commutative algebra


In this post, we shall accomplish the goal stated earlier: we shall show that a formally smooth morphism of noetherian rings (which is essentially of finite type) is flat. We shall even get an equivalence: flatness together with smoothness on the fibers will be both necessary and sufficient to ensure that a given such morphism is formally smooth.

In order to do this, we shall use a refinement of the criterion in the first post for when a quotient of a formally smooth algebra is formally smooth. We shall need a bit of local algebra to do this, but the reward will be a very convenient Jacobian criterion, which will then enable us to prove (using the results from last time on lifting flatness from the fibers) the final characterization of smoothness.

3. The Jacobian criterion

So now we want a characterization of when a morphism is smooth. Let us motivate this with an analogy from standard differential topology. Consider real-valued functions {f_1, \dots, f_p \in C^{\infty}(\mathbb{R}^n)}. Now, if {f_1, f_2, \dots, f_p} are such that their gradients {\nabla f_i} form a matrix of rank {p}, then we can define a manifold near zero which is the common zero set of all the {f_i}. We are going to give a relative version of this in the algebraic setting.

Recall that a map of rings {A \rightarrow B} is essentially of finite presentation if {B} is the localization of a finitely presented {A}-algebra.

Proposition 5 Let {(A, \mathfrak{m}) \rightarrow (B, \mathfrak{n})} be a local homomorphism of local rings such that {B} is essentially of finite presentation. Suppose {B = (A[X_1, \dots, X_n])_{\mathfrak{q}}/I} for some finitely generated ideal {I \subset A[X_1, \dots, X_n]_{\mathfrak{q}}}, where {\mathfrak{q}} is a prime ideal in the polynomial ring.Then {I/I^2} is generated as a {B}-module by polynomials {f_1, \dots, f_k \in A[X_1, \dots, X_n]} whose Jacobian matrix has maximal rank in {B/\mathfrak{n}} if and only if {B} is formally smooth over {A}. In this case, {I/I^2} is even freely generated by the {f_i}. (more…)

Ultimately, we are headed towards a characterization of formal smoothness for reasonable morphisms (e.g. the types one encounters in classical algebraic geometry): we want to show that they are precisely the flat morphisms whose fibers are smooth varieties. This will be a much more usable criterion in practice (formal smoothness is given by a somewhat abstract lifting property, but checking that a concrete variety is smooth is much easier).  This is the intuition between smoothness: one should think of a flat map is a “continuously varying” family of fibers, and one wishes the fibers to be regular. This corresponds to the fact from differential topology that a submersion has submanifolds as its fibers.

It is actually far from obvious that a formally smooth (and finitely presented) morphism is even flat. Ultimately, the idea of the proof is going to be write the ring as a quotient of a localization of a polynomial ring. The advantage is that this auxiliary ring will be clearly flat, and it will also have fibers that are regular local rings.  In a regular local ring, we have a large supply of regular sequences, and the point is that we will be able to lift the regularity of these sequences from the fiber to the full ring.

Thus we shall use the following piece of local algebra.

Theorem Let {(A, \mathfrak{m}) \rightarrow (B, \mathfrak{n})} be a local homomorphism of local noetherian rings. Let {M} be a finitely generated {B}-module, which is flat over {A}.

Let {f \in B}. Then the following are equivalent:

  1. {M/fM} is flat over {A} and {f: M \rightarrow M} is injective.
  2. {f: M \otimes k \rightarrow M \otimes k} is injective where {k = A/\mathfrak{m}}.


This is a useful criterion of checking when an element is {M}-regular by checking on the fiber. That is, what really matters is that we can deduce the first statement from the second. (more…)

I’ll now say a few words on formal smoothness. This happens to be closely related to the theory of the cotangent complex (namely, the cotangent complex provides a clean criterion for when a morphism is formally smooth). Ultimately, I would like to aim first for the result that a formally smooth morphism of finite presentation is flat, and thus to characterize such morphisms via the geometric idea of “smoothness” (even though the algebraic version of formally smooth is pure commutative algebra).

