A Note on Cancellation of Reflexive Modules
نویسنده
چکیده
By the Quillen-Suslin theorem [Qui76, Sus76], we know that projective modules over a polynomial ring over a field are free. One way of saying this is, that if two projective modules of the same rank are stably isomorphic, then they are isomorphic. That, projective modules of given rank over polynomial rings are stably isomorphic was well known at the time Quillen and Suslin proved their theorems and this result is usually attributed to Grothendieck. This result can also be deduced from Hilbert Syzygy Theorem. This note tries to answer whether a similar cancellation occurs for reflexive modules. The question was specifically raised by M. P. Murthy. I thank him for raising this question and for the innumerable discussions which ensued. In this note we show that, in general, reflexive modules are not cancellative, without further assumptions. These assumptions under which cancellation does take place are explained in the theorem in the next section, Theorem 1.
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