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I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic theorem. The others use "nullstellensatz" in a generic way, so I'm curious to know, short of reading many dozens of papers, what feature or features historically have made a new theorem count as a new nullstellensatz.

I imagine this question will have diverse answers depending upon what features of what version of the original theorem have some natural translation or extension into a given parallel discipline (to commutative ring theory). Unless I'm lucky that one respondent has made it their business to survey this whole literature, I expect a full(ish) answer will emerge only from multiple contributions, so please don't feel shy to tell just the part of the story you know.

Community wiki?

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=) trying to get to 3000 reputation, I see. I'll bite. +1 –  Harry Gindi Jan 21 '11 at 6:39
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Yes, probably CW. As you probably know, Nullstellensatz means "zero-locus-theorem," and thus a "nullstellensatz" should prove something about the zero locus of general sets of polynomials. Hilbert's Nullstellensatz and the combinatorial nullstellensatz (capitalization?) certainly satisfy this criterion. –  Daniel Litt Jan 21 '11 at 6:53
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Thanks for biting Harry, but anyone who knows me knows I always have a million questions and was just born for MO. :) Ken Appel says that when his kids were little he would get them to give him backrubs in exchange for points. It all worked perfectly until one of his kids ask "Daddy, what are points for?" –  David Feldman Jan 21 '11 at 7:57
    
Well, congratulations on reaching 3000 anyway. You can now shut down other people's questions! –  Harry Gindi Jan 21 '11 at 8:19
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David, regarding your above comment: My introducing you to MO seems almost to repay your introducing me to mathematics. I'm very happy there is such a great venue for your questions!!! –  Jon Bannon Jan 27 '11 at 19:34
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7 Answers 7

I e-mailed Bill Lawvere a link to this question, and in particular, Tom's answer, and he asked me to post this for him:


I will try to clarify the thread associated with this name for the past 120 years. The clarification involves generalizations of the type that I will need for my research, although I have not yet proved any genuinely new result.

The classical existence theorem of Hilbert has many precise analogues, strengthenings, and generalizations. Since ‘Stellen’ means ‘places’ in German, one sees immediately that the content is the geometrical one of the existence of places in a space which satisfy given conditions. Because the conditions considered are equations between functions defined on the space, the geometry is intimately related to the algebra of such functions. However, to speak of zeroes (Nullstellen) is an unnecessary restriction, useful however in limited contexts where there are theorems available concerning factorization et cetera. The question of existence and partial answers also make sense for rigs (‘rings’ without necessarily an everywhere-defined subtraction, for example in the natural numbers or in real algebraic geometry where one seeks ‘Positivensaetze’.) The usefulness of the restriction to algebras with negatives led to the development of the technique of ideal theory, in particular to the study of the generation of the unit ideal, et cetera. However, from a more conceptual perspective the purpose of the ideals is to give rise to quotient algebras. In that light one sees that the more natural algebraic interpretation of closed subset is as a surjective algebra homomorphism from the algebra of functions on a space to another algebra; in the same spirit the role of points (i.e. the desired Stellen) is to act as general ‘evaluation’ homomorphisms from the same algebra to special algebras. (The concern about maximal ideals and prime ideals comes really from the question of which algebras are special.) The general idea is that the special algebras can be qualitatively smaller than the typical algebras, but such homomorphisms can be proved to exist nonetheless.

Garrett Birkhoff’s 1935 theorem on the ubiquity of subdirectly irreducible algebras implies a qualitative improvement in that there are even enough such generalized points to yield a monomorphic embedding of algebras, as I will explain below. People studying Universal Algebra should consult Birkhoff’s paper in which he states very clearly that his theorem was motivated by work of Hilbert and Noether in algebraic geometry, even though it applies to much more diverse kinds of algebra. There are probably much more recent results in Universal Algebra which apply still to the commutative algebra case.

Hilbert’s original theorem concerned algebraic spaces of arbitrary finite dimension defined over a ground field; he proved that if the existence of points were true for a one-dimensional space, then it would be true for nontrivial spaces of all finite dimensions. (This is suggestive of the more recent theory of O-minimal spaces.) In his ‘algebraically closed’ case, there is only one special algebra, namely the ground field itself which geometrically is the function algebra of a single bare point. However, the theorem extends easily to the case when there is no such hypothesis on the ground field, by allowing arbitrary field extensions that are finite-dimensional as vector spaces to play the role of the special spaces or punctual figure shapes. These results have been further extended (permitting parametric families of spaces) to very general ground rings that are not even fields. (These more general ground rings include all those that are finitely generated as algebras over a smaller ground ring for which the theorem is true.)

