This week I learned an interesting fact about uniform annihilators of high degree Ext modules from a paper by Iyengar and Takahashi “The Jacobian ideal…”. I dare say there are many other places in the literature to read about it. In fact, I wouldn’t mind at all if you emailed me references where I could learn more about it.<\/p>\n
The general gist of the results is that given a “good” Noetherian ring S and an ideal I ⊂ S cutting out the singular locus, then there exist integers m and i_0 such that for all i > i_0 the modules Ext^i_S(M, N) are annihilated by I^m. The key here is that m does not depend on M, N.<\/p>\n
Iyengar and Takahashi show that this an essential ingredient if you want to prove strong generation for the category of modules and the derived category D^b_{Coh}(S). I would guess that conversely strong generation of D^b_{Coh}(S) will imply some uniform vanishing of Ext’s but I didn’t try to prove it; have you?<\/p>\n
Let me explain a strategy to get a result like this. It only works if you can write S as the quotient of a regular ring and you have plenty of derivations. For example for finite type algebras over perfect fields the proposition below proves what I said above (but you can also find it in the literature of course). If you are still reading, the strategy is given in the proof of the proposition; I suggest skipping the details.<\/p>\n
Let R be a regular ring. Let R —> S = R\/J be a quotient. Assume we have f_1, …, f_c in J and derivations D_1, …, D_c : R —> R as well as an element z’ in R such that z’ J is contained in (f_1, …, f_c) + J^2. Let<\/p>\n
z = det(D_i(f_j))<\/p><\/blockquote>\n
be as in the previous blog post. Finally, let n be the integer found in the first lemma of the previous post (this integer was found in a nonconstructive manner, but I hope somebody can tell me how to make it effective in some way).<\/p>\n
Proposition:<\/strong> Let d = dim(R) < ∞. For any finite S-modules M, N we have z^{2n + 1} (z’)^{2n} annihilates Ext^{d + 1}_S(M, N).<\/p>\n
Proof.<\/strong> Denote i_* : D(S) —> D(R) the pushforward and denote i* : D(R) —> D(S) the pullback. We have Ext^{d + 1}_R(i_*M, i_*N) = 0 because R is regular of dimension d. Thus we have Ext^{d + 1}_S(i*i_*M, N) = 0. But in the previous post we have seen that up to multiplication by z^{2n + 1} (z’)^{2n} the module M is a summand of i*i_*M. This concludes the proof. EndProof.<\/strong><\/p>\n
Remark<\/strong> If S is a finite type k algebra for some perfect field k and we choose a surjection R = k[x_1, …, x_t] —> S with kernel J then for choices f_1, …, f_c in J and derivations D_1, …, D_c on R, the elements z^{2n}(z’)^{2n + 1} we obtain in S generate an ideal whose vanishing locus is the singular set of Spec(S).<\/p>\n","protected":false},"excerpt":{"rendered":"
This week I learned an interesting fact about uniform annihilators of high degree Ext modules from a paper by Iyengar and Takahashi “The Jacobian ideal…”. I dare say there are many other places in the literature to read about it. … Continue reading