Divided Power Structures 2


Today we’ll do a short post on some P.D. algebra properties and constructions. Let’s start with properties of P.D. ideals. Our first proposition is that given {(I, \gamma)} and {(J, \delta)} as two P.D. ideals in {A}, then {IJ} is a sub P.D. ideal of both {I} and {J}. This is very straightforward to check using the criterion from last time, since {IJ} is generated by the set of products {xy} where {x\in I} and {y\in J}. This proposition immediately gives us that powers of P.D. ideals are sub P.D. ideals and there is a natural choice for P.D. structure on them.

Another proposition is that given two P.D. ideals as above with the additional property that {I\cap J} is a P.D. ideal of {I} and {J} and that {\gamma} and {\delta} restrict to the same thing on the intersection, then there is a unique P.D. structure on {I+J} such that {I} and {J} are sub P.D. ideals. Proving this would require developing some techniques that would lead us too far astray. We probably won’t use this one anyway. It just gives a sense of the types of constructions that are compatible with P.D. structures.

Another construction that requires no extra effort are direct limits. If {\{A_i, I_i, \gamma_i\}} is a directed system of P.D. algebras, then {\displaystyle \left(\lim_{\rightarrow} A_i, \lim_{\rightarrow} I_i\right)} has a unique P.D. structure {\gamma} such that each natural map {(A_i, I_i, \gamma_i)\rightarrow (A, I, \gamma)} is a P.D. morphism.

Unfortunately, one common construction that doesn’t work automatically is the tensor product. It works in the following specific case. If {B} and {C} are {A}-algebras, and {I\subset B} and {J\subset C} are augmentation ideals with P.D. structures {\gamma} and {\delta} respectively, then form the ideal {K=\mathrm{ker}(B\otimes C\rightarrow B/I \otimes B/J)}. We then get that {K} has a P.D. structure {\epsilon} such that {(B, I, \gamma)\rightarrow (B\otimes C, K, \epsilon)} and {(C, J, \delta)\rightarrow (B\otimes C, K, \epsilon)} are P.D. Morphisms.

Next time we’ll start thinking about how to construct compatible P.D. structures over thickenings. Since we’ll be thinking a lot about {W_m(k)} I’ll just end this post by pointing out that {(p)\subset W_m} actually has many choices of P.D. structure. But last time we said that {(p)\subset W(k)} actually has a unique one, so our convention is going to be to choice the “canonical” P.D. structure on {(p)\subset W_m} induced from the unique one in {W(k)}.

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