We develop a new form of patching that is both far-reaching and more elementary than the previous... more We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.
We consider local-global principles for torsors under linear algebraic groups, over function fiel... more We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras.
Proceedings of the London Mathematical Society, 2016
In this manuscript, we apply patching methods to give a positive answer to the inverse differenti... more In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables).
This paper proves local-global principles for Galois cohomology groups over function fields F of ... more This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H n pF, Z{mZpn´1qq, for all n ą 1. This is motivated by work of Kato and others, where such principles were shown in related cases for n " 3. Using our results in combination with cohomological invariants, we obtain localglobal principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.
We introduce a notion of refinements in the context of patching, in order to obtain new results a... more We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results.
We consider local-global principles for torsors under linear algebraic groups, over function fiel... more We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and we show that it is finite in important cases. Moreover we obtain necessary and sufficient conditions for localglobal principles to hold. The proofs use techniques from patching. We also give new applications to quadratic forms and central simple algebras.
We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete ... more We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period-index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan's Lemma.
We study differential forms invariant under a finite reflection group over a field of arbitrary c... more We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito's freeness criterion for invariant differential 1-forms. We also discuss how twisted wedging endows the invariant forms with the structure of a free exterior algebra in certain cases. Some of the results are extended to the case of relative invariants with respect to a linear character.
This paper provides applications of patching to quadratic forms and central simple algebras over ... more This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, p = 2. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.
This paper proves local-global principles for Galois cohomology groups over function fields F of ... more This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H n pF, Z{mZpn´1qq, for all n ą 1. This is motivated by work of Kato and others, where such principles were shown in related cases for n " 3. Using our results in combination with cohomological invariants, we obtain localglobal principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.
Transactions of the American Mathematical Society, May 7, 2004
Steinberg showed that when a finite reflection group acts on a real or complex vector space of fi... more Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for arbitrary fields using a ramification formula of Benson and Crawley-Boevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants.
We develop a new form of patching that is both far-reaching and more elementary than the previous... more We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.
We consider local-global principles for torsors under linear algebraic groups, over function fiel... more We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras.
Proceedings of the London Mathematical Society, 2016
In this manuscript, we apply patching methods to give a positive answer to the inverse differenti... more In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables).
This paper proves local-global principles for Galois cohomology groups over function fields F of ... more This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H n pF, Z{mZpn´1qq, for all n ą 1. This is motivated by work of Kato and others, where such principles were shown in related cases for n " 3. Using our results in combination with cohomological invariants, we obtain localglobal principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.
We introduce a notion of refinements in the context of patching, in order to obtain new results a... more We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results.
We consider local-global principles for torsors under linear algebraic groups, over function fiel... more We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and we show that it is finite in important cases. Moreover we obtain necessary and sufficient conditions for localglobal principles to hold. The proofs use techniques from patching. We also give new applications to quadratic forms and central simple algebras.
We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete ... more We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period-index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan's Lemma.
We study differential forms invariant under a finite reflection group over a field of arbitrary c... more We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito's freeness criterion for invariant differential 1-forms. We also discuss how twisted wedging endows the invariant forms with the structure of a free exterior algebra in certain cases. Some of the results are extended to the case of relative invariants with respect to a linear character.
This paper provides applications of patching to quadratic forms and central simple algebras over ... more This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, p = 2. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.
This paper proves local-global principles for Galois cohomology groups over function fields F of ... more This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H n pF, Z{mZpn´1qq, for all n ą 1. This is motivated by work of Kato and others, where such principles were shown in related cases for n " 3. Using our results in combination with cohomological invariants, we obtain localglobal principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.
Transactions of the American Mathematical Society, May 7, 2004
Steinberg showed that when a finite reflection group acts on a real or complex vector space of fi... more Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for arbitrary fields using a ramification formula of Benson and Crawley-Boevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants.
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Papers by Julia Hartmann