Applied linear algebra by Riaz A Usmani

By Riaz A Usmani

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E// Â V . E//. 1 Now suppose x 2 V . E//. E/. E//, which gives V . E//. E// D V . E//; that is, the pullback of a closed set is closed. t u Under what conditions are two topological spaces essentially the same? We next study the analog of the notion of a group or ring isomorphism. 5. A map spaces if (i) (ii) W X ! Y is a homeomorphism of topological is a bijection and and 1 are continuous. 2 Basis for a Topological Space Two spaces are homeomorphic if there exists a homeomorphism We then write X Š Y .

Since L˝S T is in the kernel of ˛ 0 , ˛ 0 is not an injection, which proves the lemma. 2. A flat map S ! T is faithfully flat if the map ' W M ! m/ D m ˝ 1 is an injection for all S -modules M . The localization map S ! Sf may not be faithfully flat, though it can be used to build a faithfully flat map. Let ff1 ; f2 ; : : : ; fn g be a finite set of non-nilpotent elements of S , and suppose that the ideal generated by ff1 ; f2 ; : : : ; fn g is S . Then n Y the map % W S ! Sfi defined as s 7! s=1/fi / is faithfully flat.

Proof. 1 hold. For (i), let x 2 Spec A. Since x 6D A, Anx is non-empty, and thus there exists an element f 2 Anx that satisfies f 62 x. f /. 2, f is non-nilpotent. f / 2 B. g/ for f; g non-nilpotent. fg/. fg/ D ;, which is a contradiction. g/. 1. f / W f 2 Z; f 6D 0g. f / is topologically equivalent to the spectrum of a localized ring. 2. f / is homeomorphic to Spec Af , where Af is the localization S 1 A, where S is the multiplicative set f1; f; f 2 ; : : : g. Proof. f / with the subspace topology induced by Spec A.

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