By Michel Broué
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Definition. For X and X in A, we denote by TrM N (X, X ), and call relative trace, the map TrM N (X, X ) : HomB (N X, N X ) −→ HomA (X, X ) deﬁned by TrM N (X, X )(β) := εM,N (X ) ◦ M (β) ◦ ηN,M (X) : X XO ηN,M MNX εM,N Mβ / MNX If it is clear from the context what the domain and the codomain of β are, M M we will write TrM N (β) instead of TrN (X, X )(β). Furthermore, TrN (X) stands for M N TrN (X, X). Notice that the map TrM is deﬁned, as well. The following example is fundamental. Example : Induction and restriction from R.
1 that the morphisms εM,N (M (Y ))◦M (ηM,N (Y )) and M (εN,M (Y )) ◦ ηN,M (M (Y )) are the identity on M (Y ). 56. We prove the implications (i) ⇒ (ii) ⇒ (iii) ⇒ and (ii) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇒ (vii) ⇒ (i) (i) (v) (iv) . (i)⇒(ii) : trivial. (ii)⇒(iii) : We may assume that X = M (Y ). For if X is a direct summand of M (Y ), we have to morphisms p : M (Y ) → X and i : X → M (Y ) such that p ◦ i = IdX . 55, we get IdX = TrM N (N (p) ◦ β ◦ N (i)) . 57. (iii)⇒(iv) and (iii)⇒(v) : These implications follow from the deﬁnition of the relative trace, since we have IdX = TrM N (X)(β) = εM,N (X) ◦ M (β) ◦ ηN,M (X) .
The map u − h ◦ π1 is a map from P1 to the kernel of π2 . , the map α factorizes through an M -split object. The converse implication can be veriﬁed similarly. Remark. It follows from the proof of Schanuel’s lemma that (α , u, α) deﬁnes a single homotopy class of morphisms from ι π 1 1 0 → X1 −→P 1 −→X1 → 0 to ι π 2 2 0 → X2 −→P 2 −→X2 → 0 This is a particular case of a more general lemma which we will prove later on. 64. Corollary. Assume that 0 → X 1 → P1 → X → 0 0 → X 2 → P2 → X → 0 and are short exact sequences in A such that 1.
Symmetric algebras by Michel Broué