Exercise 7, § 2

Let A be a commutative ring, E an A-module, F a submodule of E, B the exterior algebra $\mathbf{A}(E)$ and $\mathfrak{z}$ the two-sided ideal of B generated by the canonical images in B of the elements of F. Let gr$^\mathfrak{z}$(B) be the graded ring associated with the ring B filtered by the $\mathfrak{z}$-adic filtration.

(a) Define a surjective canonical (non-graded) A-algebra homomorphism $(\mathbf{A}(F)) \otimes_A^\mathfrak{g} (\mathbf{A}(E/F)) \to \mathrm{gr}^\mathfrak{z}(B)$ (where we mean the skew tensor product of graded algebras (Algebra, Chapter 111).

(b) Show that, if F admits a complement in E, the homomorphism defined in (a) is an isomorphism.

(c) Take $A = \mathbf{Z}, E = 2/42, F = 22/42$; show that the homomorphism defined in A is not injective in this case.