By S. A. Amitsur, D. J. Saltman, George B. Seligman

ISBN-10: 082185013X

ISBN-13: 9780821850138

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**Extra info for Algebraists' Homage: Papers in Ring Theory and Related Topics**

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19 D(E) is the graded dual of the symmetric algebra S(F ), and S(F ) is the graded dual of the divided power algebra D(E). If one uses the notation F ∗ in place of E, the above says that D(F ∗ ) = S(F )∗gr and S(F ) = D(F ∗ )∗gr . We sketch a proof of the theorem, by giving an idea of why D(E)∗gr equals S(F ). Essentially, it all amounts to showing that the diagonal map of D(E) induces on D(E)∗gr a graded R-algebra structure corresponding to that of S(F ). Let us take f ∈ Di (E)∗ and g ∈ Dj (E)∗ , and use the diagonal map of D(E) to deﬁne an element f g ∈ Di+j (E)∗ .

Linear algebra 17 Proof If part: since R is R-projective (being R-free of rank 1), every submodule of R (= every ideal of R) is projective. Only if part: every R-projective P is a direct summand of some free R-module; by the theorem, every submodule M of P is a direct sum of ideals, hence of projective modules (by deﬁnition of hereditary ring); but a direct sum of projective modules is obviously projective. 3 Projective resolutions We push the analysis of non-free modules a little further. Given any R-module M , with a generating system {mi }i∈I , we have already constructed a free presentation of M , that is, an exact complex ϕ1 ϕ0 F1 → F0 → M → 0, with each Fj a free R-module.

Y1 (st ) · · · yt = y1s1 · · · ytst and D(E) ∼ = R[y1 , . . , yt ]. In general, however, D(E) is not a symmetric algebra. What is true, though, is that D(E) is the graded dual of a symmetric algebra. 18 It is well known that, given R-modules Mi , i ∈ N, and L, Mi , L HomR i∈N ∼ = HomR (Mi , L). i∈N In particular, the ordinary dual of a graded R-algebra A = ⊕i∈N Ai is i∈N A∗i . Hence, in order to make sure that the dual of a graded algebra is still a graded algebra, one replaces A∗ with another object, namely ⊕i∈N A∗i , which is called the graded dual of A and is denoted by A∗gr .

### Algebraists' Homage: Papers in Ring Theory and Related Topics by S. A. Amitsur, D. J. Saltman, George B. Seligman

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