By enlarging still further the category of special spaces, namely to general commutative algebras that are finite-dimensional over the ground field, and thus including geometrically not only fat points but also infinitesimal motions as expressed by nilpotent elements, there are in fact enough homomorphisms from any finitely-presented algebra (typically infinite-dimensional) to these special algebras, in the sense that given any two functions $f$ and $g$ in the algebra that are distinct, there exists such an infinitesimally variable point $x$ so that $f(x)$ is not equal to $g(x)$. Results of this general type I will refer to as a ‘strong Nullstellensatz’. It means algebraically that the given algebra is mapped monomorphically into an infinite product of special algebras. Continuing to a second stage of this resolution, the typical algebra is embedded into an inverse limit involving formal power series at each point. Birkhoff’s theorem is somewhat more precise, insisting always on subdirectly irreducible pieces, whereas the construction just sketched (a ‘coadequacy monad’)is content with subalgebras of subdirectly irreducible algebras, the homomorphisms being typically not surjective and hence more functorial.

In order to express this kind of results in a more fully geometrical way, I recall the method of analysis elaborated before 1960 by Grothendieck for fully revealing the inside of a space (contrary to spurious rumors that category theory treats objects as ‘opaque’). For simplicity, I think of the category $C$ of spaces under consideration as being embedded in a topos, but sufficient would be certain existence and exactness properties that that would imply. We assume given a small subcategory $A$ to serve as ‘figure shapes’; in the case of smooth, analytic, algebraic, real algebraic, et cetera, contexts these figure shapes would typically be taken as those for which the associated function algebras are finitely presented in their appropriate category. Then the inside of any space $X$ is the discretely fibered category $A/X \to A$, this being the functor that assigns to every figure its shape; the maps in the comma category $A/X$, namely the commutative triangles over $X$, suffice to account for all incidence relations between the figures and hence for the structure of the inside of $X$. Of course, discrete fibrations are equivalent to contravariant set-valued functors, or presheaves, but the discrete fibration formulation seems to be closer to the original geometry; in any case, these discrete fibrations over a given category $A$ constitute a topos in which we assume that $C$ is fully embedded. Grothendieck referred to this analysis of the inside of $X$ as the ‘functor of points’, which I find misleading because of the 2000-year old tradition according to which points are very special figures. Thus we assume also a subcategory $P$ of $A$. Again, there is an obvious attempt to represent any $X$ as the discrete fibration $P/X \to P$, but it is intuitively obvious that this representation will probably not be full because the cohesion has been thrown away. (I say cohesion because the classical term ‘continuity’ has been given a particular determination during the past century, which I am not considering; this classical idea is essentially the preservation of incidence relations without tearing them.) Typical examples of my ‘Axiomatic Cohesion’ are obtained by comparing the toposes generated by such a pair $(A,P)$, resulting in a quartet of functors anyone of which determines the other three by adjointness, the two downward ones expressing the idea of connected components and the other one expressing the idea of points, whereas the upward ones express the minimal completeness of the topos in the fact that any discrete space gives rise to opposite discrete and codiscrete spaces between which any space with those points sits. Here it is crucial that the term discrete be understood as ‘semi-discrete’, cohesion being relative. For example, in the original context of algebraic geometry, $P$ would typically be the opposite of the category of finite field extensions, with generated topos being Boolean, but not the category of abstract sets, except in the case of algebraically closed ground field. The epimorphicity of the map from points to components says intuitively that for a space $X$, the extent to which its component set is non-empty, is the extent to which its point set is non-empty; however, ‘set’ means an object in the lower topos and by basic internal logic, internal existence means actual existence only locally, so that the point of $X$ which is asserted to exist is not necessarily over the terminal object $1$, but rather over a finite extension field.

For innumerable reasons, it is important that the left most functor of the quartet, the set of components, should preserve finite products. This may not be true if we take $P$ too small, for example, only the terminal object $1$, attempting to use abstract sets as the lower base topos. This is one reason why I have relativized discrete to semi-discrete. The reason for emphasizing the right most functor of the quartet, namely the codiscrete space, may not be obvious, but becomes more significant if we hope to obtain a strong version of the Nullstellensatz by considering an intermediate category I of infinitesimal motions between $P$ and $A$, thus expressing some of the functors of the quartet as composites. Because of the relative completeness of toposes, the inverse limit of algebras implicit in the analog of Birkhoff’s construction can be interpreted as simply the function algebra of the subspace $sk(X)$ of $X$ obtained as the union of all infinitesimal subspaces. But there is then also the dual inclusion of this intermediate topos into the upper one, assigning to $X$ a space $cosk(X)$. A typical mathematical problem could be stylized as whether a formal function $sk(X) \to R$ can be extended to $X$ without changing $R$, or dually, whether a formal path $R \to cosk(X)$ can be lifted to $X$. The strong Nullstellensatz becomes rather a property of particular objects $R$, typically objects of $A$, but not of all objects of $C$, namely that $R$ ‘perceives’ the inclusion $sk(X) \to X$ as epimorphic (meaning that the induced map of function spaces $R^X \to R^{sk(X)}$ is monomorphic.)


