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.

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