Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

The resolution property of algebraic surfaces

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

On the base field change of P-rings and P-2 rings

One finds the following example in [3, (34, B)]:Let k be a field of characteristic p and be n-variables over k. Then if p > 0 and [k : kp] = ∞, is an n-dimensional regular local ring but not a Nagata ring. In particular it is not an excellent ring.

A few theorems on completion of excellent rings

In [2], chap. IV, 2me partie, (7.4.8), Grothendieck considered the following problem: is any m-adic completion of an excellent ring A also excellent?In [8] I proved that, if A is an algebra of finite type over an arbitrary field k, the answer is positive.