A regular local ring is a UFD. See Lemma Tag 0AG0.

# Theorem of the day

Let (A,I) be a henselian pair. Set X = Spec(A) and Z = Spec(A/I). For any torsion abelian sheaf F on X_{e´tale} we have H^q_{e´tale}(X, F) = H^q_{e´tale}(Z, F|Z). See Theorem Tag 09ZI.

Slogan: Affine analogue of the proper base change theorem (due to Gabber; can also be found in a paper by Huber)

# Lemma of the day

Let k be a field. Let G be a separated group algebraic space locally of finite type over k. There does not exist a nonconstant morphism f : P^1_k → G over Spec(k). See Lemma Tag 0AEN.

Slogan: no (complete) rational curves on groups.

# Lemma of the day

Let X be a quasi-compact and quasi-separated scheme. Let U, V be quasi-compact disjoint open subschemes of X. Then there exist a (U ∪ V)-admissible blowup b : X′ → X such that X′ is a disjoint union of open subschemes X′ = X′1 ⨿ X′2 with b^{−1}(U) ⊂ X′1 and b^{−1}(V) ⊂ X′2. See Lemma Tag 080P.

Slogan: separate irreducible components by blowing up.

# Lemma of the day

Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let C be the full subcategory of (Sch/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y. Then the blowing up b : X′→X of Z in X is a final object of C. See Lemma Tag 0806.

Slogan: Universal property of blowing up

# Lemma of the day

Let S be a scheme. Let {X_{i} → X}_{i ∈ I} be an fppf covering of algebraic spaces over S. Assume I is countable (we can allow larger index sets if we bound the size of the algebraic spaces or if we don’t worry about set theoretic issues). Then any descent datum for algebraic spaces relative to {X_{i} → X}_{i ∈ I} is effective. See Lemma Tag 0ADV.

Slogan: fppf descent data for algebraic spaces are effective.

# Example wanted

Can somebody give me an explicit example of a Noetherian local domain A which does not have resolution of singularities? In particular, I would like an example where the completion A^{*} of A defines an isolated singularity, i.e., where Spec(A^{*}) – {m^{*}} is a regular scheme?

A related question (I think) is this: Does there exist a Noetherian local ring A such that there exists a proper morpism Y —> Spec(A^{*}) whose source Y is an algebraic space which is an isomorphism over the punctured spectrum such that Y is **not** isomorphic to the base change of a proper morphism X —> Spec(A) whose source is an algebraic space?

I wanted to make an example for the second question by taking a Noetherian local ring A which does not have a resolution whose completion A^{*} is a domain and defines an isolated singularity. Then a resolution of singularities Y of A^{*} would presumably not be the base change of an X, because if so then X would presumably be a resolution for A.

Anyway, I searched for examples of this sort on the web but failed to find a relevant example. Can you help? Thanks!

[**Edit 23 July 2014:** The discussion is continued in this blog post.]

# Bounding the first betti number

For some reason I am annoyed with the use of the structure of Jacobians and abelian varieties in the proof of finiteness of the first l-adic betti nr of a curve. Here is a silly argument to get around this.

Let k be an algebraically closed field. Let X be a smooth projective curve over k. We want to prove the number of finite etale G = Z/lZ covers is finite. In fact the argument will work for any finite group G.

If not then we get p_n : Y_n —> X, n = 1, 2, 3, 4, … which are finite etale covers with Galois group G and which are pairwise nonisomorphic. By Riemann-Hurwitz each of the curves Y_n has the same genus, call it g. Choose an integer d bigger than 2g and prime to |G|. Fix closed points x_i in X where i = 1, …, d. Set D = \sum x_i as a divisor on X. For each n pick points y_{i, n} in Y_n mapping to x_i. Then we can find a rational function f_n on Y_n which has poles exactly at the set of points y_{1, n}, …, y_{d, n}. Then we get a morphism

(f_n, p_n) : Y_n —–> P^1 x X

which is birational onto its image. By our careful choice of f_n the divisor class of the image is the class of a fixed line bundle, namely L = O_{P^1}(|G|) ⊗ O_{X}(D). Thus in the linear system |L| we get an infinitude of curves whose normalization is finite etale Galois over X with group G. By standard (but nontrivial) arguments, we get an actual family of such curves which contains an infinity of our Y_n. However, there are no moduli of finite \’etale Galois covers (another standard fact). Hence infinitely many Y_n are isomorphic (as Galois covers) which is a contradiction.

Of course this is quite useless!

# Pointers to the literature

Just this morning we updated the Stacks project website (huge thanks to Johan Commelin and Pieter Belmans) so that we can now start adding precise outside references. Since the Stacks project is selfcontained we view these references more as pointers to the literature. Currently we have only four examples of this:

- Yoneda’s lemma
- Nakayam’s lemma
- definition of a birational morphism
- galois cohomology and vanishing of brauer groups

If you click on one of these links and then look in the sidebar you should find a gadget with title “References” which gives you a reference. This sidebar gadget only shows up on the page of the lemma and not on the page for the section which contains the lemma; it also does not show up on the page of lemmas for which we have not yet added references.

The hope is that you, dear reader, will help. While browsing the Stacks project you may come to a lemma for which you know a precise reference. If so, please leave a corresponding comment on the page of the lemma.

The way things are setup we can add these specific references to any lemma, proposition, theorem, remark, example, exercise, situation, equation, or definition. I stress once more that these references should be as precise as possible (i.e., point to an exact result and/or an exact page number if known). If the result in the paper referenced is different, it may be helpful to briefly explain how. Also, it is very useful if multiple references are given for a single result.

More technical information can be found in this post. At the moment parts 1, 2, 4, 5, 6, and 7 of the list there are done. Please feel free to help with parts 3, 8, and 9.

# Slow with updating comments

Dear commenters,

This is just to let you know that I’ve been a bit slow with approving the valid comments as I am getting too many spam comments. This blog gets almost 500,000 spam comments per month. Yes, this is insane as only 5 people read this blog! Most of these spam comments are filtered out automatically but some get through which is already annoying enough to make me not want to check comments. If you did leave a comment and you are annoyed that it doesn’t show up, then you can try emailing me directly.

Thanks for your patience!