algebra

# 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 $\left(\lim_{\rightarrow} A_i, \lim_{\rightarrow} I_i\right)$ 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 ${W(k)}$.