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By Shoichiro Sakai

From the reports: "This e-book is a superb and finished survey of the idea of von Neumann algebras. It comprises the entire basic result of the topic, and is a priceless reference for either the newbie and the expert." (Math. experiences) "In concept, this publication might be learn by way of a well-trained third-year graduate scholar - however the reader had greater have loads of mathematical sophistication. The expert during this and allied components will locate the wealth of contemporary effects and new methods in the course of the textual content in particular rewarding." (American Scientist) "The identify of this booklet right now indicates comparability with the 2 volumes of Dixmier and the truth that you could heavily make this comparability shows that it's a way more titanic paintings that others in this topic that have lately appeared"(BLMSoc)

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4) consist of acyclics. 4) are quasi-isomorphisms follows from the fact that f* has finite cohomological dimension on the category of quasi-coherent 6X-modules. 4. 7) f* 019+n f* ("'Ojo+n (, We Rnf*(W) [0] 0 0? 6) so that it respects the natural quasi-isomorphism Let us from f 0010 to each side (this forces 0 to be a quasi-isomorphism). qo, is a quasi-isomorphism, it follows that all mute in D(Y). 8) are quasi-isomorphisms. 4). 8) yield homotopic maps when composed with s. 8) in D(X). 8), so the for which in 8 0 P2 homotopic is D(Y).

14), the bottom map in the . > a right Yeom (9*'W*) column is obtaine& from the canon- 49 0 ical map Ye' Rn f* (W) 0 jye, R nf -e f *,V) for any 69y-module Je". W*) - canonically isomorphic to R nf (W) 0'c/, -- W*, which is a bounded below complex of injective sheaves. 14) is to try to eliminate all references to lq*c* and to reduce to a commutativity assertion involving flat sheaves, which we will then check locally. 14) by a more tractable diagram. 2. We noted above that the quasi-isomorphism -+ al* has a homotopy inverse, so applying f 0 to this shows that there is a double is * 2.

Let the global sections Xi t*Xj C ]p(pnA, Y) be the associated 'projective coordinate system' L and let t '(Uj) be the UJI is not relevant to but is our = . , = - = open where sense on cocycle U' Xj' = generates Y. The functions U01 n n ... (-I)n(n+l)/2 (dti CL E H n(pn Un, A ... so A for V dtn) / (ti = t 3- = Xj'IXO' for I < j < ..... n make t-1 (it), the 6ech fU0 Un} tn) E6n (jAl, WA) = defines an n- element iWA)- We claim that CL is independent of L. 1], whose proof appears to require this 'independence of coordinates' in the first place.

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