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    • CommentRowNumber1.
    • CommentAuthorjoe.hannon
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    At scheme, the definition of a kk-ring notates the category as k/Ringk/Ring and says it’s a pair (R,f:kR),(R,f\colon k\to R), with ff a kk-algebra homomorphism. Is this correct? What does this mean? We can obviously view RR as a kk-algebra by means of the action by ff. How can we say that ff is itself a kk-linear map? With respect to what kk-action on RR? It’s somehow circular.

    Perhaps does it mean to say something more like “a kk-ring is an object in the undercategory k/Ringk/Ring, so objects are all pairs (R,f:kR)(R,f\colon k\to R) (no restriction on ff), and morphisms are kk-algebra homomorphisms”?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    It should have said, “with ff a (commutative) ring homomorphism”.

    With that in place, RR acquires a kk-module structure by means of the formula sxf(s)xs \cdot x \coloneqq f(s)x where sks \in k and xRx \in R, and we are using the ring multiplication on RR to define the right side of the formula. Moreover, RR becomes a kk-algebra assuming, as we are, that the rings in question are commutative. In other words, each element rRr \in R induces a scalar multiplication xrxx \mapsto r x which preserves the kk-linear structure on RR, since r(sx)=rf(s)x=f(s)(rx)=s(rx)r (s \cdot x) = r f(s) x = f(s) (r x) = s \cdot (r x).

    I’ll go in and correct, since I need to do something there anyway.

    • CommentRowNumber3.
    • CommentAuthorjoe.hannon
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    Thank you, Todd. That’s what I thought. And in the preceding sentence: “category of associative kk-algebras which are rings”. Is that redundant? Are there some associative algebras which are not rings? Couldn’t we just say “category of associative kk-algebras and kk-linear homomorphisms”, and then remark that this is the standard undercategory k/Ringk/Ring? Maybe I’ll just make the edit, can you see if you like it?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    Well, I agree, it’s a somewhat odd thing to write and I’m not quite sure what the point is supposed to be, unless it’s that associativity algebras are a fortiori possibly non-commutative rings and here “ring” always means commutative ring. The business boils down to “the category of commutative kk-algebras and their homomorphisms is equivalent to the undercategory k/CRingk/CRing”, and that’s what I’d write.

    Edit: I took a quick look, and didn’t see anything wrong.

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