Infinite Perfect Field Research Articles

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category SH ( k ) in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum Σ P 1 ∞ X + , where X + = X ⨿ ⁎ , for a smooth scheme X ∈ Sm k over an infinite perfect field k , is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres ( A l × X ) / ( ( A l − 0 ) × X ) , X ∈ Sm k , is one of ingredients in the theory. In the article we extend this result to the case of a pair ( X , U ) given by a smooth affine variety X over k and an open subscheme U ⊂ X . The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum Σ P 1 ∞ ( X + / U + ) of the quotient-sheaf X + / U + .

Tamely Ramified Morphisms of Curves and Belyi’s Theorem in Positive Characteristic

AbstractWe show that every smooth projective curve over a finite field $k$ admits a finite tame morphism to the projective line over $k$. Furthermore, we construct a curve with no such map when $k$ is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama–Yasuda, and Anbar–Tutdere.

Smooth Affine Model for the Framed Correspondences Spectrum

Morel–Voevodsky’s unstable pointed motivic homotopy category H●(k) over an infinite perfect field is considered. For a smooth affine scheme Y over k, a smooth ind-scheme Fl(Y) and an open subscheme El(Y) are constructed for all l > 0, so that the motivic space Fl(Y)/El(Y) is equivalent in H●(k) to the motivic space $$ {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\sum \limits_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(Y\times {T}^l\right),\kern0.5em T=\left({\mathbbm{A}}^1/{\mathbbm{A}}^1-0\right),\kern0.5em l>0 $$ . The construction is not functorial on the category of affine schemes but is functorial on the category of so-called framed schemes constructed for this purpose.

A Motivic Segal Theorem for Pairs (Announcement)

In order to provide a new, more computation-friendly, construction of the stable motivic category SH(k), V. Voevodsyky laid the foundation of delooping motivic spaces. G. Garkusha and I. Panin based on joint works with A. Ananievsky, A. Neshitov, and A. Druzhinin made that project a reality. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field k and any k-smooth scheme X, the canonical morphism of motivic spaces $$ {C}_{\ast } Fr(X)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left({X}_{+}\right) $$ is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs (X,U), where X is a k-smooth scheme and U is its open subscheme intersecting each component of X in a nonempty subscheme. It is claimed that in this case the motivic space C*Fr((X,U)) is a Nisnevich locally connected, and the motivic space morphism $$ {C}_{\ast } Fr\left(\left(X,U\right)\right)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(X/U\right) $$ is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of S = X−U in each component of X is greater than r ≥ 0, then the simplicial sheaf C*Fr((X,U)) is locally r-connected.

A moving lemma for relative 0-cycles

We prove a moving lemma for the additive and ordinary higher Chow groups of relative $0$-cycles of regular semi-local $k$-schemes essentially of finite type over an infinite perfect field. From this, we show that the cycle classes can be represented by cycles that possess certain finiteness, surjectivity, and smoothness properties. It plays a key role in showing that the crystalline cohomology of smooth varieties can be expressed in terms of algebraic cycles.

Reconstructing rational stable motivic homotopy theory

Using a recent computation of the rational minus part of$SH(k)$by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.

“$${\mathbb {P}}^1$$P1-gluing” for local complete intersections

We prove an analogue of the Affine Horrocks’ Theorem for local complete intersection ideals of height n in R[T], where R is a regular domain of dimension d, which is essentially of finite type over an infinite perfect field of characteristic unequal to 2, and $$2n\ge d+3$$.

On the zeroth stable [formula omitted]-homotopy group of a smooth curve

We provide a cohomological interpretation of the zeroth stable A1-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable A1-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.

Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field

Let [Formula: see text] be an [Formula: see text]-finite field containing an infinite perfect field of positive characteristic. Let [Formula: see text] be a projective log canonical pair over [Formula: see text]. In this note, we show that, for a semi-ample divisor [Formula: see text] on [Formula: see text], there exists an effective [Formula: see text]-divisor [Formula: see text] such that [Formula: see text] is log canonical if there exists a log resolution of [Formula: see text].

On the complete intersection conjecture of Murthy

Suppose A=k[X1,X2,…,Xn] is a polynomial ring over a field k and I is an ideal in A. M.P. Murthy conjectured that μ(I)=μ(I/I2), where μ denotes the minimal number of generators. Recently, Fasel [3] settled this conjecture, affirmatively, when k is an infinite perfect field, with 1/2∈k (always). We are able to do the same, when k is an infinite field. In fact, we prove similar results for ideals I in a polynomial ring A=R[X], that contains a monic polynomial and R is essentially smooth algebra over an infinite field k, or R is a regular ring over a perfect field k.