Bill Lawvere (who is following this thread from the shadows) sent me another e-mail to post "in response to the concern expressed in some of the comments about the adjointness of $\pi_0$" (presumably on another answer):


The functor including (semi-)discrete spaces into general smooth or cohesive spaces is indeed right adjoint to the functor assigning the (semi-)discrete space of suitably connected components to any smooth or cohesive space. This tends to be generally true when the two categories in question are (‘gros’) toposes constructed by the usual methods and connected by induced functors and their adjoints.

Here the semidiscrete spaces are indeed more structured than the totally discrete abstract sets of Cantor but NOT in the sense of the ‘zero-dimensional spaces’ that seem to arise if one mistakenly adheres to the idea that there is a default notion of space determined entirely by Sierpinski-valued funtions. Rather the inside of a space should instead be analyzed by the covariantly associated system of figures and incidence relations, as I explain in my Palermo paper on Volterra’s Functionals. The duality is not at all a naïve metaphysical one, as illustrated dialectically by Liouville’s well-known theorem that projective space is not determined by its meager algebra of functions even though functions are a key tool in analyzing the whole topos of spaces in which it sits. Hurewicz did not submit to the default when he invented k-spaces in the late 40’s in response to the needs of analysis and of algebraic topology for map spaces satisfying the exponential laws.

It is often casually asserted (for example in Wikipedia) that schemes are topological spaces. That borders on disinformation because the functor from schemes to topological spaces does not even preserve products (hence not group objects, etc.). That is why it was such a tour de force when Grothendieck proved that fibered products even exist for schemes (on the basis of the old definition). In a colloquium talk here at Buffalo in 1973 Grothendieck forcefully advocated that the definition of scheme based on locally ringed spaces be abandoned as basic in favor of a definition based on simple gros topos constructions from which the prime ideals, open set lattices, petite sheaf of functions, etc could be recovered whenever helpful by geometrically intuitive means, but not as baggage in the working definition. (I had already reached that conclusion myself in 1966 discussions with Gabriel, which immediately suggested that similar methods could be applicable in the smooth case as well, as Grothendieck had begun to do in embryonic form already in 1960 in the analytic case.)

Geometric intuition, in the roughly ‘topological’ sense advocated by Grothendieck in his plea for Tame Topology, applies to directly in any category with suitable properties, even as general as ‘extensive’ categories. For example, the opposite of the category of Boolean algebras relates directly to the ‘algebraic geometry’ of the topos of presheaves on finite non-empty sets, whose manifold uses have been unjustly neglected. The ideal-theoretic techniques developed by Noether, Krull, Gelfand, Stone, Jacobson, et al should be used when appropriate for calculation but should not be permitted to obscure the geometric intuition.

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Pointers to more explanation of what the above is about are on the nLab, here: ncatlab.org/nlab/show/Nullstellensatz#GeneralAbstract –  Urs Schreiber Feb 11 at 1:07
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What I find intriguing is that the Nullstellensatz is underappreciated in the sense that many people appeal to a variation of it without saying (or realizing) they do.

For example, Hadamard's lemma says that if a $C^{\infty}$ function $f$defined in a convex neighbourhood of the origin vanishes at the origin, you can write it as $f=\Sigma x_ig_i $ for some $C^{\infty}$ functions $g_i $ (The proof is by elementary integration and has nothing to do with the usual tools revolving around the algebraic version) This is typically a Nullstellensatz result but I have never seen it referred to as such (a challenge: can a reader supply a reference with Hadamard's result characterized as a Nullstellensatz-type theorem?)

Added later Let me add a few words of propaganda to our $\mathcal C^\infty$ differential geometry/topology friends. If you take this point of view on Hadamard's result seriously, you will be led to associate to a hypersurface $H\subset \mathbb R^n$, given locally by equations $f_i=0$, a line bundle $L_H$ on $R^n$, with cocycle $g_{ij}=f_i/f_j$ (the quotient makes sense because of Hadamard: shades of L'Hospital ?! ). Its restriction $L_H|H$ is the normal bundle to $H$. But since $L_H$ is trivial (like all line bundles on $\mathbb R^n$), the normal bundle to $H$ is also trivial and, presto , you have proved that all hypersurfaces in $\mathbb R^n$ are orientable!

