# Divided Powers 4

Today we’ll look at the P.D. envelope of an ideal. To do this properly would take many pages of gory calculations, so we’ll be a little sketchy in order to get the idea out there. Before we do that we need to look at a construction I’ve been avoiding on purpose. Suppose ${M}$ is an ${A}$-module. Then there is a P.D. algebra ${(\Gamma_A(M), \Gamma_A^+(M), \gamma)}$ and an ${A}$-linear map ${\phi:M\rightarrow \Gamma_A^+(M)}$ satisfying the universal property that given any ${(B, J, \delta)}$ an ${A}$-P.D. algeba and ${\psi: M\rightarrow J}$ an ${A}$-linear map there is a unique P.D. map ${\overline{\psi}:(\Gamma_A(M), \Gamma_A^+(M), \gamma)\rightarrow (B, J, \delta)}$ with the property ${\overline{\psi}\circ \phi=\psi}$.

Let’s be a little more explicit what this is now. First, ${\Gamma_A(M)}$ is a graded algebra with ${\Gamma_0(M)=A}$, ${\Gamma_1(M)=M}$ and ${\Gamma^+(M)=\bigoplus_{i\geq 1} \Gamma_i(M)}$. Let’s denote ${x^{[1]}}$ for ${\phi(x)}$ and ${x^{[n]}}$ for ${\gamma_n(\phi(x))}$. In fact, by abusing notation we often just write ${[ \ ]}$ in place of ${\gamma}$ for the P.D. structure. This is because ${\Gamma_n(M)}$ is generated as an ${A}$-module by ${\{x^{[q]}=x_1^{[q_1]}\cdots x_k^{[q_k]} : \sum q_i=n, x_i\in M\}}$. This should just be thought of as a “generalized P.D. polynomial algebra”. We’ll soon see its importance. Now back to the regularly planned post.

Let ${(A, I, \gamma)}$ be a P.D. algebra and ${J}$ an ideal of ${B}$ which is an ${A}$-algebra. There exists a P.D. envelope of ${J}$ which is a ${B}$-algebra denoted ${\frak{D}_{B,\gamma}(J)}$ (or sometimes just ${\frak{D}}$ when the rest is understood) with a P.D. ideal ${(\overline{J}, [ \ ])}$ such that ${J\frak{D}_{B,\gamma}(J)\subset \overline{J}}$ compatible with ${\gamma}$ and satisfying a universal property:

For any ${B}$-algebra, ${C}$, containing an ideal ${K}$ containing ${JC}$ with a compatible (with ${\gamma}$) P.D. structure ${\delta}$, there is a unique P.D. morphism ${(\frak{D}_{B, \gamma}(J), \overline{J}, [ \ ])\rightarrow (C, K, \delta)}$ which makes the diagram commute:

${\begin{matrix} (B, J) & \rightarrow & (\frak{D}, \overline{J}) \\ \uparrow & \searrow & \downarrow \\ (A, I) & \rightarrow & (C, K) \end{matrix}}$

Where that vertical right arrow should be dotted. Like I said, the existence of this thing takes a lot of tedious calculations. You basically show by brute force that it exists in a few special cases and then reduce the general case to these. These calculations actually bear out a few interesting things that we’ll just list:

First, ${\frak{D}_{B, \gamma}(J)=\frak{D}_{B, \gamma}(J+IB)}$. Next, if our structure map actually factors ${A\rightarrow A'\rightarrow B}$ and we have an extension to ${\gamma'}$ on ${A'}$, then ${\frak{D}_{B, \gamma}(J)=\frak{D}_{B, \gamma'}(J)}$, which essentially tells us that it really is an “envelope” or some sort of minimal construction.

More importantly, we have a more concrete way to think about the construction. Namely, ${\frak{D}}$ is generated as a ${B}$-algebra by ${\{x^{[n]}: n\geq 0, x\in J\}}$ and any set of generators of ${J}$ gives a set of P.D. generators for ${\overline{J}}$.

Let’s do two examples to get a feel for what this thing is. Let ${B=A[x_1, \ldots, x_n]}$ and ${J=(x_1, \ldots, x_n)}$, then we’ll consider the trivial P.D. structure on ${A}$ given by the ${0}$ ideal. This gives us ${\frak{D}(J)=A\langle x_1, \ldots, x_n\rangle}$ the P.D. polynomial algebra. Along the same lines but slightly more generally, suppose ${M}$ is an ${A}$-module and ${B=\mathrm{Sym}(M)}$. Let ${J}$ be the ideal of generated by the postive graded part ${J=\mathrm{Sym}^+(M)}$ and do everything with respect to the trivial P.D. structure on ${A}$ again. We get ${\frak{D}=\Gamma_A(M)}$.

The other example is that when we have ${\gamma}$ extending to ${B/J}$ with a section ${B/J\rightarrow B}$, then the compatibility condition is irrelevant. This just means that ${\frak{D}_{B, \gamma}(J)\simeq \frak{D}_{B, 0}(J)}$. This is just because the section gives us that ${\frak{D}_{B, 0}(J)=B/J\bigoplus \overline{J}}$ and an application of the universal property.

I’m not sure what to do next. I’m sure I’ve alienated all my readers with all this. At this point I could shift over and at least define crystalline cohomology. We have enough to cover some of the basic definitions and actually show that all this has a purpose. On the other hand, we definitely have a ton more properties we should do if we want to get anywhere with crystalline cohomology.