M. Tierney, in “Category theory”, (Como 1990), Lect. Notes Math. 1488, 213–236 (1991; Zbl 0748.18009)] then \(\Phi\) induces an equivalence between the corresponding categories of sheaves over the groupoid schemes \((X_0,X_1)\) and \((Y_0,Y_1)\), respectively.

In particular this applies to the affine site where the topology is the flat topology, and using the principle of faithfully flat descent the author shows that an internal equivalence of presheaves of groupoids \(\Phi:(X_0,X_1) \to (Y_0,Y_1)\) on this site also induces an equivalence between the category of quasi-coherent sheaves over \((X_0,X_1)\) and the category of quasi-coherent sheaves over \((Y_0,Y_1)\). Finally he gives a criterion for morphisms of Hopf algebroids \(f:(A;\Gamma) \to (B,\Sigma)\) which implies that the corresponding map of groupoid schemes \(f^*:(Spec(B),Spec(\Sigma)) \to (Spec(A),Spec(\Gamma))\) is an internal equivalence with respect to the flat topology, so that (by the above) \(f\) induces an equivalence between the corresponding categories of comodules over the Hopf algebroids, and therefore an isomorphism of Ext-groups \(\text{Ext}_{\Gamma}(M,N) \cong \text{Ext}_{\Sigma}(M\otimes_A B, N \otimes_A B)\). The criterion then is checked for the special case of interest to algebraic topology where \((A,\Gamma)=(v_n^{-1}BP_*/I_n, v_n^{-1}BP_*BP/I_n)\) for \(n>0\) and where \(f\) arises from classifying a homogeneous \(p\)-typical formal group law of strict height \(n\). As a consequence the author gets for any \(m\geq n\) an isomorphism of \(Ext\)-groups \(\text{Ext}_{BP_*BP}(M,N) \cong \text{Ext}_{E(m)_*E(m)}(E(m) \otimes_{BP_*} M, E(m)\otimes_{BP_*}N)\) for \(BP_*BP\)-comodules \(M\) and \(N\) on which \(v_n\) acts invertibly if either \(M\) is finitely presented or if \(N\) is the \(v_n\)-localization of a finitely presented \(I_n\)-nilpotent comodule.

This generalizes corresponding change of ring isomorphisms of H. R. Miller and D. C. Ravenel [Duke Math. J. 44 , 433–447 (1977; Zbl 0358.55019), Thm. 3.10] and M. Hovey and H. Sadofsky [J. Lond. Math. Soc (2) 60, 284–302 (1999; Zbl 0947.55013), Thm. 3.1]. The author mentions that much of the material of this article must be known to M. Hopkins. In fact, the most important special case from the point of view of algebraic topology (where \(M=N=A\)) is referred to in M. Hovey and H. Sadofsky [ loc. cit.] as a theorem of Hopkins. However, the present article (as to our knowledge) is the first article in print covering the results in this generality.

More precisely the author shows the following statements. Given a Hopf algebroid \((A,\Gamma)\) there is an equivalence of categories between the category of comodules over \((A,\Gamma)\) and the category of quasi-coherent sheaves over the sheaf of groupoids \((Spec(A),Spec(\Gamma))\). The author then shows that if a map \(\Phi:(X_0,X_1) \to (Y_0,Y_1)\) of presheaves of groupoids over an affine site is an internal equivalence in the sense of [A. Joyal and M. Tierney, in “Category theory”, (Como 1990), Lect. Notes Math. 1488, 213–236 (1991; Zbl 0748.18009)] then \(\Phi\) induces an equivalence between the corresponding categories of sheaves over the groupoid schemes \((X_0,X_1)\) and \((Y_0,Y_1)\), respectively.

In particular this applies to the affine site where the topology is the flat topology, and using the principle of faithfully flat descent the author shows that an internal equivalence of presheaves of groupoids \(\Phi:(X_0,X_1) \to (Y_0,Y_1)\) on this site also induces an equivalence between the category of quasi-coherent sheaves over \((X_0,X_1)\) and the category of quasi-coherent sheaves over \((Y_0,Y_1)\). Finally he gives a criterion for morphisms of Hopf algebroids \(f:(A;\Gamma) \to (B,\Sigma)\) which implies that the corresponding map of groupoid schemes \(f^*:(Spec(B),Spec(\Sigma)) \to (Spec(A),Spec(\Gamma))\) is an internal equivalence with respect to the flat topology, so that (by the above) \(f\) induces an equivalence between the corresponding categories of comodules over the Hopf algebroids, and therefore an isomorphism of Ext-groups \(\text{Ext}_{\Gamma}(M,N) \cong \text{Ext}_{\Sigma}(M\otimes_A B, N \otimes_A B)\). The criterion then is checked for the special case of interest to algebraic topology where \((A,\Gamma)=(v_n^{-1}BP_*/I_n, v_n^{-1}BP_*BP/I_n)\) for \(n>0\) and where \(f\) arises from classifying a homogeneous \(p\)-typical formal group law of strict height \(n\). As a consequence the author gets for any \(m\geq n\) an isomorphism of \(Ext\)-groups \(\text{Ext}_{BP_*BP}(M,N) \cong \text{Ext}_{E(m)_*E(m)}(E(m) \otimes_{BP_*} M, E(m)\otimes_{BP_*}N)\) for \(BP_*BP\)-comodules \(M\) and \(N\) on which \(v_n\) acts invertibly if either \(M\) is finitely presented or if \(N\) is the \(v_n\)-localization of a finitely presented \(I_n\)-nilpotent comodule.

This generalizes corresponding change of ring isomorphisms of H. R. Miller and D. C. Ravenel [Duke Math. J. 44 , 433–447 (1977; Zbl 0358.55019), Thm. 3.10] and M. Hovey and H. Sadofsky [J. Lond. Math. Soc (2) 60, 284–302 (1999; Zbl 0947.55013), Thm. 3.1]. The author mentions that much of the material of this article must be known to M. Hopkins. In fact, the most important special case from the point of view of algebraic topology (where \(M=N=A\)) is referred to in M. Hovey and H. Sadofsky [ loc. cit.] as a theorem of Hopkins. However, the present article (as to our knowledge) is the first article in print covering the results in this generality.

Reviewer: Michael Joachim (Münster)

##### MSC:

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

55Q51 | \(v_n\)-periodicity |