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Something like sbseminar.wordpress.com/2007/12/14/… ? –  Qiaochu Yuan Jan 21 '11 at 11:25
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Dear Qiaochu, I don't think so. Although the analogy with partitions of unity is quite interesting (and was probably mentioned in writing for the first time by Mumford in his Red Book ), the post doesn't seem to be in the spirit of Hadamard' result . I would even say that Hadamard is rather the complementary situation : what can you say of a function vanishing on the common zeroes of functions that do vanish simultaneously ( whereas the post studies the situation where they don't have a common zero). –  Georges Elencwajg Jan 21 '11 at 13:41
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For a field $k$, by a "Nullstellensatz" over $k$, I mean an explicit description of the Galois connection between subsets of $k^n$ and ideals in the polynomial ring $k[x_1,\ldots,x_n]$. See this MO question of mine for more on this perspective.

Whenever one has a ring of functions on a space $X$, there is an induced Galois connection, hence one can ask for a Nullstellensatz: see page 15 of these notes for a(n unfortunately not yet very good) description of this perspective.

In particular there are analytic Nullstellensatze, for instance. I recently found this interesting note concerning the possibility of a Nullstellensatz for the ring of continuous functions on a compact space. (It shows that things work out nicely for maximal ideals and argues fairly convincingly that there is nothing good for all prime ideals. Somehow it did not completely extinguish my hopes, though.)

Certainly the term "Nullstellensatz" has been used for things which do not come under the aegis of the above. For instance, there is Alon's Combinatorial Nullstellensatz, which I unfortunately don't understand very well at all other than as some distant relative of the Chevalley-Warning theorem.

By the way, you say that most of the 200 papers are not referring (albeit obliquely) to Hilbert's Nullstellensatz? I find that somewhat hard to believe. Can you give some examples of far off uses of the term?

(Added: searching "anywhere" for Nullstellensatz on MathSciNet gives 632 matches. I should really turn in soon, so I can't look at them much now. I was surprised by how many of the most recent ones are referring to the Combinatorial Nullstellensatz, but apparently this is only 29 papers in all. By my very cursory glance, most of them do seem to be roughly in line with what I was suggesting above.)

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Here's a pretty abstract answer. Bill Lawvere, working on his ideas about "axiomatic cohesion", often talks about a Nullstellensatz that at first sight has nothing whatsoever to do with zeros of polynomials.

Axiomatic cohesion is about pinning down the properties that a category of "spaces" should have. Here the word "space" is up for negotiation; the word "cohesion" is to indicate that a space should somehow cohere to itself. The typical situation when you have a category Sp of spaces (in whatever sense) is that there's a string of adjoint functors $$ \pi_0 \dashv D \dashv U \dashv I $$ between Sp and Set, where

  • $\pi_0$ gives the set of connected components of a space
  • $D$ gives the discrete space on a set
  • $U$ gives the set of points of a space
  • $I$ gives the indiscrete or codiscrete space on a set.

A couple of axioms are imposed on these adjunctions. Under those axioms, there are canonical natural transformations $U \to \pi_0$ and $D \to I$, and the former is an epimorphism iff the latter is a monomorphism. Lawvere calls this property (that $U \to \pi_0$ is an epimorphism) the Nullstellensatz.

In concrete terms, this says something like: for a space $X$, the quotient map from $X$ to its set of connected-components is surjective. Why call that the Nullstellensatz? I have no idea. Here's the reference (top of p.44).

In similar usage, Colin McLarty says (bottom of p.125) that a topos satisfies the Nullstellensatz if for every nonempty object $X$ there is at least one map $1 \to X$. Again I don't understand the usage. Maybe someone else will wander along and help out.