1. What is formal smoothness?

The idea of a smooth morphism in algebraic geometry is one that is surjective on the tangent space, at least if one is working with smooth varieties over an algebraically closed field. So this means that one should be able to lift tangent vectors, which are given by maps from the ring into {k[\epsilon]/\epsilon^2}.

This makes the following definition seem more plausible:

Definition 1 Let {B} be an {A}-algebra. Then {B} is formally smooth if given any {A}-algebra {D} and ideal {I \subset D } of square zero, the map

\displaystyle \hom_A(B, D) \rightarrow \hom_A(B, D/I)

is a surjection.

So this means that in any diagram

there exists a dotted arrow making the diagram commute. (more…)

As of late, I’ve been trying unsuccessfully to learn about Hilbert and Quot schemes. Nitin Nitsure’s notes (which appear in the book FGA Explained) are a good source, though they are also quite technical. I’ve been trying to read them, and the relevant parts of Mumford’s Lectures on curves on an algebraic surface. While doing so, I took a bunch of my own notes. They are right now unfinished, but they do cover the semicontinuity theorem and a small piece of the cohomology and base-change story. In addition, there is a little additional material (drafted) on applications of this to line bundles, which is more or less why Mumford develops all this in Abelian varieties.

I probably won’t get a chance to revise them further now. In fact, I’m going to take a long break from the “write lots of notes” mentality that has recently gripped me to focus on short, and ideally more-or-less self-contained posts in the future, not least because of limitations of time. This is something I’ve always had difficulty with: even my high school English teachers always had to tell me to shrink my essays. In a sense, though, the blog medium really is about pithy bites of profundity. Being short may take less time, but it’s also intellectually harder than just listing the main theorems in some small subsubsubfield and listing all the proofs in detail.

Yes, I’m still here. I just haven’t been in a blogging mood. I’ve been distracted a bit with the CRing project. I’ve also been writing a bunch of half-finished notes on Zariski’s Main Theorem and some of its applications, which I’ll eventually post.

I would now like to begin talking about the semicontinuity theorem in algebraic geometry, following Mumford’s Abelian Varieties. This result is used constantly throughout the book, mainly in showing that certain line bundles are trivial. Eventually, I’ll try to say something about this.

Let {f: X \rightarrow Y} be a proper morphism of noetherian schemes, {\mathcal{F}} a coherent sheaf on {X}. Suppose furthermore that {\mathcal{F}} is flat over {Y}; intuitively this means that the fibers {\mathcal{F}_y = \mathcal{F}  \otimes_Y k(y)} form a “nice”  family of sheaves. In this case, we are interested in how the cohomology {H^p(X_y, \mathcal{F}_y) =  H^p(X_y, \mathcal{F} \otimes k(y))} behaves as a function of {y}. We shall see that it is upper semi-continuous and, under nice circumstances, its constancy can be used to conclude that the higher direct-images are locally free.

1. The Grothendieck complex

Let us keep the hypotheses as above, but assume in addition that {Y = \mathrm{Spec} A} is affine, for some noetherian ring {A}. Consider an open affine cover {\left\{U_i\right\}} of {X}; we know, as {X} is separated, that the cohomology of {\mathcal{F}} on {X} can be computed using Cech cohomology. That is, there is a cochain complex {C^*(\mathcal{F})} of {A}-modules, associated functorially to the sheaf {\mathcal{F}}, such that

\displaystyle  H^p(X, \mathcal{F}) = H^p(C^*(\mathcal{F})),

that is, sheaf cohomology is the cohomology of this cochain complex. Furthermore, since the Cech complex is defined by taking sections over the {U_i}, we see that each term in {C^*(\mathcal{F})} is a flat {A}-module as {\mathcal{F}} is flat. Thus, we have represented the cohomology of {\mathcal{F}} in a manageable form. We now want to generalize this to affine base-changes:

Proposition 1 Hypotheses as above, there exists a cochain complex {C^*(\mathcal{F})} of flat {A}-modules, associated functorially to {\mathcal{F}}, such that for any {A}-algebra {B} with associated morphism {f: \mathrm{Spec} B  \rightarrow \mathrm{Spec}  A}, we have\displaystyle H^p(X \times_A B, \mathcal{F} \otimes_A B) =  H^p(C^*(\mathcal{F}) \otimes_A B).

