A Beckman-Quarles type theorem for finite desarguesian by Benz W.

By Benz W.

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236 (2001), 419–452. [BG1] L. D. Borsari and D. L. Gon¸¸calves, A Van Kampen type theorem for coincidence, Topology Appl. 101 (2000), 149–160. , Obstruction theory and minimal number of coincidences for maps from a com[BG2] plex into a manifold, Topol. Methods Nonlinear Anal. 21 (2003), 115–130. [Br1] R. B. S. Brooks, On removing coincidences of two maps when only one, rather than both of them, may be deformed by a homotopy, Paciﬁc J. Math. 40 (1972), 45–52. , On the sharpness of the ∆2 and ∆1 Nielsen numbers, J.

If the following diagram with exact rows in Vf 0 / E1 /E T L1 0 S L  / E1  /E / E2 /0 L2  / E2 /0 is commutative, then tr(L) = tr(L1 ) + tr(L2 ). T S Proof. 2), we can assume without loss of generality that E1 is a subspace of E and E2 = E/E1 . Consequently, we can assume that E = E1 ⊕E2 , then L1 = (L11 , L22 ) (comp. 6). The proof is completed. Now, we are going to deﬁne the generalized trace. Let us denote by V the category of vector spaces over the ﬁeld K (usually Q) and linear mappings.

If the following diagram with exact rows in Vf 0 / E1 /E T L1 0 S L  / E1  /E / E2 /0 L2  / E2 /0 is commutative, then tr(L) = tr(L1 ) + tr(L2 ). T S Proof. 2), we can assume without loss of generality that E1 is a subspace of E and E2 = E/E1 . Consequently, we can assume that E = E1 ⊕E2 , then L1 = (L11 , L22 ) (comp. 6). The proof is completed. Now, we are going to deﬁne the generalized trace. Let us denote by V the category of vector spaces over the ﬁeld K (usually Q) and linear mappings.