Change of Base for Commutative Algebras

In this paper we examine on a pair of adjoint functors $(\phi ^{\ast},\phi_{\ast})$ for a subcategory of the category of crossed modules over commutative algebras where $\phi ^{\ast}:\mathbf{XMod}$\textbf{/}$% Q\rightarrow $ $\mathbf{XMod/}P$, pullback, which enables us to move from crossed $Q$-modules to crossed $P$-modules by an algebra morphism $\phi :P\rightarrow Q$ and $\phi_{\ast}:\mathbf{XMod}$\textbf{/}$P\rightarrow $ $\mathbf{XMod/}Q$, induced. We note that this adjoint functor pair $(\phi ^{\ast},\phi_{\ast})$ makes $p:\mathbf{XMod}\rightarrow $ $k$\textbf{-Alg}into a bifibred category over $k$\textbf{-Alg}, the category of commutative algebras, where $p$ is given by $p(C,R,\partial)=R.$ Also, some examples and results on induced crossed modules are given.

💡 Research Summary

The paper investigates a pair of adjoint functors associated with a morphism of commutative k‑algebras φ : P → Q, focusing on the category XMod of crossed modules over commutative algebras. A crossed module (C,R,∂) consists of two commutative algebras C and R together with an algebra homomorphism ∂ : C → R and an action of R on C satisfying the Peiffer identities ∂(r·c)=r·∂(c) and c·c′=∂(c)·c′. Morphisms between crossed modules are pairs of algebra maps (f,g) that respect both ∂ and the action.

The first functor, the pull‑back φ⁎ : XMod/Q → XMod/P, is defined by “restriction of scalars”. Given a Q‑crossed module (C,Q,∂), one regards C as a P‑algebra via φ and keeps the same underlying set and map, i.e. φ⁎(C,Q,∂) = (C,P,∂∘φ). The R‑action on C is transferred by r·c := φ(r)·c. This construction is functorial and provides a cartesian lift for the projection functor p : XMod → k‑Alg, p(C,R,∂)=R.

The second functor, the induced functor φ_* : XMod/P → XMod/Q, is left adjoint to φ⁎. For a P‑crossed module (D,P,δ) one forms the tensor product D⊗_P Q, equips it with the natural Q‑algebra structure, and defines the boundary map ∂_ind : D⊗_P Q → Q by ∂_ind(d⊗q)=δ(d)·q. The resulting object (D⊗_P Q, Q, ∂ind) is a Q‑crossed module, denoted φ*(D). The adjunction is expressed by a natural bijection
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