Here, of course, we have abbreviated {X \times_A B} for the base-change {X  \times_{\mathrm{Spec} A} \mathrm{Spec} B}, and {\mathcal{F} \otimes_A B} for the pull-back sheaf.

Proof: We have already given most of the argument. Now if {\left\{U_i\right\}} is an affine cover of {X}, then {\left\{U_i \times_A B \right\}} is an affine cover of the scheme {X \times_A B}. Furthermore, we have that

\displaystyle  \Gamma(U_i \times_A B, \mathcal{F}\otimes_A B) = \Gamma(U_i ,  \mathcal{F}) \otimes_A B

by definition of how the pull-backs are defined. Since taking intersections of the {U_i} commutes with the base-change {\times_A B}, we see more generally that for any finite set {I},

\displaystyle  \Gamma\left( \bigcap_{i \in I} U_i \times_A B, \mathcal{F}\otimes_A B\right) =   \Gamma\left( \bigcap_{i \in I} U_i, \mathcal{F}\right) \otimes_A B.  (more…)

In classical algebraic geometry, one defines a subset of a variety over an algebraically closed field to be constructible if it is a union of locally closed subsets (in the Zariski topology). One of the basic results that one proves, which can be called “elimination theory” and is due to Chevalley, states that constructible sets are preserved under taking images: if {f: X \rightarrow Y} is a regular map and {C \subset X} is constructible, then so is {f(C)}. In general, this is the best one can say: even very nice subsets of {X} (e.g. {X} itself) need not have open or closed (or even locally closed) images.

In the theory of schemes, one can formulate a similar result. A morphism of finite type between noetherian schemes sends constructible sets into constructibles. One proves this result by making a sequence of reductions to considering the case of two integral affine schemes, and then using a general fact from commutative algebra. It turns out, however, that there is a more general form of the Chevalley theorem:

Theorem 1 Let {f: X \rightarrow Y} be a finitely presented morphism of schemes. Then if {C  \subset X} is locally constructible, so is {f(C)}.

I will explain today how one deduces this more general fact from the specific case of noetherian schemes. This will highlight a useful fact: oftentimes, general facts in algebraic geometry can be reduced to the noetherian case since, for instance, every ring is an inductive limit of noetherian rings. This can be developed systematically, as is done in EGA IV-8, but I shall not do so here.

N.B. As a result, this post is written entirely for those whom Ravi Vakil would call “non-noetherian people.” I will simply assume as known the noetherian results (which can be found easily, e.g. in Hartshorne or EGA I) and explain how they can be generalized. Nonetheless, even noetherian readers have a very good reason to care. In fact, it is through such a “finite presentation” argument that Grothendieck proves the general quasi-finite form of Zariski’s main theorem; the finite presentation trick is a very ingenious strategy, about which I hope to say more soon, that can reduce many results not only to the noetherian case, but also to the local case.

(more…)

[The present post is an announcement of the CRing project, whose official webpage is here.]

Like most mathematics students, I spend a lot of time writing stuff, for instance homework assignments and (of course) blog posts. So I have a lot of random, unorganized write-ups littered around my hard drive, which might be useful to others if organized properly, but which currently slumber idly.

Last semester, I took a fairly large amount of notes for my commutative algebra class (about 160 pages). I made the notes available on my webpage, and was pleased with the reception that they received from my classmates. After seeing Theo-Johnson Freyd’s projects, I decided that it might be a productive exercise to edit the notes I had taken into a mini-textbook. I quickly made progress, since the basic structure of the book was already set by the lectures. I decided early on that the work was going to be open source: to me, it seemed the best way to ensure that anyone who wanted could freely access and modify it.

But I think the project is bigger now. Namely, instead of an open source textbook, I want a massively collaborative open source textbook. This is to say that I don’t want it to be my work anymore, but my work as well as, and more importantly, the work of enthusiastic professors, procrastinating graduate students, nerdy high-schoolers,  or whoever else wishes to contribute. The goal is to end with an openly available textbook suitable for a beginner familiar only with elementary abstract algebra, but which will provide adequate preparation for the serious study of algebraic geometry.

