# Blog Archives

# Topic Archive: commutative algebra

# A model theory of affine n-space via differential algebra

Affine $n$-space $A_k^n$ and its algebraic equivalent, the polynomial ring $k[x_1,…,x_n]$, are basic and widely studied objects in geometry and algebra, about which we know a great deal. However, there remains a host of basic open problems (like the Jacobian conjecture, Zariski Cancellation Conjecture, Complement Problem, …) indicating that our knowledge is nonetheless quite limited. In fact, the greatest obstacle in solving the above conjectures is our inability to “pinpoint” affine space among all varieties (or $k[x]$ among all finitely generated $k$-algebras): this is the so-called Characterization Problem.

The most recent approach to these problems is via additive group actions on affine $n$-space, which corresponds on the algebraic side, to the theory of locally nilpotent derivations. Using this, for instance, N. Gupta recently showed the falsitude of the Zariski Cancellation Conjecture in positive characteristic.

From a model-theoretic point of view, the polynomial ring (in its natural ring language) is quite expressive: in characteristic zero, one can define the integers (as a subset), one can express in general that, say, Embedded Resolution of Singularities holds, etc. Of course, one of the peculiarities of model theory (and probably one of the reasons for its pariah status) is the unavoidable presence of non-standard models. In other words, a characterization problem is never solvable in model theory, unless one allows some non first-order conditions as well (e.g., cardinality in categorical theories–but most mainstream mathematicians would not be too happy about that either). But other, more intrinsic problems arise: there are elementary equivalent fields whose polynomial rings are not. So, can we find an expanded language plus a “natural” but non first-order condition, that pinpoints the standard model, i.e., $k[x]$ within the models of its theory. Or even better, since these complete theories will have unwieldy axiomatizations, can we find a (recursive?) theory, whose only model satisfying the extra non first-order condition is $k[x]$?

In view of the recent developments in algebra/geometry, to this end, I will propose in this talk some languages that include additional sorts, in particularly, a sort for derivations. This is different from the usual language of differential fields, where one only studies a fixed (or possibly finitely many) derivation: we need all of them! We also need a substitute for the notion of degree, and the corresponding group $Z$-action as power maps. To test our theories, we should verify which algebraic/geometric properties are reflected in this setup. For instance, affine $n$-space has no cohomology, which is equivalent to the exactness of the de Rham complex, and this latter statement is true in any of the proposed models. Nonetheless, this is only a preliminary analysis of the problem, and nothing too deep will yet be discussed in this talk.

# In search of a complete axiomatization for $F_p((t))$

I will give a survey of the attempts that have been made since the mid 1960’s to find a complete recursive axiomatization of the elementary theory of $F_p((t))$. This problem is still open, and I will describe the difficulties researchers have met in their search. Some new hope has been generated by Yu. Ershov’s observation that $F_p((t))$ is an “extremal” valued field. However, while his intuition was good, his definition of this notion was flawed. It has been corrected in a paper by Azgin, Kuhlmann and Pop, in which also a partial characterization of extremal fields was given. Further progress has been made in a recent manuscript, on which I will report at the AMS meeting at Rutgers. The talk at the Graduate Center will provide a detailed background from the model theoretic point of view.

The property of being an “extremal valued field” is both elementary and very natural, so it is an ideal candidate for inclusion in a (hopefully) complete recursive axiomatization for $F_p((t))$. It implies an axiom scheme that was considered previously, which describes the behavior of additive polynomials under the valuation. I will discuss why additive polynomials are crucial for the model theory of valued fields of positive characteristic.

The open problems around extremal fields provide a good source of research projects of various levels of difficulty for young researchers.

This talk is jointly sponsored by the Commutative Algebra & Algebraic Geometry Seminar and the CUNY Logic Workshop.

# Euclidean domains of arbitrarily high Euclidean rank

We will construct Euclidean domains of arbitrarily high Euclidean rank.

# Towards a model theory of Zariski-Riemann spaces of valuations

Zariski introduced the space of all valuations on a given field *K* and named it after his mentor the `Riemann manifold’. This terminology is justified because of the following two facts he proved about it: (1) one can define a (quasi-)compact topology on this space (and we honor him embracingly by calling it the Zariski-Riemann space), and (2) if *K* is the function field of a curve, then this space is isomorphic to a non-singular curve with the same function field. Hence (2) gives resolution of singularities in dimension one, and he then used fact (1) to show the same in dimension two. Abhyankar then followed suit by proving that in dimension two, any point (=valuation) in this space is attainable by blowing ups, but his student Shannon showed that this failed in dimension three and higher. This latter negative result somehow ended the Zariski project of using the space of all valuations to prove resolution of singularities in higher dimensions.

However, there is a resurgence of this space and its use in resolution of singularities during the last decades, and often, these results are combined with model-theoretic techniques (Kuhlmann, Knaf, Pop, Cutkosky, Cossart, Piltant, Teissier, Scanlon,…). However, the model-theoretic setting always departs from Robinson’s point of view of a valued field: a field together with a valuation. However, the Zariski-Riemann space talks about not just one valuation, but all, so that we need a new framework. I will present some preliminary remarks of how this could be done using either a simple-minded one-sorted language or a more sophisticated two-sorted language. As a simple application of the one-sorted case, I will reprove fact (1) by simply relating it to the compactness of the Stone space of types.

# Mordell-Lang and Manin-Mumford in positive characteristic, revisited

We give a reduction of function field Mordell-Lang to function field Manin-Mumford, in positive characteristic. The upshot is another account of or proof of function field Mordell-Lang in positive characteristic, avoiding the recourse to difficult results on Zariski geometries.

(This work is joint with Benoist and Bouscaren.)

# State of affairs in local uniformization and valuation theory in positive characteristic

This will be an informal talk summarizing roughly where we stand concerning the problem of local uniformization in positive characteristic. I will discuss the structure of valued algebraic function fields and the main open problems that are of particular interest for local uniformization. These include the dehenselization problem (an analogue of Temkin’s “decompletion”) as well as the question when the existence of a rational place implies that the ground field is existentially closed in the function field.

If time permits I will also discuss a stunning result about the badness of valuations in positive characteristic (due to Anna Blaszczok), and state a result (by Anna and myself) and an open question that are of interest for recent work by Koen Struyve et al. on “Euclidean buildings.”