arXiv subjects

Benjamin Briggs

Publications and source records attributed to Benjamin Briggs.

22 records · Page 2Linked to original sources

arXiv2021

Locally complete intersection maps and the proxy small property

It is proved that a map $φ\colon R\to S$ of commutative noetherian rings that is essentially of finite type and flat is locally complete intersection if and only $S$ is proxy small as a bimodule. This means that the thick subcategory generated by $S$ as a module over the enveloping algebra $S\otimes_RS$ contains a perfect complex supported fully on the diagonal ideal. This is in the spirit of the classical result that $φ$ is smooth if and only if $S$ is small as a bimodule, that is to say, it is itself equivalent to a perfect complex. The geometric analogue, dealing with maps between schemes, is also established. Applications include simpler proofs of factorization theorems for locally complete intersection maps.

math.AC↗

arXiv2020

Matrix Factorisations Arising From Well-Generated Complex Reflection Groups

We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally identify invariant vector fields with vector fields on the orbit space, for the action of a duality group. As another application, we construct matrix factorisations of the highest degree basic invariant which give free resolutions of the module of Kähler differentials of the coinvariant algebra $A$ associated to such a reflection group. From this one can explicitly calculate the dimension of each graded piece of $Ω_{A/\mathbb{C}}$ and of ${\rm Der}_{\mathbb{C}}(A,A)$, adding a new formula to the numerology of reflection groups. This applies for instance when $A$ is the cohomology of any complete flag manifold, and hence has geometric consequences.

math.RA↗

arXiv2020

Hochschild cohomology of twisted tensor products

For a tensor product of algebras twisted by a bicharacter, we completely describe its Hochschild cohomology, as a Gerstenhaber algebra, in terms of the Hochschild cohomology of its component parts. This description generalizes a result of Bergh and Oppermann. It allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology for a number of new examples.

math.RA↗

arXiv2017

The A-infinity Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category

For A a dg (or A-infinity) algebra and M a module over A, we study the image of the characteristic morphism $χ_M: HH^*(A, A) \to Ext_A(M, M)$ and its interaction with the higher structure on the Yoneda algebra $Ext_A(M, M)$. To this end, we introduce and study a notion of A-infinity centre for minimal A-infinity algebras, agreeing with the usual centre in the case that there is no higher structure. We show that the image of $χ_M$ lands in the A-infinity centre of $Ext_A(M, M)$. When A is augmented over k, we show (under mild connectedness assumptions) that the morphism $χ_k: HH^*(A, A) \to Ext_A(k,k)$ into the Koszul dual algebra lands exactly onto the A-infinity centre, generalising the situation from the Koszul case established by Buchweitz, Green, Snashall and Solberg. We give techniques for computing A-infinity centres, hence for computing the image of the characteristic morphism, and provide worked-out examples. We further study applications to topology. In particular we relate the A-infinity centre of the Pontryagin algebra to a wrong way map coming from the homology of the free loop space, first studied by Chas and Sullivan.

math.RT↗