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Brauer Groups, Class Groups and Maximal Orders for a Krull Scheme

  • Heisook Lee (a1) and Morris Orzech (a1)

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In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C 1C 2 could be related by a tidy exact sequence

It was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].

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References

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1. Auslander, B., The Brauer group of a ringed space, J. Algebra 4 (1966), 220273.
2. Auslander, B., Central separable algebras which are locally endomorphism rings of free modules, Proc. Amer. Math. Soc. 80 (1971), 395404.
3. Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 124.
4. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367409.
5. Bass, H., Algebraic K-theory (W. A. Benjamin, N.Y., 1968).
6. Childs, L. N., Garfinkel, G. and Orzech, M., On the Brauer group and factoriality of normal domains, J. Pure and Applied Algebra 6 (1975), 111123.
7. DeMeyer, F. and Ingraham, E., Separable algebras over commutative rings Lecture Notes in Mathematics 181 (Springer-Verlag, Berlin, 1971).
8. A., Frôhlich and C. T. C., Wall, Graded monoidal categories, Composito Math. (1974), 229285.
9. Fossum, R., Maximal orders over Krull domains, J. Algebra 10 (1968), 321332.
10. Fossum, R., The divisor class group of a Krull domain (Springer-Verlag, New York, 1973).
11. Grothendieck, A., Le groupe de Brauer, II, in Dix exposées sur la cohomologie des schémas (North-Holland, 1968).
12. Hartshorne, R., Algebraic geometry (Springer-Verlag, N.Y., 1977).
13. Orzech, M., Brauer groups and class groups for a Krull domain, in Brauer groups in ring theory and algebraic geometry Lecture Notes in Mathematics 917 (Springer- Verlag, Berlin, 1981).
14. Orzech, M., Divisorial modules and Krull morphisms, J. Pure and Applied Algebra 25 (1982), to appear.
15. Orzech, M. and Small, C., The Brauer group of commutative rings (Marcel Dekker, N.Y., 1975).
16. Pareigis, B., The Brauer group of a monoidal category, in Brauer groups Lecture Notes in Mathematics 549 (Springer-Verlag, Berlin, 1976).
17. Yuan, S., Reflexive modules and algebra class groups over Noetherian integrally closed domains, J. Algebra 32 (1974), 405417.
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Brauer Groups, Class Groups and Maximal Orders for a Krull Scheme

  • Heisook Lee (a1) and Morris Orzech (a1)

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