On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme

Let k be an infinite perfect field and let X = SpecR, where R is the localization of a k-algebra B of finite type with respect to a regular prime ideal p ∈ SpecB. LetD be a divisor onX with a decompositionD = ∪i=1Di into its irreducible components Di such that each Di is nonsingular and they intersect transversally. Consider Y = X − D, a deleted neighbourhood of the closed point x0 of X . Let E be a vector bundle on Y . In this set up, E is trivial in the following cases : (i) dim(X) ≤ 3 (Gabber [Ga1]) ; (ii) t = 1 (Bhatwadekar-Rao [BR]) ; (iii) rankE ≥ min{[dimR 2 ], t} (Rao [Ra1]). The above exposition is borrowed from a paper of Nisnevich [Ni], which contains some beautiful applications of these ‘purity theorems’. In the same set up, now let y ∈ Y be a closed point and let my be the corresponding maximal ideal of OY,y. It is natural to ask whether my is a complete intersection. Observing that the conormal bundle my/my is trivial, we consider a more general situation : Let I be the defining ideal of a closed subscheme Z of Y such that the conormal bundle I/I2 (defined on Z) is trivial. Is I a complete intersection? The questions we are addressing in this paper are a little stronger in nature : Can a given basis of I/I2 be lifted to a minimal set of generators of I? We remark that with the hypothesis on X , each component Di is defined by a single equation fi which vanishes at the closed point x0 of X . Let m be the maximal ideal at x0. The condition of transversality of intersection of Di’s is equivalent to saying that the images of f1, · · · , ft in the A/m-vector space m/m2 are linearly independent. Keeping this discussion in mind, we now give an excerpt of our main results in commutative algebraic terms (see 4.2, 4.3, 4.4).

On generically tame algebras over perfect fields

Given a generically tame finite-dimensional algebra Λ over an infinite perfect field, we give, for each natural number d, parametrizations of the indecomposable Λ-modules with dimension d similar to those occurring for the algebraically closed field case. We parametrize over bounded principal ideal domains, instead of over rational algebras.

On a conjecture of Vorst

Let k be an infinite perfect field of positive characteristic such that strong resolution of singularities holds over k. We prove that a localization of a d-dimensional commutative k-algebra R of finite type is K d+1-regular if and only if it is regular. This partially affirms a conjecture of Vorst.

The cl-core of an ideal

AbstractWe expand the notion of core to cl-core for Nakayama closures cl. In the characteristic p > 0 setting, when cl is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core(I) = core(I), if I is an ideal in a one-dimensional domain with infinite residue field or if I is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen–Macaulay normal local domain with infinite perfect residue field, if the analytic spread, ℓ, is equal to the *-spread and I is Gℓ and weakly-(ℓ − 1)-residually S2. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *-reductions. We also determine that the *-core of a tightly closed ideal in certain one-dimensional semigroup rings is tightly closed and therefore integrally closed.

On the vanishing of negative K-groups

Let k be an infinite perfect field of positive characteristic p and assume that strong resolution of singularities holds over k. We prove that, if X is a d-dimensional noetherian scheme whose underlying reduced scheme is essentially of finite type over the field k, then the negative K-group K_q(X) vanishes for every q < -d. This partially affirms a conjecture of Weibel.

On the negative K-theory of schemes in finite characteristic

We show that if X is a d-dimensional scheme of finite type over an infinite perfect field k of characteristic p > 0 , then K i ( X ) = 0 and X is K i -regular for i < − d − 2 whenever the resolution of singularities holds over k. This proves the K-dimension conjecture of Weibel [C. Weibel, K-theory and analytic isomorphisms, Invent. Math. 61 (1980) 177–197, 2.9] (except for − d − 1 ⩽ i ⩽ − d − 2 ) in all characteristics, assuming the resolution of singularities.

An adelic resolution for homology sheaves

We propose a generalization of the ordinary idele group by constructing certain adelic complexes for sheaves of -groups on schemes. Such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated with the presheaf of a homology theory with certain natural axioms satisfied, in particular, by -theory. In this case it is proved that the adelic complex provides a flabby resolution for this sheaf on smooth varieties over an infinite perfect field and that the natural morphism to the Gersten complex is a quasi-isomorphism. The main advantage of the new adelic resolution is that it is contravariant and multiplicative. In particular, this enables us to reprove that the intersection in Chow groups coincides (up to a sign) with the natural product in the corresponding -cohomology groups. Also, we show that the Weil pairing can be expressed as a Massey triple product in -cohomology groups with certain indices.

A Question of Nori: Projective Generation of Ideals

Let A be a smooth affine domain of dimension d over an infinite perfect field k and let n be an integer such that 2n ≥ d + 3. Let I ⊂A[T] be an ideal of height n. Assume that I = (f 1 ,...,f n ) + (I 2 T). Under these assumptions, it is proved in this paper that I = (g 1 ,...,g n ) with f i - g i ⊂ (I 2 T), thus settling a question of Nori affirmatively.

Computing the Primary Decomposition of Zero-dimensional Ideals

Let K be an infinite perfect computable field and let I⊆K [ x ] be a zero-dimensional ideal represented by a Gröbner basis. We derive a new algorithm for computing the reduced primary decomposition of I using only standard linear algebra and univariate polynomial factorization techniques. In practice, the algorithm generally works in finite fields of large characteristic as well.

Base field extensions and generic modules over finite dimensional algebras

Let R be a finite dimensional algebra over a field k. It is shown that a finite separable field extension of k preserves and respects generic tameness of R. Moreover if k is an infinite perfect field then the extension of k to its algebraic closure preserves generic tameness. As a corollary we get that if R is generically tame then for every number m all but a finite number of indecomposable R-modules of length m are DTr-periodic.