Update: Peter Johnstone has just published a paper all about this. He prefers the term "punctual local connectedness" to "Nullstellensatz". Here it is: http://tac.mta.ca/tac/volumes/25/3/25-03abs.html

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>In similar usage, Colin McLarty says (bottom of p.125) that a topos satisfies the Nullstellensatz if for every nonempty object there is at least one map . Again I don't understand the usage. Maybe someone else will wander along and help out. I suppose that the analogy goes like this, briefly: an ideal is morally non-empty if it doesn't equal the unit ideal. Then the "weak nullstellensatz" provides a homomorphism from the polynomial ring to the ground field $F$, and passing to spectra so the arrows go the other way, you get a $1(=Spec(F))\rightarrow X$ map. –  David Feldman Jan 21 '11 at 7:49
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To elaborate on David's comment: let $k$ be an alg. closed field, $S$ the category of $k$-schemes of finite type, $T$ the topos of Zariski sheaves on $S$. Then "$1$" is represented by $\mathrm{Spec}(k)$. Hilbert's Nullstellensatz says that if $U$ is a nonempty object of $S$ then there is a morphism $1\to U$. This immediately extends to every nonempty object of $T$. –  Laurent Moret-Bailly Jan 21 '11 at 8:01
    
But do Laurent's and David's comments justify it? They justify McLarty's usage, but I don't see that they justify Lawvere's. (The two usages are almost certainly related, as McLarty is very familiar with Lawvere's work.) Doubtless Lawvere did have a reason. I just don't know what it was. –  Tom Leinster Jan 21 '11 at 13:58
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I think Lawvere's criteria is justified in the same way: Doesn't it just say that every connected component is inhabited by actual point? –  Steven Gubkin Jan 21 '11 at 14:24
    
@Kevin: And yet, Sylvester, with all due respect, is no Newton. This applies to the current situation, mutatis mutandis. –  Igor Rivin Jan 21 '11 at 17:08
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Let's give this a try. A "nullstellensatz" is about the existence of zeros of polynomials, otherwise said, about the solutions of a certain "equation". So every time someone thinks that some result guarantees the existence of solutions, you have a nullstellensatz. This applies to solutions of systems of polynomial equations over an algebraically closed field (Hilbert's nullstellensatz), or to existence of solutions for any such system if you allow extending the field where you can take the solution (the "easy" nullstellensatz in modern algebraic geometry). Note that the first one fixes the behavior of the maximal spectrum, while the second one refers to the prime spectrum. It should work in any situation with a Galois connection, to characterize when some subsystem of the "algebraic part" corresponds to something non empty on the "geometric part". This is just a though and perhaps it does not apply to the examples you have in mind.

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I wanted to post a comment to Tom's answer, but couldn't figure out how to do it, so I add it as an answer...

I think you can interpret the classical Nullstellensatz in this language as follows: Take $\mathbf{Sp}$ to be the opposite of the category of reduced finitely generated $k$-algebras, and extend it to have arbitrary co-products (so a full sub-category of $k$-schemes). Take $\pi_0$ to assign to each algebra the set of maximal ideals, $D(A)=\coprod_Aspec(k)$, $U(X)$ the set of $k$-points. The map from $U$ to $\pi_0$ assigns to a $k$-point the corresponding kernel. Then the usual Nullstellensatz says that this map is surjective if $k$ is algebraically closed.

I didn't think what $I$ should be in this case...

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Regarding the commenting: I also was surprised by this when first arriving to the site. There is a minimal number of rep-points one needs to be allowed to comment. When you reach this, there will be a link 'add comment' below the question and each answer. –  quid Jan 21 '11 at 17:10
    
Thanks. Looks like I can comment on my own posts, at least... –  Moshe Jan 21 '11 at 20:27
    
Actually, this is wrong, $D$ is not right adjoint to $\pi_0$, maybe only if we restrict to $0$-dimensional schemes... –  Moshe Jan 21 '11 at 22:39
    
$\pi_0$ should be the set of idempotent prime ideals. $U$ should be the set of maximal ideals. –  Will Sawin Feb 10 at 21:41
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Reyes,Reyes,Zolfaghari discuss the nullstellen terminology in 'generic figures and their glueings' (2004,pp.204ff). They define that a cat $\mathcal A$ with initial and terminal object satisfies the nullstellensatz when every non-initial $X$ has a point $1\rightarrow X$. They show that for a presheaf topos the nullstellensatz implies the existence of the right adjoint $I$ to the section functor $U$. They also have a theorem attributed to Lawvere that in a presheaf topos a single nullstelle $1\rightarrow X$ for representable $X$ already implies the nullstellensatz. The parallel to the Hilbert nullstellensatz is that a nullstelle $p$ in $k^n$ for a polynomial ideal $I\subseteq k[x_1,..,x_n]$ amounts to homomorphism $k[x_1,..,x_n]/I\rightarrow k$ which in the opposite geometric category is a morphism from $k=1$. As a reference for the theorem they attribute to Lawvere, Reyes and Zolfaghari in their Como 1990 proceedings paper give the 1986 'generalized spaces' -paper of Lawvere (TAC-reprint). There a remark points out that the right adjoint exists for the Zariski spectrum of a local ring.

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