So, I present you the CRing project. (more…)

All right. I am now inclined to switch topics a little (I am looking forward to saying a few words about local cohomology), so I will sketch a few details in the present post. The goal is to compute the sheaf cohomology groups of the canonical line bundles on projective space. The argument will follow EGA III.2; Hartshorne does essentially the same thing (namely, analysis of the Cech complex) but without the Koszul machinery, so his approach seems more opaque to me.

Now, let us compute the cohomology of projective space {X = \mathop{\mathbb P}^n_A} over a ring {A}. Note that {X} is quasi-compact and separated, so we can compute the Cech cohomology by the above machinery. That is, we will use the Koszul-Cech connection discussed two days ago. In particular, we will consider the quasi-coherent sheaf

\displaystyle  \mathcal{H}=\bigoplus_{m \in \mathbb{Z}} \mathcal{O}(m)

(more…)

The next big application of the Koszul complex and this general machinery that I have in mind is to projective space. Namely, consider a ring {A}, and an integer {n \in \mathbb{Z}_{\geq 0}}. We have the {A}-scheme {\mathop{\mathbb  P}^n_A = \mathrm{Proj} A[x_0, \dots, x_n]}. Recall that on it, we have canonical line bundles {\mathcal{O}(m)} for each {m \in \mathbb{Z}}, which come from homogeneous localization of the {A[x_0, \dots,  x_n]}-modules obtained from {A[x_0, \dots, x_n]} itself by twisting the degrees by {m}. When {A} is a field, the only line bundles on it are of this form. (I am not sure if this is true in general. I think it will be true, but perhaps someone can confirm.)

It will be useful to compute the cohomology of these line bundles. For one thing, this will lead to Serre duality, from a very convenient isomorphism that will spring up. For another, we will see that they are finitely generated over {A}. This is far from obvious. The scheme {\mathop{\mathbb P}^n_A} is not finite over {A}, and a priori this is not expected.

But to start, let’s think more abstractly. Let {X} be any quasi-compact, quasi-separated scheme; we’ll assume this for reasons below. Let {\mathcal{L}} be a line bundle on {X}, and {\mathcal{F}} an arbitrary quasi-coherent sheaf. We can consider the twists {\mathcal{F} \otimes \mathcal{L}^{\otimes m}} for any {m \in  \mathbb{Z}}. This is a bunch of sheaves, but it is something more.

Let us package these sheaves together. Namely, let us consider the sheaves:

\displaystyle  \bigoplus  \mathcal{L}^{\otimes m}, \quad \mathcal{H}=\bigoplus \mathcal{F} \otimes  \mathcal{L}^{\otimes m}

(more…)

So last time, when (partially) computing the cohomology of affine space, we used a fact about the Koszul complex. Namely, I claimed that the Koszul complex is acyclic when the elements in question generate the unit ideal. This was swept under the rug, and logically I should have covered that before getting to yesterday’s bit of algebraic geometry. So today, I will backtrack into the elementary properties of the Koszul complex, and prove a more general claim.

0.9. A chain-homotopy on the Koszul complex

Before proceeding, we need to invoke a basic fact about the Koszul complex. If {K_*(\mathbf{f})} is a Koszul complex, then multiplication by anything in {(\mathbf{f})} is chain-homotopic to zero. In particular, if {\mathbf{f}} generates the unit ideal, then {K_*(\mathbf{f})} is homotopically trivial, thus exact. This is one reason we should restrict our definition of “regular sequence” (as we do) to sequences that do not generate the unit ideal, or the connection with the exactness of the Koszul complex wouldn’t work as well.

Proposition 33 Let {g \in (\mathbf{f})}. Then the multiplication by {g} map {K_*(\mathbf{f})  \rightarrow K_*(\mathbf{f})} is chain-homotopic to zero.

Proof: Let {\mathbf{f} = (f_1, \dots, f_r)} and let {g = \sum g_i f_i}. Then there is a vector {q_g = (g_1, \dots, g_r) \in R^r}. We can define a map of degree one

\displaystyle  H: K_*(\mathbf{f}) \rightarrow K_*(\mathbf{f})

(more…)

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