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Let $X$ be a complex scheme acted on by an affine algebraic group $G$. We prove that the Atiyah class of a $G$-equivariant perfect complex on $X$, as constructed by Huybrechts and Thomas, is $G$-equivariant in a precise sense. As an application, we show that, if $G$ is reductive, the obstruction theory on the fine relative moduli space $M\rightarrow B$ of simple perfect complexes on a $G$-invariant smooth projective family $Y\rightarrow B$ is $G$-equivariant. The results contained here are meant to suggest how to check the equivariance of the natural obstruction theories on a wide variety of moduli spaces equipped with a torus action, arising for instance in Donaldson–Thomas theory and Vafa–Witten theory.

Dinucleoside and tetranucleoside analogues with integrated backbone and nucleobase represent a novel type of oligonucleotide foldamers. They are characterized by a linker between C(5?) of one nucleoside moiety and C(8) of an adjacent adenosine, or C(6) of an adjacent uridine. Solutions in chloroform or chloroform/DMSO of these partially protected di-, or tetranucleoside analogues associate. The strength of the association depends on the nature of the linker, the presence, or absence of a hydroxymethyl group on the terminal nucleobase, other protecting groups, and intramolecular hydrogen bonds. The thiomethylene-linked dimers were studied in detail; one of them associates strongly enough that a melting temperature can be determined, evidencing base stacking in chloroform solution. Some dimers form organogels. Two 2?,3?-O-isopropylidene protected self-complementary tetramers were prepared. Their duplexes are conformationally homogeneous. One of the tetramers was characterised in detail; it associates via hydrogen bonds of the Watson-Crick type to form a partial A-type helix with all bases in a syn-conformation and a rather large twist angle. Also studied were di- and tetranucleotide analogues possessing an all-carbon linker; they were derived from (hydroxy)propynylene linked dimers by deoxygenation, partial reduction to the (E)- and (Z)-propenylene linked analogues, and further hydrogenation to propanylene-linked analogues. The propargylic hydroxy group forms an intramolecular hydrogen bond to N(3) of the adenine moiety of the same unit; the resulting conformers associate weakly. Deoxygenation leads to foldamers that associate much more strongly. A (Z)-propenylene-linked dimer also associates strongly, its (E) isomer less so, and the propanylene-linked analogues associate weakly. The results show that backbone-base integration defines a novel structural relation between backbone and nucleobases that favours selective pairing and leads to oligonucleotide foldamers.